11

What is Image Segmentation

There are many definitions Three common There are many definitions Three common

ones areones are

Image Segmentation is the process of Image Segmentation is the process of

isolating objects of interest from the rest isolating objects of interest from the rest

of the scene (of the scene (Castleman Castleman ))

Image segmentation is the process of Image segmentation is the process of

partitioning partitioning (( 分 割分 割 ) ) an image into non-an image into non-

intersecting region such that each region intersecting region such that each region

is homogeneous is homogeneous (( 相似的相似的 ) ) and the union of and the union of

no two adjacent regions is homogeneous no two adjacent regions is homogeneous

((Pal Pal ))

22

What is Image Segmentation

Image segmentation is to divide an image Image segmentation is to divide an image

into parts that have a strong correlation into parts that have a strong correlation

with objects or areas of the real world with objects or areas of the real world

contained in the image (contained in the image (Watt Watt ))

In brief sIn brief segmentation is to subdivide an egmentation is to subdivide an

imageimage into its constituentinto its constituent ( ( 组成的组成的 )) regions regions

or objectsor objects

―Segmentation should stop when theSegmentation should stop when the

objects of interest in an application objects of interest in an application

havehave been isolatedbeen isolated

33

Image Processing Flow based onImage Processing Flow based on

Image SegmentationImage Segmentation

Image Segmentation

InputInputImageImage

ObjectObjectImageImage

FeatureFeatureVectorVector

ObjectObjectTypeType

FeatureExtraction

Classification

44

Why is it difficult

In generalIn general autonomousautonomous ( ( 自 主 的自 主 的 ))

segmentation is one segmentation is one of of the most difficult the most difficult

tasks in image processing It tasks in image processing It isis difficult difficult

because of because of many reasonmany reasonss Here are some Here are some

typical obstaclestypical obstacles Non-uniform illuminationNon-uniform illumination

No control of the environmentNo control of the environment

Inadequate model of the object of interestInadequate model of the object of interest

NoiseNoise

etcetc

55

Segmentation methods can be divided into three groups according to the dominant features they employ

Segmentation based on global knowledge about an image

―The knowledge is usually represented by a histogram of image features

Edge-based segmentations

―Utilizing edge detection processes to find a closed boundary so that an inside and an outside can be defined

Region-based segmentations

―This techniques proceed by dividing the image into regions that exhibit similar properties

Principal approachesPrincipal approaches

66

2 basis properties of intensity 2 basis properties of intensity valuesvalues

Segmentation algorithms generally are baseSegmentation algorithms generally are based on one of 2 basis properties of intensity vad on one of 2 basis properties of intensity valueslues DiscontinuityDiscontinuity

―to partition an image based on abrupt (to partition an image based on abrupt ( 突然突然的的 ) changes in intensity (such as edges)) changes in intensity (such as edges)

SimilaritySimilarity

―to partition an image into regions that are simto partition an image into regions that are similar according to a set of predefined criteriailar according to a set of predefined criteria

77

Detection of DiscontinuitiesDetection of Discontinuities

detect the three basic types of gray detect the three basic types of gray

level discontinuitieslevel discontinuities

points lines edgespoints lines edges

the common way is to run a mask the common way is to run a mask

through the imagethrough the image

88

ContentsContents

ThresholdingThresholding

Point DetectionPoint Detection

Line DetectionLine Detection

Edge-based SegmentationEdge-based Segmentation

Region-based SegmentationRegion-based Segmentation

99

71 Thresholding71 Thresholding

Thresholding is a labeling operation Thresholding is a labeling operation

on a gray scale image that on a gray scale image that

distinguishes pixels of a higher distinguishes pixels of a higher

intensity from pixels with a lower intensity from pixels with a lower

intensity valueintensity value

The output of thresholding usuallyThe output of thresholding usually is is a a

binary imagebinary image

This technique is particularly useful This technique is particularly useful

for scenes which contain for scenes which contain solid objects solid objects

on a uniform contrasting backgroundon a uniform contrasting background

1010

Classification of ThresholdingClassification of Thresholding

Thresholding can be viewed as an operation Thresholding can be viewed as an operation that involves tests against a function T of ththat involves tests against a function T of the forme form

where p(xy) denotes some local property of this where p(xy) denotes some local property of this pointpoint

T T x y p x y f x y

1111

Classification of ThresholdingClassification of Thresholding

When T depends onWhen T depends on only f(xy) only on gray-level valuesonly f(xy) only on gray-level values

1048638 1048638 Global Global thresholdingthresholding

both f(xy) and p(xy) on gray-level values and itboth f(xy) and p(xy) on gray-level values and its neighborss neighbors

Local Local thresholdingthresholding

x and y (in addition)x and y (in addition)

Dynamic Dynamic thresholdingthresholding

T T x y p x y f x y

1212

Basic Global ThresholdingBasic Global Thresholding

Original imageOriginal image HistogramHistogram

SolutionSolution use T midway between the max and use T midway between the max and

min gray levelsmin gray levels

SolutionSolution use T midway between the max and use T midway between the max and

min gray levelsmin gray levels

See ldquobasic_global_threSee ldquobasic_global_thremrdquomrdquo

1313

Basic Global ThresholdingBasic Global Thresholding

Let light objects in dark backgroundLet light objects in dark background

To extract the objectsTo extract the objects Select a ldquoTrdquo that separates the objects from thSelect a ldquoTrdquo that separates the objects from th

e backgrounde background

ie any (xy) for which f(xy)gtT is an object point ie any (xy) for which f(xy)gtT is an object point

A thresholded imageA thresholded image

1

0

if f x y T backgroundg x y

if f x y T foreground

1414

Heuristic Global ThresholdingHeuristic Global Thresholding

11 Select an initial estimate for TSelect an initial estimate for T

22 Segment the image using T This will produce two Segment the image using T This will produce two groups of pixels Ggroups of pixels G11 consisting of all pixels with gra consisting of all pixels with gray level values lt T and Gy level values lt T and G22 consisting of pixels with g consisting of pixels with gray level values ray level values T T

33 Compute the average gray level values μCompute the average gray level values μ11 and μ and μ22 fo for the pixels in regions Gr the pixels in regions G11 and G and G22

44 Compute a new threshold value T=05(μCompute a new threshold value T=05(μ11+μ+μ22))

55 Repeat steps 2 through 4 until the difference in T iRepeat steps 2 through 4 until the difference in T in successive iterations is smaller than a predefinen successive iterations is smaller than a predefined parameter Td parameter T00 seerdquohhellipmrdquo seerdquohhellipmrdquo

1515

Basic Adaptive ThresholdingBasic Adaptive Thresholding

subdivide original image into small areassubdivide original image into small areas

utilize a different threshold to segment each utilize a different threshold to segment each subimagessubimages

since the threshold used for each pixel depesince the threshold used for each pixel depends on the location of the pixel in terms of tnds on the location of the pixel in terms of the subimages this type of thresholding is ahe subimages this type of thresholding is adaptivedaptive

See ldquoAdaptive_thresholdmrdquoSee ldquoAdaptive_thresholdmrdquo

1616

Multilevel ThresholdingMultilevel Thresholding

a point (xy) belongs toa point (xy) belongs to an object class if Tan object class if T11 ltlt f(xy) le T f(xy) le T22

another object class if f(xy) another object class if f(xy) gtgt T T22

to background if f(xy) le Tto background if f(xy) le T11

1717

The histogram of an image containing two pThe histogram of an image containing two principal brightness regions can be considererincipal brightness regions can be considered an estimate of the brightness probability d an estimate of the brightness probability density function p(z)density function p(z) p(z) is the sum (or mixture) of two unimodal (p(z) is the sum (or mixture) of two unimodal ( 单单峰的峰的 ) densities (one for light one for dark region) densities (one for light one for dark regions)s)

1 1 2 2

1 2 1

p z P p z P p z

P P

1818

Optimal ThresholdingOptimal Thresholding

If the form of the If the form of the

densities is densities is

known or known or

assumed in assumed in

terms of terms of

minimum error minimum error

determining an determining an

optimal optimal

threshold for threshold for

segmenting the segmenting the

image is image is

possiblepossible

1 1 2 2

1 2 1

p z P p z P p z

P P

1919

Optimal ThresholdingOptimal Thresholding

Probability of erroneouslyProbability of erroneously

2020

Optimal ThresholdingOptimal Thresholding

Minimum errorMinimum error Differentiating ( Differentiating ( 微分微分 ) E(T) with respect to T (usi) E(T) with respect to T (usi

ng Leibnizrsquos rule) and equating the result to 0ng Leibnizrsquos rule) and equating the result to 0

find T which makesfind T which makes

2121

Optimal ThresholdingOptimal Thresholding

Minimum errorMinimum error

Specially if Specially if PP11 = P = P22 then the optimum then the optimum

threshold is where the curve pthreshold is where the curve p11(z) and (z) and

pp22(z) intersect(z) intersect

2222

Optimal ThresholdingOptimal Thresholding

For exampleFor example

Let Let PDF=Gaussian densityPDF=Gaussian density p p11(z) and (z) and

pp22(z)(z)

where μwhere μ11 and σ and σ1122 are the mean and are the mean and

variance of the Gaussian density of one variance of the Gaussian density of one

objectobject

μμ22 and σ and σ2222 are the mean and variance of are the mean and variance of

the Gaussian density of the other objectthe Gaussian density of the other object

2323

Optimal ThresholdingOptimal Thresholding

Quadratic equation (Quadratic equation (二次方程二次方程 ))

2424

Problems of ThresholdingProblems of Thresholding

Original imageOriginal image Thresholded imageThresholded image

2525

Problems of ThresholdingProblems of Thresholding

(a)(a) Exact threshold Exact threshold

segmentationsegmentation

(b)(b) Threshold too lowThreshold too low

(c)(c) Threshold too Threshold too

highhigh

2626

72 Point Detection72 Point Detection

a point has been detected at the a point has been detected at the

location on which the mark is location on which the mark is

centered ifcentered if

|R|geT|R|geT

where T is a nonnegative thresholdwhere T is a nonnegative threshold

R is the sum of products of the R is the sum of products of the

coefficients with the gray levels contained coefficients with the gray levels contained

in the region encompassed by the markin the region encompassed by the mark

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

2727

72 Point Detection72 Point Detection

Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

2828

ExampleExample

2929

73 Line Detection73 Line Detection

Horizontal mask will result with max Horizontal mask will result with max

response when a line passed through the response when a line passed through the

middle row of the mask with a constant middle row of the mask with a constant

backgroundbackground

the similar idea is used with other masksthe similar idea is used with other masks

Note the preferred direction of each mask Note the preferred direction of each mask

is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

than other possible directionsthan other possible directions

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

3030

Idea 1 of Line DetectionIdea 1 of Line Detection

Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

if at a certain point in the imageif at a certain point in the image

|Ri||Ri|gtgt|Rj||Rj|

for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

3131

Idea 2 of Line DetectionIdea 2 of Line Detection

Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

3232

ExampleExample

3333

74 Edge-based 74 Edge-based SegmentationSegmentation

Edge-based segmentations rely on edges found in an image by edge detecting operators

these edges mark image locations of discontinuities in gray level

Edge detection is the most common approach for detecting meaningful discontinuities in gray level

There are a large group of methods based on information about edges in the image

3434

What is edgeWhat is edge

Edge is where change occurs Change is measured by derivative in 1D

―Biggest change derivative has maximum magnitude

Or 2nd derivative is zero we discuss approaches for implementing

―first-order derivative (Gradient operator)

―second-order derivative (Laplacian operator)

―we have introduced both derivatives in chapter 3

―Here we will talk only about their properties for edge detection

3535

What is edgeWhat is edge

In other wordsIn other words an edge is a set of an edge is a set of

connected pixelsconnected pixels

that lie on the boundary between two that lie on the boundary between two

regions with relatively distinct gray-level regions with relatively distinct gray-level

propertiesproperties

Note edge vs boundaryNote edge vs boundary

―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

―whereas a region boundary owing to whereas a region boundary owing to

the way it is defined is a more global the way it is defined is a more global

ideaidea

3636

Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

because of because of

optics optics

sampling sampling

image image

acquisition acquisition

imperfectionimperfection

3737

Thick and Thin EdgeThick and Thin Edge

The slope of the ramp is inversely The slope of the ramp is inversely

proportional to the degree of blurring in the proportional to the degree of blurring in the

edgeedge

Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

pathpath

Instead an edge point now is any point Instead an edge point now is any point

contained in the ramp and contained in the ramp and an edge would an edge would

then be a set of such points that are then be a set of such points that are

connectedconnected

The thickness is determined by the length of the The thickness is determined by the length of the

rampramp

The length is determined by the slope which is in The length is determined by the slope which is in

turn determined by the degree of blurringturn determined by the degree of blurring

Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

edges tend to be thinedges tend to be thin

3838

First and Second derivatives (First and Second derivatives ( 导数导数 ))

the signs of the the signs of the

derivatives would be derivatives would be

reversed for an edge reversed for an edge

that transitions from that transitions from

light to darklight to dark

First First derivatderivatee

SeconSecond d derivatderivatee

Gray-Gray-level level profileprofile

3939

Second derivativesSecond derivatives

an undesirable featurean undesirable feature

produces 2 values for every edge in an produces 2 values for every edge in an

imageimage

zero-crossing propertyzero-crossing property

an imaginary straight line joining the an imaginary straight line joining the

extreme positive and negative values of extreme positive and negative values of

the second derivative would cross zero the second derivative would cross zero

near the midpoint of the edgenear the midpoint of the edge

quite useful for locating the centers of quite useful for locating the centers of

thick edgesthick edges

4040

Basic idea of edge detectionBasic idea of edge detection

A profile is defined perpendicularly to A profile is defined perpendicularly to

the edge direction and the results are the edge direction and the results are

interpretedinterpreted

The magnitude of the first derivative is The magnitude of the first derivative is

used to detect an edge (if a point is on a used to detect an edge (if a point is on a

ramp)ramp)

The sign of the second derivative can The sign of the second derivative can

determine whether an edge pixel is on the determine whether an edge pixel is on the

dark or light side of an edgedark or light side of an edge

4141

Review of First DerivateReview of First Derivate

Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

Sobel operators 3Sobel operators 333

6 5 8 5x yG z z G z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

9 5 8 6x yG z z G z z 1 0 0 1

0 1 1 0

1 0 0 1

0 1 1 0

7 8 9 1 2 3

3 6 9 1 4 7

2 2

2 2

x

y

G z z z z z z

G z z z z z z

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

x yf G G

4242

Edge direction and strengthEdge direction and strength

Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

2 2x yf G G

4343

Gradient MasksGradient Masks

1 0 0 1

0 1 1 0

Roberts

1 0 0 1

0 1 1 0

Roberts

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

4444

Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
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22

What is Image Segmentation

Image segmentation is to divide an image Image segmentation is to divide an image

into parts that have a strong correlation into parts that have a strong correlation

with objects or areas of the real world with objects or areas of the real world

contained in the image (contained in the image (Watt Watt ))

In brief sIn brief segmentation is to subdivide an egmentation is to subdivide an

imageimage into its constituentinto its constituent ( ( 组成的组成的 )) regions regions

or objectsor objects

―Segmentation should stop when theSegmentation should stop when the

objects of interest in an application objects of interest in an application

havehave been isolatedbeen isolated

33

Image Processing Flow based onImage Processing Flow based on

Image SegmentationImage Segmentation

Image Segmentation

InputInputImageImage

ObjectObjectImageImage

FeatureFeatureVectorVector

ObjectObjectTypeType

FeatureExtraction

Classification

44

Why is it difficult

In generalIn general autonomousautonomous ( ( 自 主 的自 主 的 ))

segmentation is one segmentation is one of of the most difficult the most difficult

tasks in image processing It tasks in image processing It isis difficult difficult

because of because of many reasonmany reasonss Here are some Here are some

typical obstaclestypical obstacles Non-uniform illuminationNon-uniform illumination

No control of the environmentNo control of the environment

Inadequate model of the object of interestInadequate model of the object of interest

NoiseNoise

etcetc

55

Segmentation methods can be divided into three groups according to the dominant features they employ

Segmentation based on global knowledge about an image

―The knowledge is usually represented by a histogram of image features

Edge-based segmentations

―Utilizing edge detection processes to find a closed boundary so that an inside and an outside can be defined

Region-based segmentations

―This techniques proceed by dividing the image into regions that exhibit similar properties

Principal approachesPrincipal approaches

66

2 basis properties of intensity 2 basis properties of intensity valuesvalues

Segmentation algorithms generally are baseSegmentation algorithms generally are based on one of 2 basis properties of intensity vad on one of 2 basis properties of intensity valueslues DiscontinuityDiscontinuity

―to partition an image based on abrupt (to partition an image based on abrupt ( 突然突然的的 ) changes in intensity (such as edges)) changes in intensity (such as edges)

SimilaritySimilarity

―to partition an image into regions that are simto partition an image into regions that are similar according to a set of predefined criteriailar according to a set of predefined criteria

77

Detection of DiscontinuitiesDetection of Discontinuities

detect the three basic types of gray detect the three basic types of gray

level discontinuitieslevel discontinuities

points lines edgespoints lines edges

the common way is to run a mask the common way is to run a mask

through the imagethrough the image

88

ContentsContents

ThresholdingThresholding

Point DetectionPoint Detection

Line DetectionLine Detection

Edge-based SegmentationEdge-based Segmentation

Region-based SegmentationRegion-based Segmentation

99

71 Thresholding71 Thresholding

Thresholding is a labeling operation Thresholding is a labeling operation

on a gray scale image that on a gray scale image that

distinguishes pixels of a higher distinguishes pixels of a higher

intensity from pixels with a lower intensity from pixels with a lower

intensity valueintensity value

The output of thresholding usuallyThe output of thresholding usually is is a a

binary imagebinary image

This technique is particularly useful This technique is particularly useful

for scenes which contain for scenes which contain solid objects solid objects

on a uniform contrasting backgroundon a uniform contrasting background

1010

Classification of ThresholdingClassification of Thresholding

Thresholding can be viewed as an operation Thresholding can be viewed as an operation that involves tests against a function T of ththat involves tests against a function T of the forme form

where p(xy) denotes some local property of this where p(xy) denotes some local property of this pointpoint

T T x y p x y f x y

1111

Classification of ThresholdingClassification of Thresholding

When T depends onWhen T depends on only f(xy) only on gray-level valuesonly f(xy) only on gray-level values

1048638 1048638 Global Global thresholdingthresholding

both f(xy) and p(xy) on gray-level values and itboth f(xy) and p(xy) on gray-level values and its neighborss neighbors

Local Local thresholdingthresholding

x and y (in addition)x and y (in addition)

Dynamic Dynamic thresholdingthresholding

T T x y p x y f x y

1212

Basic Global ThresholdingBasic Global Thresholding

Original imageOriginal image HistogramHistogram

SolutionSolution use T midway between the max and use T midway between the max and

min gray levelsmin gray levels

SolutionSolution use T midway between the max and use T midway between the max and

min gray levelsmin gray levels

See ldquobasic_global_threSee ldquobasic_global_thremrdquomrdquo

1313

Basic Global ThresholdingBasic Global Thresholding

Let light objects in dark backgroundLet light objects in dark background

To extract the objectsTo extract the objects Select a ldquoTrdquo that separates the objects from thSelect a ldquoTrdquo that separates the objects from th

e backgrounde background

ie any (xy) for which f(xy)gtT is an object point ie any (xy) for which f(xy)gtT is an object point

A thresholded imageA thresholded image

1

0

if f x y T backgroundg x y

if f x y T foreground

1414

Heuristic Global ThresholdingHeuristic Global Thresholding

11 Select an initial estimate for TSelect an initial estimate for T

22 Segment the image using T This will produce two Segment the image using T This will produce two groups of pixels Ggroups of pixels G11 consisting of all pixels with gra consisting of all pixels with gray level values lt T and Gy level values lt T and G22 consisting of pixels with g consisting of pixels with gray level values ray level values T T

33 Compute the average gray level values μCompute the average gray level values μ11 and μ and μ22 fo for the pixels in regions Gr the pixels in regions G11 and G and G22

44 Compute a new threshold value T=05(μCompute a new threshold value T=05(μ11+μ+μ22))

55 Repeat steps 2 through 4 until the difference in T iRepeat steps 2 through 4 until the difference in T in successive iterations is smaller than a predefinen successive iterations is smaller than a predefined parameter Td parameter T00 seerdquohhellipmrdquo seerdquohhellipmrdquo

1515

Basic Adaptive ThresholdingBasic Adaptive Thresholding

subdivide original image into small areassubdivide original image into small areas

utilize a different threshold to segment each utilize a different threshold to segment each subimagessubimages

since the threshold used for each pixel depesince the threshold used for each pixel depends on the location of the pixel in terms of tnds on the location of the pixel in terms of the subimages this type of thresholding is ahe subimages this type of thresholding is adaptivedaptive

See ldquoAdaptive_thresholdmrdquoSee ldquoAdaptive_thresholdmrdquo

1616

Multilevel ThresholdingMultilevel Thresholding

a point (xy) belongs toa point (xy) belongs to an object class if Tan object class if T11 ltlt f(xy) le T f(xy) le T22

another object class if f(xy) another object class if f(xy) gtgt T T22

to background if f(xy) le Tto background if f(xy) le T11

1717

The histogram of an image containing two pThe histogram of an image containing two principal brightness regions can be considererincipal brightness regions can be considered an estimate of the brightness probability d an estimate of the brightness probability density function p(z)density function p(z) p(z) is the sum (or mixture) of two unimodal (p(z) is the sum (or mixture) of two unimodal ( 单单峰的峰的 ) densities (one for light one for dark region) densities (one for light one for dark regions)s)

1 1 2 2

1 2 1

p z P p z P p z

P P

1818

Optimal ThresholdingOptimal Thresholding

If the form of the If the form of the

densities is densities is

known or known or

assumed in assumed in

terms of terms of

minimum error minimum error

determining an determining an

optimal optimal

threshold for threshold for

segmenting the segmenting the

image is image is

possiblepossible

1 1 2 2

1 2 1

p z P p z P p z

P P

1919

Optimal ThresholdingOptimal Thresholding

Probability of erroneouslyProbability of erroneously

2020

Optimal ThresholdingOptimal Thresholding

Minimum errorMinimum error Differentiating ( Differentiating ( 微分微分 ) E(T) with respect to T (usi) E(T) with respect to T (usi

ng Leibnizrsquos rule) and equating the result to 0ng Leibnizrsquos rule) and equating the result to 0

find T which makesfind T which makes

2121

Optimal ThresholdingOptimal Thresholding

Minimum errorMinimum error

Specially if Specially if PP11 = P = P22 then the optimum then the optimum

threshold is where the curve pthreshold is where the curve p11(z) and (z) and

pp22(z) intersect(z) intersect

2222

Optimal ThresholdingOptimal Thresholding

For exampleFor example

Let Let PDF=Gaussian densityPDF=Gaussian density p p11(z) and (z) and

pp22(z)(z)

where μwhere μ11 and σ and σ1122 are the mean and are the mean and

variance of the Gaussian density of one variance of the Gaussian density of one

objectobject

μμ22 and σ and σ2222 are the mean and variance of are the mean and variance of

the Gaussian density of the other objectthe Gaussian density of the other object

2323

Optimal ThresholdingOptimal Thresholding

Quadratic equation (Quadratic equation (二次方程二次方程 ))

2424

Problems of ThresholdingProblems of Thresholding

Original imageOriginal image Thresholded imageThresholded image

2525

Problems of ThresholdingProblems of Thresholding

(a)(a) Exact threshold Exact threshold

segmentationsegmentation

(b)(b) Threshold too lowThreshold too low

(c)(c) Threshold too Threshold too

highhigh

2626

72 Point Detection72 Point Detection

a point has been detected at the a point has been detected at the

location on which the mark is location on which the mark is

centered ifcentered if

|R|geT|R|geT

where T is a nonnegative thresholdwhere T is a nonnegative threshold

R is the sum of products of the R is the sum of products of the

coefficients with the gray levels contained coefficients with the gray levels contained

in the region encompassed by the markin the region encompassed by the mark

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

2727

72 Point Detection72 Point Detection

Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

2828

ExampleExample

2929

73 Line Detection73 Line Detection

Horizontal mask will result with max Horizontal mask will result with max

response when a line passed through the response when a line passed through the

middle row of the mask with a constant middle row of the mask with a constant

backgroundbackground

the similar idea is used with other masksthe similar idea is used with other masks

Note the preferred direction of each mask Note the preferred direction of each mask

is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

than other possible directionsthan other possible directions

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

3030

Idea 1 of Line DetectionIdea 1 of Line Detection

Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

if at a certain point in the imageif at a certain point in the image

|Ri||Ri|gtgt|Rj||Rj|

for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

3131

Idea 2 of Line DetectionIdea 2 of Line Detection

Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

3232

ExampleExample

3333

74 Edge-based 74 Edge-based SegmentationSegmentation

Edge-based segmentations rely on edges found in an image by edge detecting operators

these edges mark image locations of discontinuities in gray level

Edge detection is the most common approach for detecting meaningful discontinuities in gray level

There are a large group of methods based on information about edges in the image

3434

What is edgeWhat is edge

Edge is where change occurs Change is measured by derivative in 1D

―Biggest change derivative has maximum magnitude

Or 2nd derivative is zero we discuss approaches for implementing

―first-order derivative (Gradient operator)

―second-order derivative (Laplacian operator)

―we have introduced both derivatives in chapter 3

―Here we will talk only about their properties for edge detection

3535

What is edgeWhat is edge

In other wordsIn other words an edge is a set of an edge is a set of

connected pixelsconnected pixels

that lie on the boundary between two that lie on the boundary between two

regions with relatively distinct gray-level regions with relatively distinct gray-level

propertiesproperties

Note edge vs boundaryNote edge vs boundary

―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

―whereas a region boundary owing to whereas a region boundary owing to

the way it is defined is a more global the way it is defined is a more global

ideaidea

3636

Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

because of because of

optics optics

sampling sampling

image image

acquisition acquisition

imperfectionimperfection

3737

Thick and Thin EdgeThick and Thin Edge

The slope of the ramp is inversely The slope of the ramp is inversely

proportional to the degree of blurring in the proportional to the degree of blurring in the

edgeedge

Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

pathpath

Instead an edge point now is any point Instead an edge point now is any point

contained in the ramp and contained in the ramp and an edge would an edge would

then be a set of such points that are then be a set of such points that are

connectedconnected

The thickness is determined by the length of the The thickness is determined by the length of the

rampramp

The length is determined by the slope which is in The length is determined by the slope which is in

turn determined by the degree of blurringturn determined by the degree of blurring

Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

edges tend to be thinedges tend to be thin

3838

First and Second derivatives (First and Second derivatives ( 导数导数 ))

the signs of the the signs of the

derivatives would be derivatives would be

reversed for an edge reversed for an edge

that transitions from that transitions from

light to darklight to dark

First First derivatderivatee

SeconSecond d derivatderivatee

Gray-Gray-level level profileprofile

3939

Second derivativesSecond derivatives

an undesirable featurean undesirable feature

produces 2 values for every edge in an produces 2 values for every edge in an

imageimage

zero-crossing propertyzero-crossing property

an imaginary straight line joining the an imaginary straight line joining the

extreme positive and negative values of extreme positive and negative values of

the second derivative would cross zero the second derivative would cross zero

near the midpoint of the edgenear the midpoint of the edge

quite useful for locating the centers of quite useful for locating the centers of

thick edgesthick edges

4040

Basic idea of edge detectionBasic idea of edge detection

A profile is defined perpendicularly to A profile is defined perpendicularly to

the edge direction and the results are the edge direction and the results are

interpretedinterpreted

The magnitude of the first derivative is The magnitude of the first derivative is

used to detect an edge (if a point is on a used to detect an edge (if a point is on a

ramp)ramp)

The sign of the second derivative can The sign of the second derivative can

determine whether an edge pixel is on the determine whether an edge pixel is on the

dark or light side of an edgedark or light side of an edge

4141

Review of First DerivateReview of First Derivate

Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

Sobel operators 3Sobel operators 333

6 5 8 5x yG z z G z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

9 5 8 6x yG z z G z z 1 0 0 1

0 1 1 0

1 0 0 1

0 1 1 0

7 8 9 1 2 3

3 6 9 1 4 7

2 2

2 2

x

y

G z z z z z z

G z z z z z z

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

x yf G G

4242

Edge direction and strengthEdge direction and strength

Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

2 2x yf G G

4343

Gradient MasksGradient Masks

1 0 0 1

0 1 1 0

Roberts

1 0 0 1

0 1 1 0

Roberts

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

4444

Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

33

Image Processing Flow based onImage Processing Flow based on

Image SegmentationImage Segmentation

Image Segmentation

InputInputImageImage

ObjectObjectImageImage

FeatureFeatureVectorVector

ObjectObjectTypeType

FeatureExtraction

Classification

44

Why is it difficult

In generalIn general autonomousautonomous ( ( 自 主 的自 主 的 ))

segmentation is one segmentation is one of of the most difficult the most difficult

tasks in image processing It tasks in image processing It isis difficult difficult

because of because of many reasonmany reasonss Here are some Here are some

typical obstaclestypical obstacles Non-uniform illuminationNon-uniform illumination

No control of the environmentNo control of the environment

Inadequate model of the object of interestInadequate model of the object of interest

NoiseNoise

etcetc

55

Segmentation methods can be divided into three groups according to the dominant features they employ

Segmentation based on global knowledge about an image

―The knowledge is usually represented by a histogram of image features

Edge-based segmentations

―Utilizing edge detection processes to find a closed boundary so that an inside and an outside can be defined

Region-based segmentations

―This techniques proceed by dividing the image into regions that exhibit similar properties

Principal approachesPrincipal approaches

66

2 basis properties of intensity 2 basis properties of intensity valuesvalues

Segmentation algorithms generally are baseSegmentation algorithms generally are based on one of 2 basis properties of intensity vad on one of 2 basis properties of intensity valueslues DiscontinuityDiscontinuity

―to partition an image based on abrupt (to partition an image based on abrupt ( 突然突然的的 ) changes in intensity (such as edges)) changes in intensity (such as edges)

SimilaritySimilarity

―to partition an image into regions that are simto partition an image into regions that are similar according to a set of predefined criteriailar according to a set of predefined criteria

77

Detection of DiscontinuitiesDetection of Discontinuities

detect the three basic types of gray detect the three basic types of gray

level discontinuitieslevel discontinuities

points lines edgespoints lines edges

the common way is to run a mask the common way is to run a mask

through the imagethrough the image

88

ContentsContents

ThresholdingThresholding

Point DetectionPoint Detection

Line DetectionLine Detection

Edge-based SegmentationEdge-based Segmentation

Region-based SegmentationRegion-based Segmentation

99

71 Thresholding71 Thresholding

Thresholding is a labeling operation Thresholding is a labeling operation

on a gray scale image that on a gray scale image that

distinguishes pixels of a higher distinguishes pixels of a higher

intensity from pixels with a lower intensity from pixels with a lower

intensity valueintensity value

The output of thresholding usuallyThe output of thresholding usually is is a a

binary imagebinary image

This technique is particularly useful This technique is particularly useful

for scenes which contain for scenes which contain solid objects solid objects

on a uniform contrasting backgroundon a uniform contrasting background

1010

Classification of ThresholdingClassification of Thresholding

Thresholding can be viewed as an operation Thresholding can be viewed as an operation that involves tests against a function T of ththat involves tests against a function T of the forme form

where p(xy) denotes some local property of this where p(xy) denotes some local property of this pointpoint

T T x y p x y f x y

1111

Classification of ThresholdingClassification of Thresholding

When T depends onWhen T depends on only f(xy) only on gray-level valuesonly f(xy) only on gray-level values

1048638 1048638 Global Global thresholdingthresholding

both f(xy) and p(xy) on gray-level values and itboth f(xy) and p(xy) on gray-level values and its neighborss neighbors

Local Local thresholdingthresholding

x and y (in addition)x and y (in addition)

Dynamic Dynamic thresholdingthresholding

T T x y p x y f x y

1212

Basic Global ThresholdingBasic Global Thresholding

Original imageOriginal image HistogramHistogram

SolutionSolution use T midway between the max and use T midway between the max and

min gray levelsmin gray levels

SolutionSolution use T midway between the max and use T midway between the max and

min gray levelsmin gray levels

See ldquobasic_global_threSee ldquobasic_global_thremrdquomrdquo

1313

Basic Global ThresholdingBasic Global Thresholding

Let light objects in dark backgroundLet light objects in dark background

To extract the objectsTo extract the objects Select a ldquoTrdquo that separates the objects from thSelect a ldquoTrdquo that separates the objects from th

e backgrounde background

ie any (xy) for which f(xy)gtT is an object point ie any (xy) for which f(xy)gtT is an object point

A thresholded imageA thresholded image

1

0

if f x y T backgroundg x y

if f x y T foreground

1414

Heuristic Global ThresholdingHeuristic Global Thresholding

11 Select an initial estimate for TSelect an initial estimate for T

22 Segment the image using T This will produce two Segment the image using T This will produce two groups of pixels Ggroups of pixels G11 consisting of all pixels with gra consisting of all pixels with gray level values lt T and Gy level values lt T and G22 consisting of pixels with g consisting of pixels with gray level values ray level values T T

33 Compute the average gray level values μCompute the average gray level values μ11 and μ and μ22 fo for the pixels in regions Gr the pixels in regions G11 and G and G22

44 Compute a new threshold value T=05(μCompute a new threshold value T=05(μ11+μ+μ22))

55 Repeat steps 2 through 4 until the difference in T iRepeat steps 2 through 4 until the difference in T in successive iterations is smaller than a predefinen successive iterations is smaller than a predefined parameter Td parameter T00 seerdquohhellipmrdquo seerdquohhellipmrdquo

1515

Basic Adaptive ThresholdingBasic Adaptive Thresholding

subdivide original image into small areassubdivide original image into small areas

utilize a different threshold to segment each utilize a different threshold to segment each subimagessubimages

since the threshold used for each pixel depesince the threshold used for each pixel depends on the location of the pixel in terms of tnds on the location of the pixel in terms of the subimages this type of thresholding is ahe subimages this type of thresholding is adaptivedaptive

See ldquoAdaptive_thresholdmrdquoSee ldquoAdaptive_thresholdmrdquo

1616

Multilevel ThresholdingMultilevel Thresholding

a point (xy) belongs toa point (xy) belongs to an object class if Tan object class if T11 ltlt f(xy) le T f(xy) le T22

another object class if f(xy) another object class if f(xy) gtgt T T22

to background if f(xy) le Tto background if f(xy) le T11

1717

The histogram of an image containing two pThe histogram of an image containing two principal brightness regions can be considererincipal brightness regions can be considered an estimate of the brightness probability d an estimate of the brightness probability density function p(z)density function p(z) p(z) is the sum (or mixture) of two unimodal (p(z) is the sum (or mixture) of two unimodal ( 单单峰的峰的 ) densities (one for light one for dark region) densities (one for light one for dark regions)s)

1 1 2 2

1 2 1

p z P p z P p z

P P

1818

Optimal ThresholdingOptimal Thresholding

If the form of the If the form of the

densities is densities is

known or known or

assumed in assumed in

terms of terms of

minimum error minimum error

determining an determining an

optimal optimal

threshold for threshold for

segmenting the segmenting the

image is image is

possiblepossible

1 1 2 2

1 2 1

p z P p z P p z

P P

1919

Optimal ThresholdingOptimal Thresholding

Probability of erroneouslyProbability of erroneously

2020

Optimal ThresholdingOptimal Thresholding

Minimum errorMinimum error Differentiating ( Differentiating ( 微分微分 ) E(T) with respect to T (usi) E(T) with respect to T (usi

ng Leibnizrsquos rule) and equating the result to 0ng Leibnizrsquos rule) and equating the result to 0

find T which makesfind T which makes

2121

Optimal ThresholdingOptimal Thresholding

Minimum errorMinimum error

Specially if Specially if PP11 = P = P22 then the optimum then the optimum

threshold is where the curve pthreshold is where the curve p11(z) and (z) and

pp22(z) intersect(z) intersect

2222

Optimal ThresholdingOptimal Thresholding

For exampleFor example

Let Let PDF=Gaussian densityPDF=Gaussian density p p11(z) and (z) and

pp22(z)(z)

where μwhere μ11 and σ and σ1122 are the mean and are the mean and

variance of the Gaussian density of one variance of the Gaussian density of one

objectobject

μμ22 and σ and σ2222 are the mean and variance of are the mean and variance of

the Gaussian density of the other objectthe Gaussian density of the other object

2323

Optimal ThresholdingOptimal Thresholding

Quadratic equation (Quadratic equation (二次方程二次方程 ))

2424

Problems of ThresholdingProblems of Thresholding

Original imageOriginal image Thresholded imageThresholded image

2525

Problems of ThresholdingProblems of Thresholding

(a)(a) Exact threshold Exact threshold

segmentationsegmentation

(b)(b) Threshold too lowThreshold too low

(c)(c) Threshold too Threshold too

highhigh

2626

72 Point Detection72 Point Detection

a point has been detected at the a point has been detected at the

location on which the mark is location on which the mark is

centered ifcentered if

|R|geT|R|geT

where T is a nonnegative thresholdwhere T is a nonnegative threshold

R is the sum of products of the R is the sum of products of the

coefficients with the gray levels contained coefficients with the gray levels contained

in the region encompassed by the markin the region encompassed by the mark

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

2727

72 Point Detection72 Point Detection

Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

2828

ExampleExample

2929

73 Line Detection73 Line Detection

Horizontal mask will result with max Horizontal mask will result with max

response when a line passed through the response when a line passed through the

middle row of the mask with a constant middle row of the mask with a constant

backgroundbackground

the similar idea is used with other masksthe similar idea is used with other masks

Note the preferred direction of each mask Note the preferred direction of each mask

is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

than other possible directionsthan other possible directions

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

3030

Idea 1 of Line DetectionIdea 1 of Line Detection

Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

if at a certain point in the imageif at a certain point in the image

|Ri||Ri|gtgt|Rj||Rj|

for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

3131

Idea 2 of Line DetectionIdea 2 of Line Detection

Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

3232

ExampleExample

3333

74 Edge-based 74 Edge-based SegmentationSegmentation

Edge-based segmentations rely on edges found in an image by edge detecting operators

these edges mark image locations of discontinuities in gray level

Edge detection is the most common approach for detecting meaningful discontinuities in gray level

There are a large group of methods based on information about edges in the image

3434

What is edgeWhat is edge

Edge is where change occurs Change is measured by derivative in 1D

―Biggest change derivative has maximum magnitude

Or 2nd derivative is zero we discuss approaches for implementing

―first-order derivative (Gradient operator)

―second-order derivative (Laplacian operator)

―we have introduced both derivatives in chapter 3

―Here we will talk only about their properties for edge detection

3535

What is edgeWhat is edge

In other wordsIn other words an edge is a set of an edge is a set of

connected pixelsconnected pixels

that lie on the boundary between two that lie on the boundary between two

regions with relatively distinct gray-level regions with relatively distinct gray-level

propertiesproperties

Note edge vs boundaryNote edge vs boundary

―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

―whereas a region boundary owing to whereas a region boundary owing to

the way it is defined is a more global the way it is defined is a more global

ideaidea

3636

Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

because of because of

optics optics

sampling sampling

image image

acquisition acquisition

imperfectionimperfection

3737

Thick and Thin EdgeThick and Thin Edge

The slope of the ramp is inversely The slope of the ramp is inversely

proportional to the degree of blurring in the proportional to the degree of blurring in the

edgeedge

Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

pathpath

Instead an edge point now is any point Instead an edge point now is any point

contained in the ramp and contained in the ramp and an edge would an edge would

then be a set of such points that are then be a set of such points that are

connectedconnected

The thickness is determined by the length of the The thickness is determined by the length of the

rampramp

The length is determined by the slope which is in The length is determined by the slope which is in

turn determined by the degree of blurringturn determined by the degree of blurring

Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

edges tend to be thinedges tend to be thin

3838

First and Second derivatives (First and Second derivatives ( 导数导数 ))

the signs of the the signs of the

derivatives would be derivatives would be

reversed for an edge reversed for an edge

that transitions from that transitions from

light to darklight to dark

First First derivatderivatee

SeconSecond d derivatderivatee

Gray-Gray-level level profileprofile

3939

Second derivativesSecond derivatives

an undesirable featurean undesirable feature

produces 2 values for every edge in an produces 2 values for every edge in an

imageimage

zero-crossing propertyzero-crossing property

an imaginary straight line joining the an imaginary straight line joining the

extreme positive and negative values of extreme positive and negative values of

the second derivative would cross zero the second derivative would cross zero

near the midpoint of the edgenear the midpoint of the edge

quite useful for locating the centers of quite useful for locating the centers of

thick edgesthick edges

4040

Basic idea of edge detectionBasic idea of edge detection

A profile is defined perpendicularly to A profile is defined perpendicularly to

the edge direction and the results are the edge direction and the results are

interpretedinterpreted

The magnitude of the first derivative is The magnitude of the first derivative is

used to detect an edge (if a point is on a used to detect an edge (if a point is on a

ramp)ramp)

The sign of the second derivative can The sign of the second derivative can

determine whether an edge pixel is on the determine whether an edge pixel is on the

dark or light side of an edgedark or light side of an edge

4141

Review of First DerivateReview of First Derivate

Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

Sobel operators 3Sobel operators 333

6 5 8 5x yG z z G z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

9 5 8 6x yG z z G z z 1 0 0 1

0 1 1 0

1 0 0 1

0 1 1 0

7 8 9 1 2 3

3 6 9 1 4 7

2 2

2 2

x

y

G z z z z z z

G z z z z z z

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

x yf G G

4242

Edge direction and strengthEdge direction and strength

Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

2 2x yf G G

4343

Gradient MasksGradient Masks

1 0 0 1

0 1 1 0

Roberts

1 0 0 1

0 1 1 0

Roberts

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

4444

Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

44

Why is it difficult

In generalIn general autonomousautonomous ( ( 自 主 的自 主 的 ))

segmentation is one segmentation is one of of the most difficult the most difficult

tasks in image processing It tasks in image processing It isis difficult difficult

because of because of many reasonmany reasonss Here are some Here are some

typical obstaclestypical obstacles Non-uniform illuminationNon-uniform illumination

No control of the environmentNo control of the environment

Inadequate model of the object of interestInadequate model of the object of interest

NoiseNoise

etcetc

55

Segmentation methods can be divided into three groups according to the dominant features they employ

Segmentation based on global knowledge about an image

―The knowledge is usually represented by a histogram of image features

Edge-based segmentations

―Utilizing edge detection processes to find a closed boundary so that an inside and an outside can be defined

Region-based segmentations

―This techniques proceed by dividing the image into regions that exhibit similar properties

Principal approachesPrincipal approaches

66

2 basis properties of intensity 2 basis properties of intensity valuesvalues

Segmentation algorithms generally are baseSegmentation algorithms generally are based on one of 2 basis properties of intensity vad on one of 2 basis properties of intensity valueslues DiscontinuityDiscontinuity

―to partition an image based on abrupt (to partition an image based on abrupt ( 突然突然的的 ) changes in intensity (such as edges)) changes in intensity (such as edges)

SimilaritySimilarity

―to partition an image into regions that are simto partition an image into regions that are similar according to a set of predefined criteriailar according to a set of predefined criteria

77

Detection of DiscontinuitiesDetection of Discontinuities

detect the three basic types of gray detect the three basic types of gray

level discontinuitieslevel discontinuities

points lines edgespoints lines edges

the common way is to run a mask the common way is to run a mask

through the imagethrough the image

88

ContentsContents

ThresholdingThresholding

Point DetectionPoint Detection

Line DetectionLine Detection

Edge-based SegmentationEdge-based Segmentation

Region-based SegmentationRegion-based Segmentation

99

71 Thresholding71 Thresholding

Thresholding is a labeling operation Thresholding is a labeling operation

on a gray scale image that on a gray scale image that

distinguishes pixels of a higher distinguishes pixels of a higher

intensity from pixels with a lower intensity from pixels with a lower

intensity valueintensity value

The output of thresholding usuallyThe output of thresholding usually is is a a

binary imagebinary image

This technique is particularly useful This technique is particularly useful

for scenes which contain for scenes which contain solid objects solid objects

on a uniform contrasting backgroundon a uniform contrasting background

1010

Classification of ThresholdingClassification of Thresholding

Thresholding can be viewed as an operation Thresholding can be viewed as an operation that involves tests against a function T of ththat involves tests against a function T of the forme form

where p(xy) denotes some local property of this where p(xy) denotes some local property of this pointpoint

T T x y p x y f x y

1111

Classification of ThresholdingClassification of Thresholding

When T depends onWhen T depends on only f(xy) only on gray-level valuesonly f(xy) only on gray-level values

1048638 1048638 Global Global thresholdingthresholding

both f(xy) and p(xy) on gray-level values and itboth f(xy) and p(xy) on gray-level values and its neighborss neighbors

Local Local thresholdingthresholding

x and y (in addition)x and y (in addition)

Dynamic Dynamic thresholdingthresholding

T T x y p x y f x y

1212

Basic Global ThresholdingBasic Global Thresholding

Original imageOriginal image HistogramHistogram

SolutionSolution use T midway between the max and use T midway between the max and

min gray levelsmin gray levels

SolutionSolution use T midway between the max and use T midway between the max and

min gray levelsmin gray levels

See ldquobasic_global_threSee ldquobasic_global_thremrdquomrdquo

1313

Basic Global ThresholdingBasic Global Thresholding

Let light objects in dark backgroundLet light objects in dark background

To extract the objectsTo extract the objects Select a ldquoTrdquo that separates the objects from thSelect a ldquoTrdquo that separates the objects from th

e backgrounde background

ie any (xy) for which f(xy)gtT is an object point ie any (xy) for which f(xy)gtT is an object point

A thresholded imageA thresholded image

1

0

if f x y T backgroundg x y

if f x y T foreground

1414

Heuristic Global ThresholdingHeuristic Global Thresholding

11 Select an initial estimate for TSelect an initial estimate for T

22 Segment the image using T This will produce two Segment the image using T This will produce two groups of pixels Ggroups of pixels G11 consisting of all pixels with gra consisting of all pixels with gray level values lt T and Gy level values lt T and G22 consisting of pixels with g consisting of pixels with gray level values ray level values T T

33 Compute the average gray level values μCompute the average gray level values μ11 and μ and μ22 fo for the pixels in regions Gr the pixels in regions G11 and G and G22

44 Compute a new threshold value T=05(μCompute a new threshold value T=05(μ11+μ+μ22))

55 Repeat steps 2 through 4 until the difference in T iRepeat steps 2 through 4 until the difference in T in successive iterations is smaller than a predefinen successive iterations is smaller than a predefined parameter Td parameter T00 seerdquohhellipmrdquo seerdquohhellipmrdquo

1515

Basic Adaptive ThresholdingBasic Adaptive Thresholding

subdivide original image into small areassubdivide original image into small areas

utilize a different threshold to segment each utilize a different threshold to segment each subimagessubimages

since the threshold used for each pixel depesince the threshold used for each pixel depends on the location of the pixel in terms of tnds on the location of the pixel in terms of the subimages this type of thresholding is ahe subimages this type of thresholding is adaptivedaptive

See ldquoAdaptive_thresholdmrdquoSee ldquoAdaptive_thresholdmrdquo

1616

Multilevel ThresholdingMultilevel Thresholding

a point (xy) belongs toa point (xy) belongs to an object class if Tan object class if T11 ltlt f(xy) le T f(xy) le T22

another object class if f(xy) another object class if f(xy) gtgt T T22

to background if f(xy) le Tto background if f(xy) le T11

1717

The histogram of an image containing two pThe histogram of an image containing two principal brightness regions can be considererincipal brightness regions can be considered an estimate of the brightness probability d an estimate of the brightness probability density function p(z)density function p(z) p(z) is the sum (or mixture) of two unimodal (p(z) is the sum (or mixture) of two unimodal ( 单单峰的峰的 ) densities (one for light one for dark region) densities (one for light one for dark regions)s)

1 1 2 2

1 2 1

p z P p z P p z

P P

1818

Optimal ThresholdingOptimal Thresholding

If the form of the If the form of the

densities is densities is

known or known or

assumed in assumed in

terms of terms of

minimum error minimum error

determining an determining an

optimal optimal

threshold for threshold for

segmenting the segmenting the

image is image is

possiblepossible

1 1 2 2

1 2 1

p z P p z P p z

P P

1919

Optimal ThresholdingOptimal Thresholding

Probability of erroneouslyProbability of erroneously

2020

Optimal ThresholdingOptimal Thresholding

Minimum errorMinimum error Differentiating ( Differentiating ( 微分微分 ) E(T) with respect to T (usi) E(T) with respect to T (usi

ng Leibnizrsquos rule) and equating the result to 0ng Leibnizrsquos rule) and equating the result to 0

find T which makesfind T which makes

2121

Optimal ThresholdingOptimal Thresholding

Minimum errorMinimum error

Specially if Specially if PP11 = P = P22 then the optimum then the optimum

threshold is where the curve pthreshold is where the curve p11(z) and (z) and

pp22(z) intersect(z) intersect

2222

Optimal ThresholdingOptimal Thresholding

For exampleFor example

Let Let PDF=Gaussian densityPDF=Gaussian density p p11(z) and (z) and

pp22(z)(z)

where μwhere μ11 and σ and σ1122 are the mean and are the mean and

variance of the Gaussian density of one variance of the Gaussian density of one

objectobject

μμ22 and σ and σ2222 are the mean and variance of are the mean and variance of

the Gaussian density of the other objectthe Gaussian density of the other object

2323

Optimal ThresholdingOptimal Thresholding

Quadratic equation (Quadratic equation (二次方程二次方程 ))

2424

Problems of ThresholdingProblems of Thresholding

Original imageOriginal image Thresholded imageThresholded image

2525

Problems of ThresholdingProblems of Thresholding

(a)(a) Exact threshold Exact threshold

segmentationsegmentation

(b)(b) Threshold too lowThreshold too low

(c)(c) Threshold too Threshold too

highhigh

2626

72 Point Detection72 Point Detection

a point has been detected at the a point has been detected at the

location on which the mark is location on which the mark is

centered ifcentered if

|R|geT|R|geT

where T is a nonnegative thresholdwhere T is a nonnegative threshold

R is the sum of products of the R is the sum of products of the

coefficients with the gray levels contained coefficients with the gray levels contained

in the region encompassed by the markin the region encompassed by the mark

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

2727

72 Point Detection72 Point Detection

Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

2828

ExampleExample

2929

73 Line Detection73 Line Detection

Horizontal mask will result with max Horizontal mask will result with max

response when a line passed through the response when a line passed through the

middle row of the mask with a constant middle row of the mask with a constant

backgroundbackground

the similar idea is used with other masksthe similar idea is used with other masks

Note the preferred direction of each mask Note the preferred direction of each mask

is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

than other possible directionsthan other possible directions

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

3030

Idea 1 of Line DetectionIdea 1 of Line Detection

Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

if at a certain point in the imageif at a certain point in the image

|Ri||Ri|gtgt|Rj||Rj|

for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

3131

Idea 2 of Line DetectionIdea 2 of Line Detection

Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

3232

ExampleExample

3333

74 Edge-based 74 Edge-based SegmentationSegmentation

Edge-based segmentations rely on edges found in an image by edge detecting operators

these edges mark image locations of discontinuities in gray level

Edge detection is the most common approach for detecting meaningful discontinuities in gray level

There are a large group of methods based on information about edges in the image

3434

What is edgeWhat is edge

Edge is where change occurs Change is measured by derivative in 1D

―Biggest change derivative has maximum magnitude

Or 2nd derivative is zero we discuss approaches for implementing

―first-order derivative (Gradient operator)

―second-order derivative (Laplacian operator)

―we have introduced both derivatives in chapter 3

―Here we will talk only about their properties for edge detection

3535

What is edgeWhat is edge

In other wordsIn other words an edge is a set of an edge is a set of

connected pixelsconnected pixels

that lie on the boundary between two that lie on the boundary between two

regions with relatively distinct gray-level regions with relatively distinct gray-level

propertiesproperties

Note edge vs boundaryNote edge vs boundary

―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

―whereas a region boundary owing to whereas a region boundary owing to

the way it is defined is a more global the way it is defined is a more global

ideaidea

3636

Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

because of because of

optics optics

sampling sampling

image image

acquisition acquisition

imperfectionimperfection

3737

Thick and Thin EdgeThick and Thin Edge

The slope of the ramp is inversely The slope of the ramp is inversely

proportional to the degree of blurring in the proportional to the degree of blurring in the

edgeedge

Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

pathpath

Instead an edge point now is any point Instead an edge point now is any point

contained in the ramp and contained in the ramp and an edge would an edge would

then be a set of such points that are then be a set of such points that are

connectedconnected

The thickness is determined by the length of the The thickness is determined by the length of the

rampramp

The length is determined by the slope which is in The length is determined by the slope which is in

turn determined by the degree of blurringturn determined by the degree of blurring

Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

edges tend to be thinedges tend to be thin

3838

First and Second derivatives (First and Second derivatives ( 导数导数 ))

the signs of the the signs of the

derivatives would be derivatives would be

reversed for an edge reversed for an edge

that transitions from that transitions from

light to darklight to dark

First First derivatderivatee

SeconSecond d derivatderivatee

Gray-Gray-level level profileprofile

3939

Second derivativesSecond derivatives

an undesirable featurean undesirable feature

produces 2 values for every edge in an produces 2 values for every edge in an

imageimage

zero-crossing propertyzero-crossing property

an imaginary straight line joining the an imaginary straight line joining the

extreme positive and negative values of extreme positive and negative values of

the second derivative would cross zero the second derivative would cross zero

near the midpoint of the edgenear the midpoint of the edge

quite useful for locating the centers of quite useful for locating the centers of

thick edgesthick edges

4040

Basic idea of edge detectionBasic idea of edge detection

A profile is defined perpendicularly to A profile is defined perpendicularly to

the edge direction and the results are the edge direction and the results are

interpretedinterpreted

The magnitude of the first derivative is The magnitude of the first derivative is

used to detect an edge (if a point is on a used to detect an edge (if a point is on a

ramp)ramp)

The sign of the second derivative can The sign of the second derivative can

determine whether an edge pixel is on the determine whether an edge pixel is on the

dark or light side of an edgedark or light side of an edge

4141

Review of First DerivateReview of First Derivate

Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

Sobel operators 3Sobel operators 333

6 5 8 5x yG z z G z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

9 5 8 6x yG z z G z z 1 0 0 1

0 1 1 0

1 0 0 1

0 1 1 0

7 8 9 1 2 3

3 6 9 1 4 7

2 2

2 2

x

y

G z z z z z z

G z z z z z z

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

x yf G G

4242

Edge direction and strengthEdge direction and strength

Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

2 2x yf G G

4343

Gradient MasksGradient Masks

1 0 0 1

0 1 1 0

Roberts

1 0 0 1

0 1 1 0

Roberts

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

4444

Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

55

Segmentation methods can be divided into three groups according to the dominant features they employ

Segmentation based on global knowledge about an image

―The knowledge is usually represented by a histogram of image features

Edge-based segmentations

―Utilizing edge detection processes to find a closed boundary so that an inside and an outside can be defined

Region-based segmentations

―This techniques proceed by dividing the image into regions that exhibit similar properties

Principal approachesPrincipal approaches

66

2 basis properties of intensity 2 basis properties of intensity valuesvalues

Segmentation algorithms generally are baseSegmentation algorithms generally are based on one of 2 basis properties of intensity vad on one of 2 basis properties of intensity valueslues DiscontinuityDiscontinuity

―to partition an image based on abrupt (to partition an image based on abrupt ( 突然突然的的 ) changes in intensity (such as edges)) changes in intensity (such as edges)

SimilaritySimilarity

―to partition an image into regions that are simto partition an image into regions that are similar according to a set of predefined criteriailar according to a set of predefined criteria

77

Detection of DiscontinuitiesDetection of Discontinuities

detect the three basic types of gray detect the three basic types of gray

level discontinuitieslevel discontinuities

points lines edgespoints lines edges

the common way is to run a mask the common way is to run a mask

through the imagethrough the image

88

ContentsContents

ThresholdingThresholding

Point DetectionPoint Detection

Line DetectionLine Detection

Edge-based SegmentationEdge-based Segmentation

Region-based SegmentationRegion-based Segmentation

99

71 Thresholding71 Thresholding

Thresholding is a labeling operation Thresholding is a labeling operation

on a gray scale image that on a gray scale image that

distinguishes pixels of a higher distinguishes pixels of a higher

intensity from pixels with a lower intensity from pixels with a lower

intensity valueintensity value

The output of thresholding usuallyThe output of thresholding usually is is a a

binary imagebinary image

This technique is particularly useful This technique is particularly useful

for scenes which contain for scenes which contain solid objects solid objects

on a uniform contrasting backgroundon a uniform contrasting background

1010

Classification of ThresholdingClassification of Thresholding

Thresholding can be viewed as an operation Thresholding can be viewed as an operation that involves tests against a function T of ththat involves tests against a function T of the forme form

where p(xy) denotes some local property of this where p(xy) denotes some local property of this pointpoint

T T x y p x y f x y

1111

Classification of ThresholdingClassification of Thresholding

When T depends onWhen T depends on only f(xy) only on gray-level valuesonly f(xy) only on gray-level values

1048638 1048638 Global Global thresholdingthresholding

both f(xy) and p(xy) on gray-level values and itboth f(xy) and p(xy) on gray-level values and its neighborss neighbors

Local Local thresholdingthresholding

x and y (in addition)x and y (in addition)

Dynamic Dynamic thresholdingthresholding

T T x y p x y f x y

1212

Basic Global ThresholdingBasic Global Thresholding

Original imageOriginal image HistogramHistogram

SolutionSolution use T midway between the max and use T midway between the max and

min gray levelsmin gray levels

SolutionSolution use T midway between the max and use T midway between the max and

min gray levelsmin gray levels

See ldquobasic_global_threSee ldquobasic_global_thremrdquomrdquo

1313

Basic Global ThresholdingBasic Global Thresholding

Let light objects in dark backgroundLet light objects in dark background

To extract the objectsTo extract the objects Select a ldquoTrdquo that separates the objects from thSelect a ldquoTrdquo that separates the objects from th

e backgrounde background

ie any (xy) for which f(xy)gtT is an object point ie any (xy) for which f(xy)gtT is an object point

A thresholded imageA thresholded image

1

0

if f x y T backgroundg x y

if f x y T foreground

1414

Heuristic Global ThresholdingHeuristic Global Thresholding

11 Select an initial estimate for TSelect an initial estimate for T

22 Segment the image using T This will produce two Segment the image using T This will produce two groups of pixels Ggroups of pixels G11 consisting of all pixels with gra consisting of all pixels with gray level values lt T and Gy level values lt T and G22 consisting of pixels with g consisting of pixels with gray level values ray level values T T

33 Compute the average gray level values μCompute the average gray level values μ11 and μ and μ22 fo for the pixels in regions Gr the pixels in regions G11 and G and G22

44 Compute a new threshold value T=05(μCompute a new threshold value T=05(μ11+μ+μ22))

55 Repeat steps 2 through 4 until the difference in T iRepeat steps 2 through 4 until the difference in T in successive iterations is smaller than a predefinen successive iterations is smaller than a predefined parameter Td parameter T00 seerdquohhellipmrdquo seerdquohhellipmrdquo

1515

Basic Adaptive ThresholdingBasic Adaptive Thresholding

subdivide original image into small areassubdivide original image into small areas

utilize a different threshold to segment each utilize a different threshold to segment each subimagessubimages

since the threshold used for each pixel depesince the threshold used for each pixel depends on the location of the pixel in terms of tnds on the location of the pixel in terms of the subimages this type of thresholding is ahe subimages this type of thresholding is adaptivedaptive

See ldquoAdaptive_thresholdmrdquoSee ldquoAdaptive_thresholdmrdquo

1616

Multilevel ThresholdingMultilevel Thresholding

a point (xy) belongs toa point (xy) belongs to an object class if Tan object class if T11 ltlt f(xy) le T f(xy) le T22

another object class if f(xy) another object class if f(xy) gtgt T T22

to background if f(xy) le Tto background if f(xy) le T11

1717

The histogram of an image containing two pThe histogram of an image containing two principal brightness regions can be considererincipal brightness regions can be considered an estimate of the brightness probability d an estimate of the brightness probability density function p(z)density function p(z) p(z) is the sum (or mixture) of two unimodal (p(z) is the sum (or mixture) of two unimodal ( 单单峰的峰的 ) densities (one for light one for dark region) densities (one for light one for dark regions)s)

1 1 2 2

1 2 1

p z P p z P p z

P P

1818

Optimal ThresholdingOptimal Thresholding

If the form of the If the form of the

densities is densities is

known or known or

assumed in assumed in

terms of terms of

minimum error minimum error

determining an determining an

optimal optimal

threshold for threshold for

segmenting the segmenting the

image is image is

possiblepossible

1 1 2 2

1 2 1

p z P p z P p z

P P

1919

Optimal ThresholdingOptimal Thresholding

Probability of erroneouslyProbability of erroneously

2020

Optimal ThresholdingOptimal Thresholding

Minimum errorMinimum error Differentiating ( Differentiating ( 微分微分 ) E(T) with respect to T (usi) E(T) with respect to T (usi

ng Leibnizrsquos rule) and equating the result to 0ng Leibnizrsquos rule) and equating the result to 0

find T which makesfind T which makes

2121

Optimal ThresholdingOptimal Thresholding

Minimum errorMinimum error

Specially if Specially if PP11 = P = P22 then the optimum then the optimum

threshold is where the curve pthreshold is where the curve p11(z) and (z) and

pp22(z) intersect(z) intersect

2222

Optimal ThresholdingOptimal Thresholding

For exampleFor example

Let Let PDF=Gaussian densityPDF=Gaussian density p p11(z) and (z) and

pp22(z)(z)

where μwhere μ11 and σ and σ1122 are the mean and are the mean and

variance of the Gaussian density of one variance of the Gaussian density of one

objectobject

μμ22 and σ and σ2222 are the mean and variance of are the mean and variance of

the Gaussian density of the other objectthe Gaussian density of the other object

2323

Optimal ThresholdingOptimal Thresholding

Quadratic equation (Quadratic equation (二次方程二次方程 ))

2424

Problems of ThresholdingProblems of Thresholding

Original imageOriginal image Thresholded imageThresholded image

2525

Problems of ThresholdingProblems of Thresholding

(a)(a) Exact threshold Exact threshold

segmentationsegmentation

(b)(b) Threshold too lowThreshold too low

(c)(c) Threshold too Threshold too

highhigh

2626

72 Point Detection72 Point Detection

a point has been detected at the a point has been detected at the

location on which the mark is location on which the mark is

centered ifcentered if

|R|geT|R|geT

where T is a nonnegative thresholdwhere T is a nonnegative threshold

R is the sum of products of the R is the sum of products of the

coefficients with the gray levels contained coefficients with the gray levels contained

in the region encompassed by the markin the region encompassed by the mark

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

2727

72 Point Detection72 Point Detection

Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

2828

ExampleExample

2929

73 Line Detection73 Line Detection

Horizontal mask will result with max Horizontal mask will result with max

response when a line passed through the response when a line passed through the

middle row of the mask with a constant middle row of the mask with a constant

backgroundbackground

the similar idea is used with other masksthe similar idea is used with other masks

Note the preferred direction of each mask Note the preferred direction of each mask

is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

than other possible directionsthan other possible directions

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

3030

Idea 1 of Line DetectionIdea 1 of Line Detection

Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

if at a certain point in the imageif at a certain point in the image

|Ri||Ri|gtgt|Rj||Rj|

for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

3131

Idea 2 of Line DetectionIdea 2 of Line Detection

Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

3232

ExampleExample

3333

74 Edge-based 74 Edge-based SegmentationSegmentation

Edge-based segmentations rely on edges found in an image by edge detecting operators

these edges mark image locations of discontinuities in gray level

Edge detection is the most common approach for detecting meaningful discontinuities in gray level

There are a large group of methods based on information about edges in the image

3434

What is edgeWhat is edge

Edge is where change occurs Change is measured by derivative in 1D

―Biggest change derivative has maximum magnitude

Or 2nd derivative is zero we discuss approaches for implementing

―first-order derivative (Gradient operator)

―second-order derivative (Laplacian operator)

―we have introduced both derivatives in chapter 3

―Here we will talk only about their properties for edge detection

3535

What is edgeWhat is edge

In other wordsIn other words an edge is a set of an edge is a set of

connected pixelsconnected pixels

that lie on the boundary between two that lie on the boundary between two

regions with relatively distinct gray-level regions with relatively distinct gray-level

propertiesproperties

Note edge vs boundaryNote edge vs boundary

―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

―whereas a region boundary owing to whereas a region boundary owing to

the way it is defined is a more global the way it is defined is a more global

ideaidea

3636

Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

because of because of

optics optics

sampling sampling

image image

acquisition acquisition

imperfectionimperfection

3737

Thick and Thin EdgeThick and Thin Edge

The slope of the ramp is inversely The slope of the ramp is inversely

proportional to the degree of blurring in the proportional to the degree of blurring in the

edgeedge

Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

pathpath

Instead an edge point now is any point Instead an edge point now is any point

contained in the ramp and contained in the ramp and an edge would an edge would

then be a set of such points that are then be a set of such points that are

connectedconnected

The thickness is determined by the length of the The thickness is determined by the length of the

rampramp

The length is determined by the slope which is in The length is determined by the slope which is in

turn determined by the degree of blurringturn determined by the degree of blurring

Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

edges tend to be thinedges tend to be thin

3838

First and Second derivatives (First and Second derivatives ( 导数导数 ))

the signs of the the signs of the

derivatives would be derivatives would be

reversed for an edge reversed for an edge

that transitions from that transitions from

light to darklight to dark

First First derivatderivatee

SeconSecond d derivatderivatee

Gray-Gray-level level profileprofile

3939

Second derivativesSecond derivatives

an undesirable featurean undesirable feature

produces 2 values for every edge in an produces 2 values for every edge in an

imageimage

zero-crossing propertyzero-crossing property

an imaginary straight line joining the an imaginary straight line joining the

extreme positive and negative values of extreme positive and negative values of

the second derivative would cross zero the second derivative would cross zero

near the midpoint of the edgenear the midpoint of the edge

quite useful for locating the centers of quite useful for locating the centers of

thick edgesthick edges

4040

Basic idea of edge detectionBasic idea of edge detection

A profile is defined perpendicularly to A profile is defined perpendicularly to

the edge direction and the results are the edge direction and the results are

interpretedinterpreted

The magnitude of the first derivative is The magnitude of the first derivative is

used to detect an edge (if a point is on a used to detect an edge (if a point is on a

ramp)ramp)

The sign of the second derivative can The sign of the second derivative can

determine whether an edge pixel is on the determine whether an edge pixel is on the

dark or light side of an edgedark or light side of an edge

4141

Review of First DerivateReview of First Derivate

Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

Sobel operators 3Sobel operators 333

6 5 8 5x yG z z G z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

9 5 8 6x yG z z G z z 1 0 0 1

0 1 1 0

1 0 0 1

0 1 1 0

7 8 9 1 2 3

3 6 9 1 4 7

2 2

2 2

x

y

G z z z z z z

G z z z z z z

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

x yf G G

4242

Edge direction and strengthEdge direction and strength

Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

2 2x yf G G

4343

Gradient MasksGradient Masks

1 0 0 1

0 1 1 0

Roberts

1 0 0 1

0 1 1 0

Roberts

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

4444

Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

66

2 basis properties of intensity 2 basis properties of intensity valuesvalues

Segmentation algorithms generally are baseSegmentation algorithms generally are based on one of 2 basis properties of intensity vad on one of 2 basis properties of intensity valueslues DiscontinuityDiscontinuity

―to partition an image based on abrupt (to partition an image based on abrupt ( 突然突然的的 ) changes in intensity (such as edges)) changes in intensity (such as edges)

SimilaritySimilarity

―to partition an image into regions that are simto partition an image into regions that are similar according to a set of predefined criteriailar according to a set of predefined criteria

77

Detection of DiscontinuitiesDetection of Discontinuities

detect the three basic types of gray detect the three basic types of gray

level discontinuitieslevel discontinuities

points lines edgespoints lines edges

the common way is to run a mask the common way is to run a mask

through the imagethrough the image

88

ContentsContents

ThresholdingThresholding

Point DetectionPoint Detection

Line DetectionLine Detection

Edge-based SegmentationEdge-based Segmentation

Region-based SegmentationRegion-based Segmentation

99

71 Thresholding71 Thresholding

Thresholding is a labeling operation Thresholding is a labeling operation

on a gray scale image that on a gray scale image that

distinguishes pixels of a higher distinguishes pixels of a higher

intensity from pixels with a lower intensity from pixels with a lower

intensity valueintensity value

The output of thresholding usuallyThe output of thresholding usually is is a a

binary imagebinary image

This technique is particularly useful This technique is particularly useful

for scenes which contain for scenes which contain solid objects solid objects

on a uniform contrasting backgroundon a uniform contrasting background

1010

Classification of ThresholdingClassification of Thresholding

Thresholding can be viewed as an operation Thresholding can be viewed as an operation that involves tests against a function T of ththat involves tests against a function T of the forme form

where p(xy) denotes some local property of this where p(xy) denotes some local property of this pointpoint

T T x y p x y f x y

1111

Classification of ThresholdingClassification of Thresholding

When T depends onWhen T depends on only f(xy) only on gray-level valuesonly f(xy) only on gray-level values

1048638 1048638 Global Global thresholdingthresholding

both f(xy) and p(xy) on gray-level values and itboth f(xy) and p(xy) on gray-level values and its neighborss neighbors

Local Local thresholdingthresholding

x and y (in addition)x and y (in addition)

Dynamic Dynamic thresholdingthresholding

T T x y p x y f x y

1212

Basic Global ThresholdingBasic Global Thresholding

Original imageOriginal image HistogramHistogram

SolutionSolution use T midway between the max and use T midway between the max and

min gray levelsmin gray levels

SolutionSolution use T midway between the max and use T midway between the max and

min gray levelsmin gray levels

See ldquobasic_global_threSee ldquobasic_global_thremrdquomrdquo

1313

Basic Global ThresholdingBasic Global Thresholding

Let light objects in dark backgroundLet light objects in dark background

To extract the objectsTo extract the objects Select a ldquoTrdquo that separates the objects from thSelect a ldquoTrdquo that separates the objects from th

e backgrounde background

ie any (xy) for which f(xy)gtT is an object point ie any (xy) for which f(xy)gtT is an object point

A thresholded imageA thresholded image

1

0

if f x y T backgroundg x y

if f x y T foreground

1414

Heuristic Global ThresholdingHeuristic Global Thresholding

11 Select an initial estimate for TSelect an initial estimate for T

22 Segment the image using T This will produce two Segment the image using T This will produce two groups of pixels Ggroups of pixels G11 consisting of all pixels with gra consisting of all pixels with gray level values lt T and Gy level values lt T and G22 consisting of pixels with g consisting of pixels with gray level values ray level values T T

33 Compute the average gray level values μCompute the average gray level values μ11 and μ and μ22 fo for the pixels in regions Gr the pixels in regions G11 and G and G22

44 Compute a new threshold value T=05(μCompute a new threshold value T=05(μ11+μ+μ22))

55 Repeat steps 2 through 4 until the difference in T iRepeat steps 2 through 4 until the difference in T in successive iterations is smaller than a predefinen successive iterations is smaller than a predefined parameter Td parameter T00 seerdquohhellipmrdquo seerdquohhellipmrdquo

1515

Basic Adaptive ThresholdingBasic Adaptive Thresholding

subdivide original image into small areassubdivide original image into small areas

utilize a different threshold to segment each utilize a different threshold to segment each subimagessubimages

since the threshold used for each pixel depesince the threshold used for each pixel depends on the location of the pixel in terms of tnds on the location of the pixel in terms of the subimages this type of thresholding is ahe subimages this type of thresholding is adaptivedaptive

See ldquoAdaptive_thresholdmrdquoSee ldquoAdaptive_thresholdmrdquo

1616

Multilevel ThresholdingMultilevel Thresholding

a point (xy) belongs toa point (xy) belongs to an object class if Tan object class if T11 ltlt f(xy) le T f(xy) le T22

another object class if f(xy) another object class if f(xy) gtgt T T22

to background if f(xy) le Tto background if f(xy) le T11

1717

The histogram of an image containing two pThe histogram of an image containing two principal brightness regions can be considererincipal brightness regions can be considered an estimate of the brightness probability d an estimate of the brightness probability density function p(z)density function p(z) p(z) is the sum (or mixture) of two unimodal (p(z) is the sum (or mixture) of two unimodal ( 单单峰的峰的 ) densities (one for light one for dark region) densities (one for light one for dark regions)s)

1 1 2 2

1 2 1

p z P p z P p z

P P

1818

Optimal ThresholdingOptimal Thresholding

If the form of the If the form of the

densities is densities is

known or known or

assumed in assumed in

terms of terms of

minimum error minimum error

determining an determining an

optimal optimal

threshold for threshold for

segmenting the segmenting the

image is image is

possiblepossible

1 1 2 2

1 2 1

p z P p z P p z

P P

1919

Optimal ThresholdingOptimal Thresholding

Probability of erroneouslyProbability of erroneously

2020

Optimal ThresholdingOptimal Thresholding

Minimum errorMinimum error Differentiating ( Differentiating ( 微分微分 ) E(T) with respect to T (usi) E(T) with respect to T (usi

ng Leibnizrsquos rule) and equating the result to 0ng Leibnizrsquos rule) and equating the result to 0

find T which makesfind T which makes

2121

Optimal ThresholdingOptimal Thresholding

Minimum errorMinimum error

Specially if Specially if PP11 = P = P22 then the optimum then the optimum

threshold is where the curve pthreshold is where the curve p11(z) and (z) and

pp22(z) intersect(z) intersect

2222

Optimal ThresholdingOptimal Thresholding

For exampleFor example

Let Let PDF=Gaussian densityPDF=Gaussian density p p11(z) and (z) and

pp22(z)(z)

where μwhere μ11 and σ and σ1122 are the mean and are the mean and

variance of the Gaussian density of one variance of the Gaussian density of one

objectobject

μμ22 and σ and σ2222 are the mean and variance of are the mean and variance of

the Gaussian density of the other objectthe Gaussian density of the other object

2323

Optimal ThresholdingOptimal Thresholding

Quadratic equation (Quadratic equation (二次方程二次方程 ))

2424

Problems of ThresholdingProblems of Thresholding

Original imageOriginal image Thresholded imageThresholded image

2525

Problems of ThresholdingProblems of Thresholding

(a)(a) Exact threshold Exact threshold

segmentationsegmentation

(b)(b) Threshold too lowThreshold too low

(c)(c) Threshold too Threshold too

highhigh

2626

72 Point Detection72 Point Detection

a point has been detected at the a point has been detected at the

location on which the mark is location on which the mark is

centered ifcentered if

|R|geT|R|geT

where T is a nonnegative thresholdwhere T is a nonnegative threshold

R is the sum of products of the R is the sum of products of the

coefficients with the gray levels contained coefficients with the gray levels contained

in the region encompassed by the markin the region encompassed by the mark

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

2727

72 Point Detection72 Point Detection

Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

2828

ExampleExample

2929

73 Line Detection73 Line Detection

Horizontal mask will result with max Horizontal mask will result with max

response when a line passed through the response when a line passed through the

middle row of the mask with a constant middle row of the mask with a constant

backgroundbackground

the similar idea is used with other masksthe similar idea is used with other masks

Note the preferred direction of each mask Note the preferred direction of each mask

is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

than other possible directionsthan other possible directions

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

3030

Idea 1 of Line DetectionIdea 1 of Line Detection

Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

if at a certain point in the imageif at a certain point in the image

|Ri||Ri|gtgt|Rj||Rj|

for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

3131

Idea 2 of Line DetectionIdea 2 of Line Detection

Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

3232

ExampleExample

3333

74 Edge-based 74 Edge-based SegmentationSegmentation

Edge-based segmentations rely on edges found in an image by edge detecting operators

these edges mark image locations of discontinuities in gray level

Edge detection is the most common approach for detecting meaningful discontinuities in gray level

There are a large group of methods based on information about edges in the image

3434

What is edgeWhat is edge

Edge is where change occurs Change is measured by derivative in 1D

―Biggest change derivative has maximum magnitude

Or 2nd derivative is zero we discuss approaches for implementing

―first-order derivative (Gradient operator)

―second-order derivative (Laplacian operator)

―we have introduced both derivatives in chapter 3

―Here we will talk only about their properties for edge detection

3535

What is edgeWhat is edge

In other wordsIn other words an edge is a set of an edge is a set of

connected pixelsconnected pixels

that lie on the boundary between two that lie on the boundary between two

regions with relatively distinct gray-level regions with relatively distinct gray-level

propertiesproperties

Note edge vs boundaryNote edge vs boundary

―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

―whereas a region boundary owing to whereas a region boundary owing to

the way it is defined is a more global the way it is defined is a more global

ideaidea

3636

Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

because of because of

optics optics

sampling sampling

image image

acquisition acquisition

imperfectionimperfection

3737

Thick and Thin EdgeThick and Thin Edge

The slope of the ramp is inversely The slope of the ramp is inversely

proportional to the degree of blurring in the proportional to the degree of blurring in the

edgeedge

Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

pathpath

Instead an edge point now is any point Instead an edge point now is any point

contained in the ramp and contained in the ramp and an edge would an edge would

then be a set of such points that are then be a set of such points that are

connectedconnected

The thickness is determined by the length of the The thickness is determined by the length of the

rampramp

The length is determined by the slope which is in The length is determined by the slope which is in

turn determined by the degree of blurringturn determined by the degree of blurring

Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

edges tend to be thinedges tend to be thin

3838

First and Second derivatives (First and Second derivatives ( 导数导数 ))

the signs of the the signs of the

derivatives would be derivatives would be

reversed for an edge reversed for an edge

that transitions from that transitions from

light to darklight to dark

First First derivatderivatee

SeconSecond d derivatderivatee

Gray-Gray-level level profileprofile

3939

Second derivativesSecond derivatives

an undesirable featurean undesirable feature

produces 2 values for every edge in an produces 2 values for every edge in an

imageimage

zero-crossing propertyzero-crossing property

an imaginary straight line joining the an imaginary straight line joining the

extreme positive and negative values of extreme positive and negative values of

the second derivative would cross zero the second derivative would cross zero

near the midpoint of the edgenear the midpoint of the edge

quite useful for locating the centers of quite useful for locating the centers of

thick edgesthick edges

4040

Basic idea of edge detectionBasic idea of edge detection

A profile is defined perpendicularly to A profile is defined perpendicularly to

the edge direction and the results are the edge direction and the results are

interpretedinterpreted

The magnitude of the first derivative is The magnitude of the first derivative is

used to detect an edge (if a point is on a used to detect an edge (if a point is on a

ramp)ramp)

The sign of the second derivative can The sign of the second derivative can

determine whether an edge pixel is on the determine whether an edge pixel is on the

dark or light side of an edgedark or light side of an edge

4141

Review of First DerivateReview of First Derivate

Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

Sobel operators 3Sobel operators 333

6 5 8 5x yG z z G z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

9 5 8 6x yG z z G z z 1 0 0 1

0 1 1 0

1 0 0 1

0 1 1 0

7 8 9 1 2 3

3 6 9 1 4 7

2 2

2 2

x

y

G z z z z z z

G z z z z z z

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

x yf G G

4242

Edge direction and strengthEdge direction and strength

Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

2 2x yf G G

4343

Gradient MasksGradient Masks

1 0 0 1

0 1 1 0

Roberts

1 0 0 1

0 1 1 0

Roberts

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

4444

Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
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• Slide 53
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• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

77

Detection of DiscontinuitiesDetection of Discontinuities

detect the three basic types of gray detect the three basic types of gray

level discontinuitieslevel discontinuities

points lines edgespoints lines edges

the common way is to run a mask the common way is to run a mask

through the imagethrough the image

88

ContentsContents

ThresholdingThresholding

Point DetectionPoint Detection

Line DetectionLine Detection

Edge-based SegmentationEdge-based Segmentation

Region-based SegmentationRegion-based Segmentation

99

71 Thresholding71 Thresholding

Thresholding is a labeling operation Thresholding is a labeling operation

on a gray scale image that on a gray scale image that

distinguishes pixels of a higher distinguishes pixels of a higher

intensity from pixels with a lower intensity from pixels with a lower

intensity valueintensity value

The output of thresholding usuallyThe output of thresholding usually is is a a

binary imagebinary image

This technique is particularly useful This technique is particularly useful

for scenes which contain for scenes which contain solid objects solid objects

on a uniform contrasting backgroundon a uniform contrasting background

1010

Classification of ThresholdingClassification of Thresholding

Thresholding can be viewed as an operation Thresholding can be viewed as an operation that involves tests against a function T of ththat involves tests against a function T of the forme form

where p(xy) denotes some local property of this where p(xy) denotes some local property of this pointpoint

T T x y p x y f x y

1111

Classification of ThresholdingClassification of Thresholding

When T depends onWhen T depends on only f(xy) only on gray-level valuesonly f(xy) only on gray-level values

1048638 1048638 Global Global thresholdingthresholding

both f(xy) and p(xy) on gray-level values and itboth f(xy) and p(xy) on gray-level values and its neighborss neighbors

Local Local thresholdingthresholding

x and y (in addition)x and y (in addition)

Dynamic Dynamic thresholdingthresholding

T T x y p x y f x y

1212

Basic Global ThresholdingBasic Global Thresholding

Original imageOriginal image HistogramHistogram

SolutionSolution use T midway between the max and use T midway between the max and

min gray levelsmin gray levels

SolutionSolution use T midway between the max and use T midway between the max and

min gray levelsmin gray levels

See ldquobasic_global_threSee ldquobasic_global_thremrdquomrdquo

1313

Basic Global ThresholdingBasic Global Thresholding

Let light objects in dark backgroundLet light objects in dark background

To extract the objectsTo extract the objects Select a ldquoTrdquo that separates the objects from thSelect a ldquoTrdquo that separates the objects from th

e backgrounde background

ie any (xy) for which f(xy)gtT is an object point ie any (xy) for which f(xy)gtT is an object point

A thresholded imageA thresholded image

1

0

if f x y T backgroundg x y

if f x y T foreground

1414

Heuristic Global ThresholdingHeuristic Global Thresholding

11 Select an initial estimate for TSelect an initial estimate for T

22 Segment the image using T This will produce two Segment the image using T This will produce two groups of pixels Ggroups of pixels G11 consisting of all pixels with gra consisting of all pixels with gray level values lt T and Gy level values lt T and G22 consisting of pixels with g consisting of pixels with gray level values ray level values T T

33 Compute the average gray level values μCompute the average gray level values μ11 and μ and μ22 fo for the pixels in regions Gr the pixels in regions G11 and G and G22

44 Compute a new threshold value T=05(μCompute a new threshold value T=05(μ11+μ+μ22))

55 Repeat steps 2 through 4 until the difference in T iRepeat steps 2 through 4 until the difference in T in successive iterations is smaller than a predefinen successive iterations is smaller than a predefined parameter Td parameter T00 seerdquohhellipmrdquo seerdquohhellipmrdquo

1515

Basic Adaptive ThresholdingBasic Adaptive Thresholding

subdivide original image into small areassubdivide original image into small areas

utilize a different threshold to segment each utilize a different threshold to segment each subimagessubimages

since the threshold used for each pixel depesince the threshold used for each pixel depends on the location of the pixel in terms of tnds on the location of the pixel in terms of the subimages this type of thresholding is ahe subimages this type of thresholding is adaptivedaptive

See ldquoAdaptive_thresholdmrdquoSee ldquoAdaptive_thresholdmrdquo

1616

Multilevel ThresholdingMultilevel Thresholding

a point (xy) belongs toa point (xy) belongs to an object class if Tan object class if T11 ltlt f(xy) le T f(xy) le T22

another object class if f(xy) another object class if f(xy) gtgt T T22

to background if f(xy) le Tto background if f(xy) le T11

1717

The histogram of an image containing two pThe histogram of an image containing two principal brightness regions can be considererincipal brightness regions can be considered an estimate of the brightness probability d an estimate of the brightness probability density function p(z)density function p(z) p(z) is the sum (or mixture) of two unimodal (p(z) is the sum (or mixture) of two unimodal ( 单单峰的峰的 ) densities (one for light one for dark region) densities (one for light one for dark regions)s)

1 1 2 2

1 2 1

p z P p z P p z

P P

1818

Optimal ThresholdingOptimal Thresholding

If the form of the If the form of the

densities is densities is

known or known or

assumed in assumed in

terms of terms of

minimum error minimum error

determining an determining an

optimal optimal

threshold for threshold for

segmenting the segmenting the

image is image is

possiblepossible

1 1 2 2

1 2 1

p z P p z P p z

P P

1919

Optimal ThresholdingOptimal Thresholding

Probability of erroneouslyProbability of erroneously

2020

Optimal ThresholdingOptimal Thresholding

Minimum errorMinimum error Differentiating ( Differentiating ( 微分微分 ) E(T) with respect to T (usi) E(T) with respect to T (usi

ng Leibnizrsquos rule) and equating the result to 0ng Leibnizrsquos rule) and equating the result to 0

find T which makesfind T which makes

2121

Optimal ThresholdingOptimal Thresholding

Minimum errorMinimum error

Specially if Specially if PP11 = P = P22 then the optimum then the optimum

threshold is where the curve pthreshold is where the curve p11(z) and (z) and

pp22(z) intersect(z) intersect

2222

Optimal ThresholdingOptimal Thresholding

For exampleFor example

Let Let PDF=Gaussian densityPDF=Gaussian density p p11(z) and (z) and

pp22(z)(z)

where μwhere μ11 and σ and σ1122 are the mean and are the mean and

variance of the Gaussian density of one variance of the Gaussian density of one

objectobject

μμ22 and σ and σ2222 are the mean and variance of are the mean and variance of

the Gaussian density of the other objectthe Gaussian density of the other object

2323

Optimal ThresholdingOptimal Thresholding

Quadratic equation (Quadratic equation (二次方程二次方程 ))

2424

Problems of ThresholdingProblems of Thresholding

Original imageOriginal image Thresholded imageThresholded image

2525

Problems of ThresholdingProblems of Thresholding

(a)(a) Exact threshold Exact threshold

segmentationsegmentation

(b)(b) Threshold too lowThreshold too low

(c)(c) Threshold too Threshold too

highhigh

2626

72 Point Detection72 Point Detection

a point has been detected at the a point has been detected at the

location on which the mark is location on which the mark is

centered ifcentered if

|R|geT|R|geT

where T is a nonnegative thresholdwhere T is a nonnegative threshold

R is the sum of products of the R is the sum of products of the

coefficients with the gray levels contained coefficients with the gray levels contained

in the region encompassed by the markin the region encompassed by the mark

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

2727

72 Point Detection72 Point Detection

Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

2828

ExampleExample

2929

73 Line Detection73 Line Detection

Horizontal mask will result with max Horizontal mask will result with max

response when a line passed through the response when a line passed through the

middle row of the mask with a constant middle row of the mask with a constant

backgroundbackground

the similar idea is used with other masksthe similar idea is used with other masks

Note the preferred direction of each mask Note the preferred direction of each mask

is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

than other possible directionsthan other possible directions

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

3030

Idea 1 of Line DetectionIdea 1 of Line Detection

Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

if at a certain point in the imageif at a certain point in the image

|Ri||Ri|gtgt|Rj||Rj|

for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

3131

Idea 2 of Line DetectionIdea 2 of Line Detection

Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

3232

ExampleExample

3333

74 Edge-based 74 Edge-based SegmentationSegmentation

Edge-based segmentations rely on edges found in an image by edge detecting operators

these edges mark image locations of discontinuities in gray level

Edge detection is the most common approach for detecting meaningful discontinuities in gray level

There are a large group of methods based on information about edges in the image

3434

What is edgeWhat is edge

Edge is where change occurs Change is measured by derivative in 1D

―Biggest change derivative has maximum magnitude

Or 2nd derivative is zero we discuss approaches for implementing

―first-order derivative (Gradient operator)

―second-order derivative (Laplacian operator)

―we have introduced both derivatives in chapter 3

―Here we will talk only about their properties for edge detection

3535

What is edgeWhat is edge

In other wordsIn other words an edge is a set of an edge is a set of

connected pixelsconnected pixels

that lie on the boundary between two that lie on the boundary between two

regions with relatively distinct gray-level regions with relatively distinct gray-level

propertiesproperties

Note edge vs boundaryNote edge vs boundary

―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

―whereas a region boundary owing to whereas a region boundary owing to

the way it is defined is a more global the way it is defined is a more global

ideaidea

3636

Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

because of because of

optics optics

sampling sampling

image image

acquisition acquisition

imperfectionimperfection

3737

Thick and Thin EdgeThick and Thin Edge

The slope of the ramp is inversely The slope of the ramp is inversely

proportional to the degree of blurring in the proportional to the degree of blurring in the

edgeedge

Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

pathpath

Instead an edge point now is any point Instead an edge point now is any point

contained in the ramp and contained in the ramp and an edge would an edge would

then be a set of such points that are then be a set of such points that are

connectedconnected

The thickness is determined by the length of the The thickness is determined by the length of the

rampramp

The length is determined by the slope which is in The length is determined by the slope which is in

turn determined by the degree of blurringturn determined by the degree of blurring

Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

edges tend to be thinedges tend to be thin

3838

First and Second derivatives (First and Second derivatives ( 导数导数 ))

the signs of the the signs of the

derivatives would be derivatives would be

reversed for an edge reversed for an edge

that transitions from that transitions from

light to darklight to dark

First First derivatderivatee

SeconSecond d derivatderivatee

Gray-Gray-level level profileprofile

3939

Second derivativesSecond derivatives

an undesirable featurean undesirable feature

produces 2 values for every edge in an produces 2 values for every edge in an

imageimage

zero-crossing propertyzero-crossing property

an imaginary straight line joining the an imaginary straight line joining the

extreme positive and negative values of extreme positive and negative values of

the second derivative would cross zero the second derivative would cross zero

near the midpoint of the edgenear the midpoint of the edge

quite useful for locating the centers of quite useful for locating the centers of

thick edgesthick edges

4040

Basic idea of edge detectionBasic idea of edge detection

A profile is defined perpendicularly to A profile is defined perpendicularly to

the edge direction and the results are the edge direction and the results are

interpretedinterpreted

The magnitude of the first derivative is The magnitude of the first derivative is

used to detect an edge (if a point is on a used to detect an edge (if a point is on a

ramp)ramp)

The sign of the second derivative can The sign of the second derivative can

determine whether an edge pixel is on the determine whether an edge pixel is on the

dark or light side of an edgedark or light side of an edge

4141

Review of First DerivateReview of First Derivate

Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

Sobel operators 3Sobel operators 333

6 5 8 5x yG z z G z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

9 5 8 6x yG z z G z z 1 0 0 1

0 1 1 0

1 0 0 1

0 1 1 0

7 8 9 1 2 3

3 6 9 1 4 7

2 2

2 2

x

y

G z z z z z z

G z z z z z z

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

x yf G G

4242

Edge direction and strengthEdge direction and strength

Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

2 2x yf G G

4343

Gradient MasksGradient Masks

1 0 0 1

0 1 1 0

Roberts

1 0 0 1

0 1 1 0

Roberts

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

4444

Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

88

ContentsContents

ThresholdingThresholding

Point DetectionPoint Detection

Line DetectionLine Detection

Edge-based SegmentationEdge-based Segmentation

Region-based SegmentationRegion-based Segmentation

99

71 Thresholding71 Thresholding

Thresholding is a labeling operation Thresholding is a labeling operation

on a gray scale image that on a gray scale image that

distinguishes pixels of a higher distinguishes pixels of a higher

intensity from pixels with a lower intensity from pixels with a lower

intensity valueintensity value

The output of thresholding usuallyThe output of thresholding usually is is a a

binary imagebinary image

This technique is particularly useful This technique is particularly useful

for scenes which contain for scenes which contain solid objects solid objects

on a uniform contrasting backgroundon a uniform contrasting background

1010

Classification of ThresholdingClassification of Thresholding

Thresholding can be viewed as an operation Thresholding can be viewed as an operation that involves tests against a function T of ththat involves tests against a function T of the forme form

where p(xy) denotes some local property of this where p(xy) denotes some local property of this pointpoint

T T x y p x y f x y

1111

Classification of ThresholdingClassification of Thresholding

When T depends onWhen T depends on only f(xy) only on gray-level valuesonly f(xy) only on gray-level values

1048638 1048638 Global Global thresholdingthresholding

both f(xy) and p(xy) on gray-level values and itboth f(xy) and p(xy) on gray-level values and its neighborss neighbors

Local Local thresholdingthresholding

x and y (in addition)x and y (in addition)

Dynamic Dynamic thresholdingthresholding

T T x y p x y f x y

1212

Basic Global ThresholdingBasic Global Thresholding

Original imageOriginal image HistogramHistogram

SolutionSolution use T midway between the max and use T midway between the max and

min gray levelsmin gray levels

SolutionSolution use T midway between the max and use T midway between the max and

min gray levelsmin gray levels

See ldquobasic_global_threSee ldquobasic_global_thremrdquomrdquo

1313

Basic Global ThresholdingBasic Global Thresholding

Let light objects in dark backgroundLet light objects in dark background

To extract the objectsTo extract the objects Select a ldquoTrdquo that separates the objects from thSelect a ldquoTrdquo that separates the objects from th

e backgrounde background

ie any (xy) for which f(xy)gtT is an object point ie any (xy) for which f(xy)gtT is an object point

A thresholded imageA thresholded image

1

0

if f x y T backgroundg x y

if f x y T foreground

1414

Heuristic Global ThresholdingHeuristic Global Thresholding

11 Select an initial estimate for TSelect an initial estimate for T

22 Segment the image using T This will produce two Segment the image using T This will produce two groups of pixels Ggroups of pixels G11 consisting of all pixels with gra consisting of all pixels with gray level values lt T and Gy level values lt T and G22 consisting of pixels with g consisting of pixels with gray level values ray level values T T

33 Compute the average gray level values μCompute the average gray level values μ11 and μ and μ22 fo for the pixels in regions Gr the pixels in regions G11 and G and G22

44 Compute a new threshold value T=05(μCompute a new threshold value T=05(μ11+μ+μ22))

55 Repeat steps 2 through 4 until the difference in T iRepeat steps 2 through 4 until the difference in T in successive iterations is smaller than a predefinen successive iterations is smaller than a predefined parameter Td parameter T00 seerdquohhellipmrdquo seerdquohhellipmrdquo

1515

Basic Adaptive ThresholdingBasic Adaptive Thresholding

subdivide original image into small areassubdivide original image into small areas

utilize a different threshold to segment each utilize a different threshold to segment each subimagessubimages

since the threshold used for each pixel depesince the threshold used for each pixel depends on the location of the pixel in terms of tnds on the location of the pixel in terms of the subimages this type of thresholding is ahe subimages this type of thresholding is adaptivedaptive

See ldquoAdaptive_thresholdmrdquoSee ldquoAdaptive_thresholdmrdquo

1616

Multilevel ThresholdingMultilevel Thresholding

a point (xy) belongs toa point (xy) belongs to an object class if Tan object class if T11 ltlt f(xy) le T f(xy) le T22

another object class if f(xy) another object class if f(xy) gtgt T T22

to background if f(xy) le Tto background if f(xy) le T11

1717

The histogram of an image containing two pThe histogram of an image containing two principal brightness regions can be considererincipal brightness regions can be considered an estimate of the brightness probability d an estimate of the brightness probability density function p(z)density function p(z) p(z) is the sum (or mixture) of two unimodal (p(z) is the sum (or mixture) of two unimodal ( 单单峰的峰的 ) densities (one for light one for dark region) densities (one for light one for dark regions)s)

1 1 2 2

1 2 1

p z P p z P p z

P P

1818

Optimal ThresholdingOptimal Thresholding

If the form of the If the form of the

densities is densities is

known or known or

assumed in assumed in

terms of terms of

minimum error minimum error

determining an determining an

optimal optimal

threshold for threshold for

segmenting the segmenting the

image is image is

possiblepossible

1 1 2 2

1 2 1

p z P p z P p z

P P

1919

Optimal ThresholdingOptimal Thresholding

Probability of erroneouslyProbability of erroneously

2020

Optimal ThresholdingOptimal Thresholding

Minimum errorMinimum error Differentiating ( Differentiating ( 微分微分 ) E(T) with respect to T (usi) E(T) with respect to T (usi

ng Leibnizrsquos rule) and equating the result to 0ng Leibnizrsquos rule) and equating the result to 0

find T which makesfind T which makes

2121

Optimal ThresholdingOptimal Thresholding

Minimum errorMinimum error

Specially if Specially if PP11 = P = P22 then the optimum then the optimum

threshold is where the curve pthreshold is where the curve p11(z) and (z) and

pp22(z) intersect(z) intersect

2222

Optimal ThresholdingOptimal Thresholding

For exampleFor example

Let Let PDF=Gaussian densityPDF=Gaussian density p p11(z) and (z) and

pp22(z)(z)

where μwhere μ11 and σ and σ1122 are the mean and are the mean and

variance of the Gaussian density of one variance of the Gaussian density of one

objectobject

μμ22 and σ and σ2222 are the mean and variance of are the mean and variance of

the Gaussian density of the other objectthe Gaussian density of the other object

2323

Optimal ThresholdingOptimal Thresholding

Quadratic equation (Quadratic equation (二次方程二次方程 ))

2424

Problems of ThresholdingProblems of Thresholding

Original imageOriginal image Thresholded imageThresholded image

2525

Problems of ThresholdingProblems of Thresholding

(a)(a) Exact threshold Exact threshold

segmentationsegmentation

(b)(b) Threshold too lowThreshold too low

(c)(c) Threshold too Threshold too

highhigh

2626

72 Point Detection72 Point Detection

a point has been detected at the a point has been detected at the

location on which the mark is location on which the mark is

centered ifcentered if

|R|geT|R|geT

where T is a nonnegative thresholdwhere T is a nonnegative threshold

R is the sum of products of the R is the sum of products of the

coefficients with the gray levels contained coefficients with the gray levels contained

in the region encompassed by the markin the region encompassed by the mark

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

2727

72 Point Detection72 Point Detection

Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

2828

ExampleExample

2929

73 Line Detection73 Line Detection

Horizontal mask will result with max Horizontal mask will result with max

response when a line passed through the response when a line passed through the

middle row of the mask with a constant middle row of the mask with a constant

backgroundbackground

the similar idea is used with other masksthe similar idea is used with other masks

Note the preferred direction of each mask Note the preferred direction of each mask

is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

than other possible directionsthan other possible directions

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

3030

Idea 1 of Line DetectionIdea 1 of Line Detection

Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

if at a certain point in the imageif at a certain point in the image

|Ri||Ri|gtgt|Rj||Rj|

for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

3131

Idea 2 of Line DetectionIdea 2 of Line Detection

Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

3232

ExampleExample

3333

74 Edge-based 74 Edge-based SegmentationSegmentation

Edge-based segmentations rely on edges found in an image by edge detecting operators

these edges mark image locations of discontinuities in gray level

Edge detection is the most common approach for detecting meaningful discontinuities in gray level

There are a large group of methods based on information about edges in the image

3434

What is edgeWhat is edge

Edge is where change occurs Change is measured by derivative in 1D

―Biggest change derivative has maximum magnitude

Or 2nd derivative is zero we discuss approaches for implementing

―first-order derivative (Gradient operator)

―second-order derivative (Laplacian operator)

―we have introduced both derivatives in chapter 3

―Here we will talk only about their properties for edge detection

3535

What is edgeWhat is edge

In other wordsIn other words an edge is a set of an edge is a set of

connected pixelsconnected pixels

that lie on the boundary between two that lie on the boundary between two

regions with relatively distinct gray-level regions with relatively distinct gray-level

propertiesproperties

Note edge vs boundaryNote edge vs boundary

―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

―whereas a region boundary owing to whereas a region boundary owing to

the way it is defined is a more global the way it is defined is a more global

ideaidea

3636

Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

because of because of

optics optics

sampling sampling

image image

acquisition acquisition

imperfectionimperfection

3737

Thick and Thin EdgeThick and Thin Edge

The slope of the ramp is inversely The slope of the ramp is inversely

proportional to the degree of blurring in the proportional to the degree of blurring in the

edgeedge

Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

pathpath

Instead an edge point now is any point Instead an edge point now is any point

contained in the ramp and contained in the ramp and an edge would an edge would

then be a set of such points that are then be a set of such points that are

connectedconnected

The thickness is determined by the length of the The thickness is determined by the length of the

rampramp

The length is determined by the slope which is in The length is determined by the slope which is in

turn determined by the degree of blurringturn determined by the degree of blurring

Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

edges tend to be thinedges tend to be thin

3838

First and Second derivatives (First and Second derivatives ( 导数导数 ))

the signs of the the signs of the

derivatives would be derivatives would be

reversed for an edge reversed for an edge

that transitions from that transitions from

light to darklight to dark

First First derivatderivatee

SeconSecond d derivatderivatee

Gray-Gray-level level profileprofile

3939

Second derivativesSecond derivatives

an undesirable featurean undesirable feature

produces 2 values for every edge in an produces 2 values for every edge in an

imageimage

zero-crossing propertyzero-crossing property

an imaginary straight line joining the an imaginary straight line joining the

extreme positive and negative values of extreme positive and negative values of

the second derivative would cross zero the second derivative would cross zero

near the midpoint of the edgenear the midpoint of the edge

quite useful for locating the centers of quite useful for locating the centers of

thick edgesthick edges

4040

Basic idea of edge detectionBasic idea of edge detection

A profile is defined perpendicularly to A profile is defined perpendicularly to

the edge direction and the results are the edge direction and the results are

interpretedinterpreted

The magnitude of the first derivative is The magnitude of the first derivative is

used to detect an edge (if a point is on a used to detect an edge (if a point is on a

ramp)ramp)

The sign of the second derivative can The sign of the second derivative can

determine whether an edge pixel is on the determine whether an edge pixel is on the

dark or light side of an edgedark or light side of an edge

4141

Review of First DerivateReview of First Derivate

Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

Sobel operators 3Sobel operators 333

6 5 8 5x yG z z G z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

9 5 8 6x yG z z G z z 1 0 0 1

0 1 1 0

1 0 0 1

0 1 1 0

7 8 9 1 2 3

3 6 9 1 4 7

2 2

2 2

x

y

G z z z z z z

G z z z z z z

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

x yf G G

4242

Edge direction and strengthEdge direction and strength

Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

2 2x yf G G

4343

Gradient MasksGradient Masks

1 0 0 1

0 1 1 0

Roberts

1 0 0 1

0 1 1 0

Roberts

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

4444

Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

99

71 Thresholding71 Thresholding

Thresholding is a labeling operation Thresholding is a labeling operation

on a gray scale image that on a gray scale image that

distinguishes pixels of a higher distinguishes pixels of a higher

intensity from pixels with a lower intensity from pixels with a lower

intensity valueintensity value

The output of thresholding usuallyThe output of thresholding usually is is a a

binary imagebinary image

This technique is particularly useful This technique is particularly useful

for scenes which contain for scenes which contain solid objects solid objects

on a uniform contrasting backgroundon a uniform contrasting background

1010

Classification of ThresholdingClassification of Thresholding

Thresholding can be viewed as an operation Thresholding can be viewed as an operation that involves tests against a function T of ththat involves tests against a function T of the forme form

where p(xy) denotes some local property of this where p(xy) denotes some local property of this pointpoint

T T x y p x y f x y

1111

Classification of ThresholdingClassification of Thresholding

When T depends onWhen T depends on only f(xy) only on gray-level valuesonly f(xy) only on gray-level values

1048638 1048638 Global Global thresholdingthresholding

both f(xy) and p(xy) on gray-level values and itboth f(xy) and p(xy) on gray-level values and its neighborss neighbors

Local Local thresholdingthresholding

x and y (in addition)x and y (in addition)

Dynamic Dynamic thresholdingthresholding

T T x y p x y f x y

1212

Basic Global ThresholdingBasic Global Thresholding

Original imageOriginal image HistogramHistogram

SolutionSolution use T midway between the max and use T midway between the max and

min gray levelsmin gray levels

SolutionSolution use T midway between the max and use T midway between the max and

min gray levelsmin gray levels

See ldquobasic_global_threSee ldquobasic_global_thremrdquomrdquo

1313

Basic Global ThresholdingBasic Global Thresholding

Let light objects in dark backgroundLet light objects in dark background

To extract the objectsTo extract the objects Select a ldquoTrdquo that separates the objects from thSelect a ldquoTrdquo that separates the objects from th

e backgrounde background

ie any (xy) for which f(xy)gtT is an object point ie any (xy) for which f(xy)gtT is an object point

A thresholded imageA thresholded image

1

0

if f x y T backgroundg x y

if f x y T foreground

1414

Heuristic Global ThresholdingHeuristic Global Thresholding

11 Select an initial estimate for TSelect an initial estimate for T

22 Segment the image using T This will produce two Segment the image using T This will produce two groups of pixels Ggroups of pixels G11 consisting of all pixels with gra consisting of all pixels with gray level values lt T and Gy level values lt T and G22 consisting of pixels with g consisting of pixels with gray level values ray level values T T

33 Compute the average gray level values μCompute the average gray level values μ11 and μ and μ22 fo for the pixels in regions Gr the pixels in regions G11 and G and G22

44 Compute a new threshold value T=05(μCompute a new threshold value T=05(μ11+μ+μ22))

55 Repeat steps 2 through 4 until the difference in T iRepeat steps 2 through 4 until the difference in T in successive iterations is smaller than a predefinen successive iterations is smaller than a predefined parameter Td parameter T00 seerdquohhellipmrdquo seerdquohhellipmrdquo

1515

Basic Adaptive ThresholdingBasic Adaptive Thresholding

subdivide original image into small areassubdivide original image into small areas

utilize a different threshold to segment each utilize a different threshold to segment each subimagessubimages

since the threshold used for each pixel depesince the threshold used for each pixel depends on the location of the pixel in terms of tnds on the location of the pixel in terms of the subimages this type of thresholding is ahe subimages this type of thresholding is adaptivedaptive

See ldquoAdaptive_thresholdmrdquoSee ldquoAdaptive_thresholdmrdquo

1616

Multilevel ThresholdingMultilevel Thresholding

a point (xy) belongs toa point (xy) belongs to an object class if Tan object class if T11 ltlt f(xy) le T f(xy) le T22

another object class if f(xy) another object class if f(xy) gtgt T T22

to background if f(xy) le Tto background if f(xy) le T11

1717

The histogram of an image containing two pThe histogram of an image containing two principal brightness regions can be considererincipal brightness regions can be considered an estimate of the brightness probability d an estimate of the brightness probability density function p(z)density function p(z) p(z) is the sum (or mixture) of two unimodal (p(z) is the sum (or mixture) of two unimodal ( 单单峰的峰的 ) densities (one for light one for dark region) densities (one for light one for dark regions)s)

1 1 2 2

1 2 1

p z P p z P p z

P P

1818

Optimal ThresholdingOptimal Thresholding

If the form of the If the form of the

densities is densities is

known or known or

assumed in assumed in

terms of terms of

minimum error minimum error

determining an determining an

optimal optimal

threshold for threshold for

segmenting the segmenting the

image is image is

possiblepossible

1 1 2 2

1 2 1

p z P p z P p z

P P

1919

Optimal ThresholdingOptimal Thresholding

Probability of erroneouslyProbability of erroneously

2020

Optimal ThresholdingOptimal Thresholding

Minimum errorMinimum error Differentiating ( Differentiating ( 微分微分 ) E(T) with respect to T (usi) E(T) with respect to T (usi

ng Leibnizrsquos rule) and equating the result to 0ng Leibnizrsquos rule) and equating the result to 0

find T which makesfind T which makes

2121

Optimal ThresholdingOptimal Thresholding

Minimum errorMinimum error

Specially if Specially if PP11 = P = P22 then the optimum then the optimum

threshold is where the curve pthreshold is where the curve p11(z) and (z) and

pp22(z) intersect(z) intersect

2222

Optimal ThresholdingOptimal Thresholding

For exampleFor example

Let Let PDF=Gaussian densityPDF=Gaussian density p p11(z) and (z) and

pp22(z)(z)

where μwhere μ11 and σ and σ1122 are the mean and are the mean and

variance of the Gaussian density of one variance of the Gaussian density of one

objectobject

μμ22 and σ and σ2222 are the mean and variance of are the mean and variance of

the Gaussian density of the other objectthe Gaussian density of the other object

2323

Optimal ThresholdingOptimal Thresholding

Quadratic equation (Quadratic equation (二次方程二次方程 ))

2424

Problems of ThresholdingProblems of Thresholding

Original imageOriginal image Thresholded imageThresholded image

2525

Problems of ThresholdingProblems of Thresholding

(a)(a) Exact threshold Exact threshold

segmentationsegmentation

(b)(b) Threshold too lowThreshold too low

(c)(c) Threshold too Threshold too

highhigh

2626

72 Point Detection72 Point Detection

a point has been detected at the a point has been detected at the

location on which the mark is location on which the mark is

centered ifcentered if

|R|geT|R|geT

where T is a nonnegative thresholdwhere T is a nonnegative threshold

R is the sum of products of the R is the sum of products of the

coefficients with the gray levels contained coefficients with the gray levels contained

in the region encompassed by the markin the region encompassed by the mark

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

2727

72 Point Detection72 Point Detection

Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

2828

ExampleExample

2929

73 Line Detection73 Line Detection

Horizontal mask will result with max Horizontal mask will result with max

response when a line passed through the response when a line passed through the

middle row of the mask with a constant middle row of the mask with a constant

backgroundbackground

the similar idea is used with other masksthe similar idea is used with other masks

Note the preferred direction of each mask Note the preferred direction of each mask

is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

than other possible directionsthan other possible directions

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

3030

Idea 1 of Line DetectionIdea 1 of Line Detection

Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

if at a certain point in the imageif at a certain point in the image

|Ri||Ri|gtgt|Rj||Rj|

for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

3131

Idea 2 of Line DetectionIdea 2 of Line Detection

Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

3232

ExampleExample

3333

74 Edge-based 74 Edge-based SegmentationSegmentation

Edge-based segmentations rely on edges found in an image by edge detecting operators

these edges mark image locations of discontinuities in gray level

Edge detection is the most common approach for detecting meaningful discontinuities in gray level

There are a large group of methods based on information about edges in the image

3434

What is edgeWhat is edge

Edge is where change occurs Change is measured by derivative in 1D

―Biggest change derivative has maximum magnitude

Or 2nd derivative is zero we discuss approaches for implementing

―first-order derivative (Gradient operator)

―second-order derivative (Laplacian operator)

―we have introduced both derivatives in chapter 3

―Here we will talk only about their properties for edge detection

3535

What is edgeWhat is edge

In other wordsIn other words an edge is a set of an edge is a set of

connected pixelsconnected pixels

that lie on the boundary between two that lie on the boundary between two

regions with relatively distinct gray-level regions with relatively distinct gray-level

propertiesproperties

Note edge vs boundaryNote edge vs boundary

―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

―whereas a region boundary owing to whereas a region boundary owing to

the way it is defined is a more global the way it is defined is a more global

ideaidea

3636

Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

because of because of

optics optics

sampling sampling

image image

acquisition acquisition

imperfectionimperfection

3737

Thick and Thin EdgeThick and Thin Edge

The slope of the ramp is inversely The slope of the ramp is inversely

proportional to the degree of blurring in the proportional to the degree of blurring in the

edgeedge

Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

pathpath

Instead an edge point now is any point Instead an edge point now is any point

contained in the ramp and contained in the ramp and an edge would an edge would

then be a set of such points that are then be a set of such points that are

connectedconnected

The thickness is determined by the length of the The thickness is determined by the length of the

rampramp

The length is determined by the slope which is in The length is determined by the slope which is in

turn determined by the degree of blurringturn determined by the degree of blurring

Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

edges tend to be thinedges tend to be thin

3838

First and Second derivatives (First and Second derivatives ( 导数导数 ))

the signs of the the signs of the

derivatives would be derivatives would be

reversed for an edge reversed for an edge

that transitions from that transitions from

light to darklight to dark

First First derivatderivatee

SeconSecond d derivatderivatee

Gray-Gray-level level profileprofile

3939

Second derivativesSecond derivatives

an undesirable featurean undesirable feature

produces 2 values for every edge in an produces 2 values for every edge in an

imageimage

zero-crossing propertyzero-crossing property

an imaginary straight line joining the an imaginary straight line joining the

extreme positive and negative values of extreme positive and negative values of

the second derivative would cross zero the second derivative would cross zero

near the midpoint of the edgenear the midpoint of the edge

quite useful for locating the centers of quite useful for locating the centers of

thick edgesthick edges

4040

Basic idea of edge detectionBasic idea of edge detection

A profile is defined perpendicularly to A profile is defined perpendicularly to

the edge direction and the results are the edge direction and the results are

interpretedinterpreted

The magnitude of the first derivative is The magnitude of the first derivative is

used to detect an edge (if a point is on a used to detect an edge (if a point is on a

ramp)ramp)

The sign of the second derivative can The sign of the second derivative can

determine whether an edge pixel is on the determine whether an edge pixel is on the

dark or light side of an edgedark or light side of an edge

4141

Review of First DerivateReview of First Derivate

Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

Sobel operators 3Sobel operators 333

6 5 8 5x yG z z G z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

9 5 8 6x yG z z G z z 1 0 0 1

0 1 1 0

1 0 0 1

0 1 1 0

7 8 9 1 2 3

3 6 9 1 4 7

2 2

2 2

x

y

G z z z z z z

G z z z z z z

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

x yf G G

4242

Edge direction and strengthEdge direction and strength

Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

2 2x yf G G

4343

Gradient MasksGradient Masks

1 0 0 1

0 1 1 0

Roberts

1 0 0 1

0 1 1 0

Roberts

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

4444

Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

1010

Classification of ThresholdingClassification of Thresholding

Thresholding can be viewed as an operation Thresholding can be viewed as an operation that involves tests against a function T of ththat involves tests against a function T of the forme form

where p(xy) denotes some local property of this where p(xy) denotes some local property of this pointpoint

T T x y p x y f x y

1111

Classification of ThresholdingClassification of Thresholding

When T depends onWhen T depends on only f(xy) only on gray-level valuesonly f(xy) only on gray-level values

1048638 1048638 Global Global thresholdingthresholding

both f(xy) and p(xy) on gray-level values and itboth f(xy) and p(xy) on gray-level values and its neighborss neighbors

Local Local thresholdingthresholding

x and y (in addition)x and y (in addition)

Dynamic Dynamic thresholdingthresholding

T T x y p x y f x y

1212

Basic Global ThresholdingBasic Global Thresholding

Original imageOriginal image HistogramHistogram

SolutionSolution use T midway between the max and use T midway between the max and

min gray levelsmin gray levels

SolutionSolution use T midway between the max and use T midway between the max and

min gray levelsmin gray levels

See ldquobasic_global_threSee ldquobasic_global_thremrdquomrdquo

1313

Basic Global ThresholdingBasic Global Thresholding

Let light objects in dark backgroundLet light objects in dark background

To extract the objectsTo extract the objects Select a ldquoTrdquo that separates the objects from thSelect a ldquoTrdquo that separates the objects from th

e backgrounde background

ie any (xy) for which f(xy)gtT is an object point ie any (xy) for which f(xy)gtT is an object point

A thresholded imageA thresholded image

1

0

if f x y T backgroundg x y

if f x y T foreground

1414

Heuristic Global ThresholdingHeuristic Global Thresholding

11 Select an initial estimate for TSelect an initial estimate for T

22 Segment the image using T This will produce two Segment the image using T This will produce two groups of pixels Ggroups of pixels G11 consisting of all pixels with gra consisting of all pixels with gray level values lt T and Gy level values lt T and G22 consisting of pixels with g consisting of pixels with gray level values ray level values T T

33 Compute the average gray level values μCompute the average gray level values μ11 and μ and μ22 fo for the pixels in regions Gr the pixels in regions G11 and G and G22

44 Compute a new threshold value T=05(μCompute a new threshold value T=05(μ11+μ+μ22))

55 Repeat steps 2 through 4 until the difference in T iRepeat steps 2 through 4 until the difference in T in successive iterations is smaller than a predefinen successive iterations is smaller than a predefined parameter Td parameter T00 seerdquohhellipmrdquo seerdquohhellipmrdquo

1515

Basic Adaptive ThresholdingBasic Adaptive Thresholding

subdivide original image into small areassubdivide original image into small areas

utilize a different threshold to segment each utilize a different threshold to segment each subimagessubimages

since the threshold used for each pixel depesince the threshold used for each pixel depends on the location of the pixel in terms of tnds on the location of the pixel in terms of the subimages this type of thresholding is ahe subimages this type of thresholding is adaptivedaptive

See ldquoAdaptive_thresholdmrdquoSee ldquoAdaptive_thresholdmrdquo

1616

Multilevel ThresholdingMultilevel Thresholding

a point (xy) belongs toa point (xy) belongs to an object class if Tan object class if T11 ltlt f(xy) le T f(xy) le T22

another object class if f(xy) another object class if f(xy) gtgt T T22

to background if f(xy) le Tto background if f(xy) le T11

1717

The histogram of an image containing two pThe histogram of an image containing two principal brightness regions can be considererincipal brightness regions can be considered an estimate of the brightness probability d an estimate of the brightness probability density function p(z)density function p(z) p(z) is the sum (or mixture) of two unimodal (p(z) is the sum (or mixture) of two unimodal ( 单单峰的峰的 ) densities (one for light one for dark region) densities (one for light one for dark regions)s)

1 1 2 2

1 2 1

p z P p z P p z

P P

1818

Optimal ThresholdingOptimal Thresholding

If the form of the If the form of the

densities is densities is

known or known or

assumed in assumed in

terms of terms of

minimum error minimum error

determining an determining an

optimal optimal

threshold for threshold for

segmenting the segmenting the

image is image is

possiblepossible

1 1 2 2

1 2 1

p z P p z P p z

P P

1919

Optimal ThresholdingOptimal Thresholding

Probability of erroneouslyProbability of erroneously

2020

Optimal ThresholdingOptimal Thresholding

Minimum errorMinimum error Differentiating ( Differentiating ( 微分微分 ) E(T) with respect to T (usi) E(T) with respect to T (usi

ng Leibnizrsquos rule) and equating the result to 0ng Leibnizrsquos rule) and equating the result to 0

find T which makesfind T which makes

2121

Optimal ThresholdingOptimal Thresholding

Minimum errorMinimum error

Specially if Specially if PP11 = P = P22 then the optimum then the optimum

threshold is where the curve pthreshold is where the curve p11(z) and (z) and

pp22(z) intersect(z) intersect

2222

Optimal ThresholdingOptimal Thresholding

For exampleFor example

Let Let PDF=Gaussian densityPDF=Gaussian density p p11(z) and (z) and

pp22(z)(z)

where μwhere μ11 and σ and σ1122 are the mean and are the mean and

variance of the Gaussian density of one variance of the Gaussian density of one

objectobject

μμ22 and σ and σ2222 are the mean and variance of are the mean and variance of

the Gaussian density of the other objectthe Gaussian density of the other object

2323

Optimal ThresholdingOptimal Thresholding

Quadratic equation (Quadratic equation (二次方程二次方程 ))

2424

Problems of ThresholdingProblems of Thresholding

Original imageOriginal image Thresholded imageThresholded image

2525

Problems of ThresholdingProblems of Thresholding

(a)(a) Exact threshold Exact threshold

segmentationsegmentation

(b)(b) Threshold too lowThreshold too low

(c)(c) Threshold too Threshold too

highhigh

2626

72 Point Detection72 Point Detection

a point has been detected at the a point has been detected at the

location on which the mark is location on which the mark is

centered ifcentered if

|R|geT|R|geT

where T is a nonnegative thresholdwhere T is a nonnegative threshold

R is the sum of products of the R is the sum of products of the

coefficients with the gray levels contained coefficients with the gray levels contained

in the region encompassed by the markin the region encompassed by the mark

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

2727

72 Point Detection72 Point Detection

Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

2828

ExampleExample

2929

73 Line Detection73 Line Detection

Horizontal mask will result with max Horizontal mask will result with max

response when a line passed through the response when a line passed through the

middle row of the mask with a constant middle row of the mask with a constant

backgroundbackground

the similar idea is used with other masksthe similar idea is used with other masks

Note the preferred direction of each mask Note the preferred direction of each mask

is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

than other possible directionsthan other possible directions

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

3030

Idea 1 of Line DetectionIdea 1 of Line Detection

Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

if at a certain point in the imageif at a certain point in the image

|Ri||Ri|gtgt|Rj||Rj|

for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

3131

Idea 2 of Line DetectionIdea 2 of Line Detection

Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

3232

ExampleExample

3333

74 Edge-based 74 Edge-based SegmentationSegmentation

Edge-based segmentations rely on edges found in an image by edge detecting operators

these edges mark image locations of discontinuities in gray level

Edge detection is the most common approach for detecting meaningful discontinuities in gray level

There are a large group of methods based on information about edges in the image

3434

What is edgeWhat is edge

Edge is where change occurs Change is measured by derivative in 1D

―Biggest change derivative has maximum magnitude

Or 2nd derivative is zero we discuss approaches for implementing

―first-order derivative (Gradient operator)

―second-order derivative (Laplacian operator)

―we have introduced both derivatives in chapter 3

―Here we will talk only about their properties for edge detection

3535

What is edgeWhat is edge

In other wordsIn other words an edge is a set of an edge is a set of

connected pixelsconnected pixels

that lie on the boundary between two that lie on the boundary between two

regions with relatively distinct gray-level regions with relatively distinct gray-level

propertiesproperties

Note edge vs boundaryNote edge vs boundary

―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

―whereas a region boundary owing to whereas a region boundary owing to

the way it is defined is a more global the way it is defined is a more global

ideaidea

3636

Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

because of because of

optics optics

sampling sampling

image image

acquisition acquisition

imperfectionimperfection

3737

Thick and Thin EdgeThick and Thin Edge

The slope of the ramp is inversely The slope of the ramp is inversely

proportional to the degree of blurring in the proportional to the degree of blurring in the

edgeedge

Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

pathpath

Instead an edge point now is any point Instead an edge point now is any point

contained in the ramp and contained in the ramp and an edge would an edge would

then be a set of such points that are then be a set of such points that are

connectedconnected

The thickness is determined by the length of the The thickness is determined by the length of the

rampramp

The length is determined by the slope which is in The length is determined by the slope which is in

turn determined by the degree of blurringturn determined by the degree of blurring

Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

edges tend to be thinedges tend to be thin

3838

First and Second derivatives (First and Second derivatives ( 导数导数 ))

the signs of the the signs of the

derivatives would be derivatives would be

reversed for an edge reversed for an edge

that transitions from that transitions from

light to darklight to dark

First First derivatderivatee

SeconSecond d derivatderivatee

Gray-Gray-level level profileprofile

3939

Second derivativesSecond derivatives

an undesirable featurean undesirable feature

produces 2 values for every edge in an produces 2 values for every edge in an

imageimage

zero-crossing propertyzero-crossing property

an imaginary straight line joining the an imaginary straight line joining the

extreme positive and negative values of extreme positive and negative values of

the second derivative would cross zero the second derivative would cross zero

near the midpoint of the edgenear the midpoint of the edge

quite useful for locating the centers of quite useful for locating the centers of

thick edgesthick edges

4040

Basic idea of edge detectionBasic idea of edge detection

A profile is defined perpendicularly to A profile is defined perpendicularly to

the edge direction and the results are the edge direction and the results are

interpretedinterpreted

The magnitude of the first derivative is The magnitude of the first derivative is

used to detect an edge (if a point is on a used to detect an edge (if a point is on a

ramp)ramp)

The sign of the second derivative can The sign of the second derivative can

determine whether an edge pixel is on the determine whether an edge pixel is on the

dark or light side of an edgedark or light side of an edge

4141

Review of First DerivateReview of First Derivate

Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

Sobel operators 3Sobel operators 333

6 5 8 5x yG z z G z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

9 5 8 6x yG z z G z z 1 0 0 1

0 1 1 0

1 0 0 1

0 1 1 0

7 8 9 1 2 3

3 6 9 1 4 7

2 2

2 2

x

y

G z z z z z z

G z z z z z z

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

x yf G G

4242

Edge direction and strengthEdge direction and strength

Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

2 2x yf G G

4343

Gradient MasksGradient Masks

1 0 0 1

0 1 1 0

Roberts

1 0 0 1

0 1 1 0

Roberts

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

4444

Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

1111

Classification of ThresholdingClassification of Thresholding

When T depends onWhen T depends on only f(xy) only on gray-level valuesonly f(xy) only on gray-level values

1048638 1048638 Global Global thresholdingthresholding

both f(xy) and p(xy) on gray-level values and itboth f(xy) and p(xy) on gray-level values and its neighborss neighbors

Local Local thresholdingthresholding

x and y (in addition)x and y (in addition)

Dynamic Dynamic thresholdingthresholding

T T x y p x y f x y

1212

Basic Global ThresholdingBasic Global Thresholding

Original imageOriginal image HistogramHistogram

SolutionSolution use T midway between the max and use T midway between the max and

min gray levelsmin gray levels

SolutionSolution use T midway between the max and use T midway between the max and

min gray levelsmin gray levels

See ldquobasic_global_threSee ldquobasic_global_thremrdquomrdquo

1313

Basic Global ThresholdingBasic Global Thresholding

Let light objects in dark backgroundLet light objects in dark background

To extract the objectsTo extract the objects Select a ldquoTrdquo that separates the objects from thSelect a ldquoTrdquo that separates the objects from th

e backgrounde background

ie any (xy) for which f(xy)gtT is an object point ie any (xy) for which f(xy)gtT is an object point

A thresholded imageA thresholded image

1

0

if f x y T backgroundg x y

if f x y T foreground

1414

Heuristic Global ThresholdingHeuristic Global Thresholding

11 Select an initial estimate for TSelect an initial estimate for T

22 Segment the image using T This will produce two Segment the image using T This will produce two groups of pixels Ggroups of pixels G11 consisting of all pixels with gra consisting of all pixels with gray level values lt T and Gy level values lt T and G22 consisting of pixels with g consisting of pixels with gray level values ray level values T T

33 Compute the average gray level values μCompute the average gray level values μ11 and μ and μ22 fo for the pixels in regions Gr the pixels in regions G11 and G and G22

44 Compute a new threshold value T=05(μCompute a new threshold value T=05(μ11+μ+μ22))

55 Repeat steps 2 through 4 until the difference in T iRepeat steps 2 through 4 until the difference in T in successive iterations is smaller than a predefinen successive iterations is smaller than a predefined parameter Td parameter T00 seerdquohhellipmrdquo seerdquohhellipmrdquo

1515

Basic Adaptive ThresholdingBasic Adaptive Thresholding

subdivide original image into small areassubdivide original image into small areas

utilize a different threshold to segment each utilize a different threshold to segment each subimagessubimages

since the threshold used for each pixel depesince the threshold used for each pixel depends on the location of the pixel in terms of tnds on the location of the pixel in terms of the subimages this type of thresholding is ahe subimages this type of thresholding is adaptivedaptive

See ldquoAdaptive_thresholdmrdquoSee ldquoAdaptive_thresholdmrdquo

1616

Multilevel ThresholdingMultilevel Thresholding

a point (xy) belongs toa point (xy) belongs to an object class if Tan object class if T11 ltlt f(xy) le T f(xy) le T22

another object class if f(xy) another object class if f(xy) gtgt T T22

to background if f(xy) le Tto background if f(xy) le T11

1717

The histogram of an image containing two pThe histogram of an image containing two principal brightness regions can be considererincipal brightness regions can be considered an estimate of the brightness probability d an estimate of the brightness probability density function p(z)density function p(z) p(z) is the sum (or mixture) of two unimodal (p(z) is the sum (or mixture) of two unimodal ( 单单峰的峰的 ) densities (one for light one for dark region) densities (one for light one for dark regions)s)

1 1 2 2

1 2 1

p z P p z P p z

P P

1818

Optimal ThresholdingOptimal Thresholding

If the form of the If the form of the

densities is densities is

known or known or

assumed in assumed in

terms of terms of

minimum error minimum error

determining an determining an

optimal optimal

threshold for threshold for

segmenting the segmenting the

image is image is

possiblepossible

1 1 2 2

1 2 1

p z P p z P p z

P P

1919

Optimal ThresholdingOptimal Thresholding

Probability of erroneouslyProbability of erroneously

2020

Optimal ThresholdingOptimal Thresholding

Minimum errorMinimum error Differentiating ( Differentiating ( 微分微分 ) E(T) with respect to T (usi) E(T) with respect to T (usi

ng Leibnizrsquos rule) and equating the result to 0ng Leibnizrsquos rule) and equating the result to 0

find T which makesfind T which makes

2121

Optimal ThresholdingOptimal Thresholding

Minimum errorMinimum error

Specially if Specially if PP11 = P = P22 then the optimum then the optimum

threshold is where the curve pthreshold is where the curve p11(z) and (z) and

pp22(z) intersect(z) intersect

2222

Optimal ThresholdingOptimal Thresholding

For exampleFor example

Let Let PDF=Gaussian densityPDF=Gaussian density p p11(z) and (z) and

pp22(z)(z)

where μwhere μ11 and σ and σ1122 are the mean and are the mean and

variance of the Gaussian density of one variance of the Gaussian density of one

objectobject

μμ22 and σ and σ2222 are the mean and variance of are the mean and variance of

the Gaussian density of the other objectthe Gaussian density of the other object

2323

Optimal ThresholdingOptimal Thresholding

Quadratic equation (Quadratic equation (二次方程二次方程 ))

2424

Problems of ThresholdingProblems of Thresholding

Original imageOriginal image Thresholded imageThresholded image

2525

Problems of ThresholdingProblems of Thresholding

(a)(a) Exact threshold Exact threshold

segmentationsegmentation

(b)(b) Threshold too lowThreshold too low

(c)(c) Threshold too Threshold too

highhigh

2626

72 Point Detection72 Point Detection

a point has been detected at the a point has been detected at the

location on which the mark is location on which the mark is

centered ifcentered if

|R|geT|R|geT

where T is a nonnegative thresholdwhere T is a nonnegative threshold

R is the sum of products of the R is the sum of products of the

coefficients with the gray levels contained coefficients with the gray levels contained

in the region encompassed by the markin the region encompassed by the mark

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

2727

72 Point Detection72 Point Detection

Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

2828

ExampleExample

2929

73 Line Detection73 Line Detection

Horizontal mask will result with max Horizontal mask will result with max

response when a line passed through the response when a line passed through the

middle row of the mask with a constant middle row of the mask with a constant

backgroundbackground

the similar idea is used with other masksthe similar idea is used with other masks

Note the preferred direction of each mask Note the preferred direction of each mask

is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

than other possible directionsthan other possible directions

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

3030

Idea 1 of Line DetectionIdea 1 of Line Detection

Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

if at a certain point in the imageif at a certain point in the image

|Ri||Ri|gtgt|Rj||Rj|

for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

3131

Idea 2 of Line DetectionIdea 2 of Line Detection

Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

3232

ExampleExample

3333

74 Edge-based 74 Edge-based SegmentationSegmentation

Edge-based segmentations rely on edges found in an image by edge detecting operators

these edges mark image locations of discontinuities in gray level

Edge detection is the most common approach for detecting meaningful discontinuities in gray level

There are a large group of methods based on information about edges in the image

3434

What is edgeWhat is edge

Edge is where change occurs Change is measured by derivative in 1D

―Biggest change derivative has maximum magnitude

Or 2nd derivative is zero we discuss approaches for implementing

―first-order derivative (Gradient operator)

―second-order derivative (Laplacian operator)

―we have introduced both derivatives in chapter 3

―Here we will talk only about their properties for edge detection

3535

What is edgeWhat is edge

In other wordsIn other words an edge is a set of an edge is a set of

connected pixelsconnected pixels

that lie on the boundary between two that lie on the boundary between two

regions with relatively distinct gray-level regions with relatively distinct gray-level

propertiesproperties

Note edge vs boundaryNote edge vs boundary

―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

―whereas a region boundary owing to whereas a region boundary owing to

the way it is defined is a more global the way it is defined is a more global

ideaidea

3636

Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

because of because of

optics optics

sampling sampling

image image

acquisition acquisition

imperfectionimperfection

3737

Thick and Thin EdgeThick and Thin Edge

The slope of the ramp is inversely The slope of the ramp is inversely

proportional to the degree of blurring in the proportional to the degree of blurring in the

edgeedge

Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

pathpath

Instead an edge point now is any point Instead an edge point now is any point

contained in the ramp and contained in the ramp and an edge would an edge would

then be a set of such points that are then be a set of such points that are

connectedconnected

The thickness is determined by the length of the The thickness is determined by the length of the

rampramp

The length is determined by the slope which is in The length is determined by the slope which is in

turn determined by the degree of blurringturn determined by the degree of blurring

Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

edges tend to be thinedges tend to be thin

3838

First and Second derivatives (First and Second derivatives ( 导数导数 ))

the signs of the the signs of the

derivatives would be derivatives would be

reversed for an edge reversed for an edge

that transitions from that transitions from

light to darklight to dark

First First derivatderivatee

SeconSecond d derivatderivatee

Gray-Gray-level level profileprofile

3939

Second derivativesSecond derivatives

an undesirable featurean undesirable feature

produces 2 values for every edge in an produces 2 values for every edge in an

imageimage

zero-crossing propertyzero-crossing property

an imaginary straight line joining the an imaginary straight line joining the

extreme positive and negative values of extreme positive and negative values of

the second derivative would cross zero the second derivative would cross zero

near the midpoint of the edgenear the midpoint of the edge

quite useful for locating the centers of quite useful for locating the centers of

thick edgesthick edges

4040

Basic idea of edge detectionBasic idea of edge detection

A profile is defined perpendicularly to A profile is defined perpendicularly to

the edge direction and the results are the edge direction and the results are

interpretedinterpreted

The magnitude of the first derivative is The magnitude of the first derivative is

used to detect an edge (if a point is on a used to detect an edge (if a point is on a

ramp)ramp)

The sign of the second derivative can The sign of the second derivative can

determine whether an edge pixel is on the determine whether an edge pixel is on the

dark or light side of an edgedark or light side of an edge

4141

Review of First DerivateReview of First Derivate

Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

Sobel operators 3Sobel operators 333

6 5 8 5x yG z z G z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

9 5 8 6x yG z z G z z 1 0 0 1

0 1 1 0

1 0 0 1

0 1 1 0

7 8 9 1 2 3

3 6 9 1 4 7

2 2

2 2

x

y

G z z z z z z

G z z z z z z

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

x yf G G

4242

Edge direction and strengthEdge direction and strength

Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

2 2x yf G G

4343

Gradient MasksGradient Masks

1 0 0 1

0 1 1 0

Roberts

1 0 0 1

0 1 1 0

Roberts

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

4444

Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

1212

Basic Global ThresholdingBasic Global Thresholding

Original imageOriginal image HistogramHistogram

SolutionSolution use T midway between the max and use T midway between the max and

min gray levelsmin gray levels

SolutionSolution use T midway between the max and use T midway between the max and

min gray levelsmin gray levels

See ldquobasic_global_threSee ldquobasic_global_thremrdquomrdquo

1313

Basic Global ThresholdingBasic Global Thresholding

Let light objects in dark backgroundLet light objects in dark background

To extract the objectsTo extract the objects Select a ldquoTrdquo that separates the objects from thSelect a ldquoTrdquo that separates the objects from th

e backgrounde background

ie any (xy) for which f(xy)gtT is an object point ie any (xy) for which f(xy)gtT is an object point

A thresholded imageA thresholded image

1

0

if f x y T backgroundg x y

if f x y T foreground

1414

Heuristic Global ThresholdingHeuristic Global Thresholding

11 Select an initial estimate for TSelect an initial estimate for T

22 Segment the image using T This will produce two Segment the image using T This will produce two groups of pixels Ggroups of pixels G11 consisting of all pixels with gra consisting of all pixels with gray level values lt T and Gy level values lt T and G22 consisting of pixels with g consisting of pixels with gray level values ray level values T T

33 Compute the average gray level values μCompute the average gray level values μ11 and μ and μ22 fo for the pixels in regions Gr the pixels in regions G11 and G and G22

44 Compute a new threshold value T=05(μCompute a new threshold value T=05(μ11+μ+μ22))

55 Repeat steps 2 through 4 until the difference in T iRepeat steps 2 through 4 until the difference in T in successive iterations is smaller than a predefinen successive iterations is smaller than a predefined parameter Td parameter T00 seerdquohhellipmrdquo seerdquohhellipmrdquo

1515

Basic Adaptive ThresholdingBasic Adaptive Thresholding

subdivide original image into small areassubdivide original image into small areas

utilize a different threshold to segment each utilize a different threshold to segment each subimagessubimages

since the threshold used for each pixel depesince the threshold used for each pixel depends on the location of the pixel in terms of tnds on the location of the pixel in terms of the subimages this type of thresholding is ahe subimages this type of thresholding is adaptivedaptive

See ldquoAdaptive_thresholdmrdquoSee ldquoAdaptive_thresholdmrdquo

1616

Multilevel ThresholdingMultilevel Thresholding

a point (xy) belongs toa point (xy) belongs to an object class if Tan object class if T11 ltlt f(xy) le T f(xy) le T22

another object class if f(xy) another object class if f(xy) gtgt T T22

to background if f(xy) le Tto background if f(xy) le T11

1717

The histogram of an image containing two pThe histogram of an image containing two principal brightness regions can be considererincipal brightness regions can be considered an estimate of the brightness probability d an estimate of the brightness probability density function p(z)density function p(z) p(z) is the sum (or mixture) of two unimodal (p(z) is the sum (or mixture) of two unimodal ( 单单峰的峰的 ) densities (one for light one for dark region) densities (one for light one for dark regions)s)

1 1 2 2

1 2 1

p z P p z P p z

P P

1818

Optimal ThresholdingOptimal Thresholding

If the form of the If the form of the

densities is densities is

known or known or

assumed in assumed in

terms of terms of

minimum error minimum error

determining an determining an

optimal optimal

threshold for threshold for

segmenting the segmenting the

image is image is

possiblepossible

1 1 2 2

1 2 1

p z P p z P p z

P P

1919

Optimal ThresholdingOptimal Thresholding

Probability of erroneouslyProbability of erroneously

2020

Optimal ThresholdingOptimal Thresholding

Minimum errorMinimum error Differentiating ( Differentiating ( 微分微分 ) E(T) with respect to T (usi) E(T) with respect to T (usi

ng Leibnizrsquos rule) and equating the result to 0ng Leibnizrsquos rule) and equating the result to 0

find T which makesfind T which makes

2121

Optimal ThresholdingOptimal Thresholding

Minimum errorMinimum error

Specially if Specially if PP11 = P = P22 then the optimum then the optimum

threshold is where the curve pthreshold is where the curve p11(z) and (z) and

pp22(z) intersect(z) intersect

2222

Optimal ThresholdingOptimal Thresholding

For exampleFor example

Let Let PDF=Gaussian densityPDF=Gaussian density p p11(z) and (z) and

pp22(z)(z)

where μwhere μ11 and σ and σ1122 are the mean and are the mean and

variance of the Gaussian density of one variance of the Gaussian density of one

objectobject

μμ22 and σ and σ2222 are the mean and variance of are the mean and variance of

the Gaussian density of the other objectthe Gaussian density of the other object

2323

Optimal ThresholdingOptimal Thresholding

Quadratic equation (Quadratic equation (二次方程二次方程 ))

2424

Problems of ThresholdingProblems of Thresholding

Original imageOriginal image Thresholded imageThresholded image

2525

Problems of ThresholdingProblems of Thresholding

(a)(a) Exact threshold Exact threshold

segmentationsegmentation

(b)(b) Threshold too lowThreshold too low

(c)(c) Threshold too Threshold too

highhigh

2626

72 Point Detection72 Point Detection

a point has been detected at the a point has been detected at the

location on which the mark is location on which the mark is

centered ifcentered if

|R|geT|R|geT

where T is a nonnegative thresholdwhere T is a nonnegative threshold

R is the sum of products of the R is the sum of products of the

coefficients with the gray levels contained coefficients with the gray levels contained

in the region encompassed by the markin the region encompassed by the mark

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

2727

72 Point Detection72 Point Detection

Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

2828

ExampleExample

2929

73 Line Detection73 Line Detection

Horizontal mask will result with max Horizontal mask will result with max

response when a line passed through the response when a line passed through the

middle row of the mask with a constant middle row of the mask with a constant

backgroundbackground

the similar idea is used with other masksthe similar idea is used with other masks

Note the preferred direction of each mask Note the preferred direction of each mask

is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

than other possible directionsthan other possible directions

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

3030

Idea 1 of Line DetectionIdea 1 of Line Detection

Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

if at a certain point in the imageif at a certain point in the image

|Ri||Ri|gtgt|Rj||Rj|

for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

3131

Idea 2 of Line DetectionIdea 2 of Line Detection

Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

3232

ExampleExample

3333

74 Edge-based 74 Edge-based SegmentationSegmentation

Edge-based segmentations rely on edges found in an image by edge detecting operators

these edges mark image locations of discontinuities in gray level

Edge detection is the most common approach for detecting meaningful discontinuities in gray level

There are a large group of methods based on information about edges in the image

3434

What is edgeWhat is edge

Edge is where change occurs Change is measured by derivative in 1D

―Biggest change derivative has maximum magnitude

Or 2nd derivative is zero we discuss approaches for implementing

―first-order derivative (Gradient operator)

―second-order derivative (Laplacian operator)

―we have introduced both derivatives in chapter 3

―Here we will talk only about their properties for edge detection

3535

What is edgeWhat is edge

In other wordsIn other words an edge is a set of an edge is a set of

connected pixelsconnected pixels

that lie on the boundary between two that lie on the boundary between two

regions with relatively distinct gray-level regions with relatively distinct gray-level

propertiesproperties

Note edge vs boundaryNote edge vs boundary

―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

―whereas a region boundary owing to whereas a region boundary owing to

the way it is defined is a more global the way it is defined is a more global

ideaidea

3636

Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

because of because of

optics optics

sampling sampling

image image

acquisition acquisition

imperfectionimperfection

3737

Thick and Thin EdgeThick and Thin Edge

The slope of the ramp is inversely The slope of the ramp is inversely

proportional to the degree of blurring in the proportional to the degree of blurring in the

edgeedge

Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

pathpath

Instead an edge point now is any point Instead an edge point now is any point

contained in the ramp and contained in the ramp and an edge would an edge would

then be a set of such points that are then be a set of such points that are

connectedconnected

The thickness is determined by the length of the The thickness is determined by the length of the

rampramp

The length is determined by the slope which is in The length is determined by the slope which is in

turn determined by the degree of blurringturn determined by the degree of blurring

Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

edges tend to be thinedges tend to be thin

3838

First and Second derivatives (First and Second derivatives ( 导数导数 ))

the signs of the the signs of the

derivatives would be derivatives would be

reversed for an edge reversed for an edge

that transitions from that transitions from

light to darklight to dark

First First derivatderivatee

SeconSecond d derivatderivatee

Gray-Gray-level level profileprofile

3939

Second derivativesSecond derivatives

an undesirable featurean undesirable feature

produces 2 values for every edge in an produces 2 values for every edge in an

imageimage

zero-crossing propertyzero-crossing property

an imaginary straight line joining the an imaginary straight line joining the

extreme positive and negative values of extreme positive and negative values of

the second derivative would cross zero the second derivative would cross zero

near the midpoint of the edgenear the midpoint of the edge

quite useful for locating the centers of quite useful for locating the centers of

thick edgesthick edges

4040

Basic idea of edge detectionBasic idea of edge detection

A profile is defined perpendicularly to A profile is defined perpendicularly to

the edge direction and the results are the edge direction and the results are

interpretedinterpreted

The magnitude of the first derivative is The magnitude of the first derivative is

used to detect an edge (if a point is on a used to detect an edge (if a point is on a

ramp)ramp)

The sign of the second derivative can The sign of the second derivative can

determine whether an edge pixel is on the determine whether an edge pixel is on the

dark or light side of an edgedark or light side of an edge

4141

Review of First DerivateReview of First Derivate

Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

Sobel operators 3Sobel operators 333

6 5 8 5x yG z z G z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

9 5 8 6x yG z z G z z 1 0 0 1

0 1 1 0

1 0 0 1

0 1 1 0

7 8 9 1 2 3

3 6 9 1 4 7

2 2

2 2

x

y

G z z z z z z

G z z z z z z

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

x yf G G

4242

Edge direction and strengthEdge direction and strength

Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

2 2x yf G G

4343

Gradient MasksGradient Masks

1 0 0 1

0 1 1 0

Roberts

1 0 0 1

0 1 1 0

Roberts

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

4444

Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

1313

Basic Global ThresholdingBasic Global Thresholding

Let light objects in dark backgroundLet light objects in dark background

To extract the objectsTo extract the objects Select a ldquoTrdquo that separates the objects from thSelect a ldquoTrdquo that separates the objects from th

e backgrounde background

ie any (xy) for which f(xy)gtT is an object point ie any (xy) for which f(xy)gtT is an object point

A thresholded imageA thresholded image

1

0

if f x y T backgroundg x y

if f x y T foreground

1414

Heuristic Global ThresholdingHeuristic Global Thresholding

11 Select an initial estimate for TSelect an initial estimate for T

22 Segment the image using T This will produce two Segment the image using T This will produce two groups of pixels Ggroups of pixels G11 consisting of all pixels with gra consisting of all pixels with gray level values lt T and Gy level values lt T and G22 consisting of pixels with g consisting of pixels with gray level values ray level values T T

33 Compute the average gray level values μCompute the average gray level values μ11 and μ and μ22 fo for the pixels in regions Gr the pixels in regions G11 and G and G22

44 Compute a new threshold value T=05(μCompute a new threshold value T=05(μ11+μ+μ22))

55 Repeat steps 2 through 4 until the difference in T iRepeat steps 2 through 4 until the difference in T in successive iterations is smaller than a predefinen successive iterations is smaller than a predefined parameter Td parameter T00 seerdquohhellipmrdquo seerdquohhellipmrdquo

1515

Basic Adaptive ThresholdingBasic Adaptive Thresholding

subdivide original image into small areassubdivide original image into small areas

utilize a different threshold to segment each utilize a different threshold to segment each subimagessubimages

since the threshold used for each pixel depesince the threshold used for each pixel depends on the location of the pixel in terms of tnds on the location of the pixel in terms of the subimages this type of thresholding is ahe subimages this type of thresholding is adaptivedaptive

See ldquoAdaptive_thresholdmrdquoSee ldquoAdaptive_thresholdmrdquo

1616

Multilevel ThresholdingMultilevel Thresholding

a point (xy) belongs toa point (xy) belongs to an object class if Tan object class if T11 ltlt f(xy) le T f(xy) le T22

another object class if f(xy) another object class if f(xy) gtgt T T22

to background if f(xy) le Tto background if f(xy) le T11

1717

The histogram of an image containing two pThe histogram of an image containing two principal brightness regions can be considererincipal brightness regions can be considered an estimate of the brightness probability d an estimate of the brightness probability density function p(z)density function p(z) p(z) is the sum (or mixture) of two unimodal (p(z) is the sum (or mixture) of two unimodal ( 单单峰的峰的 ) densities (one for light one for dark region) densities (one for light one for dark regions)s)

1 1 2 2

1 2 1

p z P p z P p z

P P

1818

Optimal ThresholdingOptimal Thresholding

If the form of the If the form of the

densities is densities is

known or known or

assumed in assumed in

terms of terms of

minimum error minimum error

determining an determining an

optimal optimal

threshold for threshold for

segmenting the segmenting the

image is image is

possiblepossible

1 1 2 2

1 2 1

p z P p z P p z

P P

1919

Optimal ThresholdingOptimal Thresholding

Probability of erroneouslyProbability of erroneously

2020

Optimal ThresholdingOptimal Thresholding

Minimum errorMinimum error Differentiating ( Differentiating ( 微分微分 ) E(T) with respect to T (usi) E(T) with respect to T (usi

ng Leibnizrsquos rule) and equating the result to 0ng Leibnizrsquos rule) and equating the result to 0

find T which makesfind T which makes

2121

Optimal ThresholdingOptimal Thresholding

Minimum errorMinimum error

Specially if Specially if PP11 = P = P22 then the optimum then the optimum

threshold is where the curve pthreshold is where the curve p11(z) and (z) and

pp22(z) intersect(z) intersect

2222

Optimal ThresholdingOptimal Thresholding

For exampleFor example

Let Let PDF=Gaussian densityPDF=Gaussian density p p11(z) and (z) and

pp22(z)(z)

where μwhere μ11 and σ and σ1122 are the mean and are the mean and

variance of the Gaussian density of one variance of the Gaussian density of one

objectobject

μμ22 and σ and σ2222 are the mean and variance of are the mean and variance of

the Gaussian density of the other objectthe Gaussian density of the other object

2323

Optimal ThresholdingOptimal Thresholding

Quadratic equation (Quadratic equation (二次方程二次方程 ))

2424

Problems of ThresholdingProblems of Thresholding

Original imageOriginal image Thresholded imageThresholded image

2525

Problems of ThresholdingProblems of Thresholding

(a)(a) Exact threshold Exact threshold

segmentationsegmentation

(b)(b) Threshold too lowThreshold too low

(c)(c) Threshold too Threshold too

highhigh

2626

72 Point Detection72 Point Detection

a point has been detected at the a point has been detected at the

location on which the mark is location on which the mark is

centered ifcentered if

|R|geT|R|geT

where T is a nonnegative thresholdwhere T is a nonnegative threshold

R is the sum of products of the R is the sum of products of the

coefficients with the gray levels contained coefficients with the gray levels contained

in the region encompassed by the markin the region encompassed by the mark

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

2727

72 Point Detection72 Point Detection

Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

2828

ExampleExample

2929

73 Line Detection73 Line Detection

Horizontal mask will result with max Horizontal mask will result with max

response when a line passed through the response when a line passed through the

middle row of the mask with a constant middle row of the mask with a constant

backgroundbackground

the similar idea is used with other masksthe similar idea is used with other masks

Note the preferred direction of each mask Note the preferred direction of each mask

is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

than other possible directionsthan other possible directions

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

3030

Idea 1 of Line DetectionIdea 1 of Line Detection

Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

if at a certain point in the imageif at a certain point in the image

|Ri||Ri|gtgt|Rj||Rj|

for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

3131

Idea 2 of Line DetectionIdea 2 of Line Detection

Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

3232

ExampleExample

3333

74 Edge-based 74 Edge-based SegmentationSegmentation

Edge-based segmentations rely on edges found in an image by edge detecting operators

these edges mark image locations of discontinuities in gray level

Edge detection is the most common approach for detecting meaningful discontinuities in gray level

There are a large group of methods based on information about edges in the image

3434

What is edgeWhat is edge

Edge is where change occurs Change is measured by derivative in 1D

―Biggest change derivative has maximum magnitude

Or 2nd derivative is zero we discuss approaches for implementing

―first-order derivative (Gradient operator)

―second-order derivative (Laplacian operator)

―we have introduced both derivatives in chapter 3

―Here we will talk only about their properties for edge detection

3535

What is edgeWhat is edge

In other wordsIn other words an edge is a set of an edge is a set of

connected pixelsconnected pixels

that lie on the boundary between two that lie on the boundary between two

regions with relatively distinct gray-level regions with relatively distinct gray-level

propertiesproperties

Note edge vs boundaryNote edge vs boundary

―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

―whereas a region boundary owing to whereas a region boundary owing to

the way it is defined is a more global the way it is defined is a more global

ideaidea

3636

Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

because of because of

optics optics

sampling sampling

image image

acquisition acquisition

imperfectionimperfection

3737

Thick and Thin EdgeThick and Thin Edge

The slope of the ramp is inversely The slope of the ramp is inversely

proportional to the degree of blurring in the proportional to the degree of blurring in the

edgeedge

Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

pathpath

Instead an edge point now is any point Instead an edge point now is any point

contained in the ramp and contained in the ramp and an edge would an edge would

then be a set of such points that are then be a set of such points that are

connectedconnected

The thickness is determined by the length of the The thickness is determined by the length of the

rampramp

The length is determined by the slope which is in The length is determined by the slope which is in

turn determined by the degree of blurringturn determined by the degree of blurring

Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

edges tend to be thinedges tend to be thin

3838

First and Second derivatives (First and Second derivatives ( 导数导数 ))

the signs of the the signs of the

derivatives would be derivatives would be

reversed for an edge reversed for an edge

that transitions from that transitions from

light to darklight to dark

First First derivatderivatee

SeconSecond d derivatderivatee

Gray-Gray-level level profileprofile

3939

Second derivativesSecond derivatives

an undesirable featurean undesirable feature

produces 2 values for every edge in an produces 2 values for every edge in an

imageimage

zero-crossing propertyzero-crossing property

an imaginary straight line joining the an imaginary straight line joining the

extreme positive and negative values of extreme positive and negative values of

the second derivative would cross zero the second derivative would cross zero

near the midpoint of the edgenear the midpoint of the edge

quite useful for locating the centers of quite useful for locating the centers of

thick edgesthick edges

4040

Basic idea of edge detectionBasic idea of edge detection

A profile is defined perpendicularly to A profile is defined perpendicularly to

the edge direction and the results are the edge direction and the results are

interpretedinterpreted

The magnitude of the first derivative is The magnitude of the first derivative is

used to detect an edge (if a point is on a used to detect an edge (if a point is on a

ramp)ramp)

The sign of the second derivative can The sign of the second derivative can

determine whether an edge pixel is on the determine whether an edge pixel is on the

dark or light side of an edgedark or light side of an edge

4141

Review of First DerivateReview of First Derivate

Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

Sobel operators 3Sobel operators 333

6 5 8 5x yG z z G z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

9 5 8 6x yG z z G z z 1 0 0 1

0 1 1 0

1 0 0 1

0 1 1 0

7 8 9 1 2 3

3 6 9 1 4 7

2 2

2 2

x

y

G z z z z z z

G z z z z z z

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

x yf G G

4242

Edge direction and strengthEdge direction and strength

Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

2 2x yf G G

4343

Gradient MasksGradient Masks

1 0 0 1

0 1 1 0

Roberts

1 0 0 1

0 1 1 0

Roberts

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

4444

Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

1414

Heuristic Global ThresholdingHeuristic Global Thresholding

11 Select an initial estimate for TSelect an initial estimate for T

22 Segment the image using T This will produce two Segment the image using T This will produce two groups of pixels Ggroups of pixels G11 consisting of all pixels with gra consisting of all pixels with gray level values lt T and Gy level values lt T and G22 consisting of pixels with g consisting of pixels with gray level values ray level values T T

33 Compute the average gray level values μCompute the average gray level values μ11 and μ and μ22 fo for the pixels in regions Gr the pixels in regions G11 and G and G22

44 Compute a new threshold value T=05(μCompute a new threshold value T=05(μ11+μ+μ22))

55 Repeat steps 2 through 4 until the difference in T iRepeat steps 2 through 4 until the difference in T in successive iterations is smaller than a predefinen successive iterations is smaller than a predefined parameter Td parameter T00 seerdquohhellipmrdquo seerdquohhellipmrdquo

1515

Basic Adaptive ThresholdingBasic Adaptive Thresholding

subdivide original image into small areassubdivide original image into small areas

utilize a different threshold to segment each utilize a different threshold to segment each subimagessubimages

since the threshold used for each pixel depesince the threshold used for each pixel depends on the location of the pixel in terms of tnds on the location of the pixel in terms of the subimages this type of thresholding is ahe subimages this type of thresholding is adaptivedaptive

See ldquoAdaptive_thresholdmrdquoSee ldquoAdaptive_thresholdmrdquo

1616

Multilevel ThresholdingMultilevel Thresholding

a point (xy) belongs toa point (xy) belongs to an object class if Tan object class if T11 ltlt f(xy) le T f(xy) le T22

another object class if f(xy) another object class if f(xy) gtgt T T22

to background if f(xy) le Tto background if f(xy) le T11

1717

The histogram of an image containing two pThe histogram of an image containing two principal brightness regions can be considererincipal brightness regions can be considered an estimate of the brightness probability d an estimate of the brightness probability density function p(z)density function p(z) p(z) is the sum (or mixture) of two unimodal (p(z) is the sum (or mixture) of two unimodal ( 单单峰的峰的 ) densities (one for light one for dark region) densities (one for light one for dark regions)s)

1 1 2 2

1 2 1

p z P p z P p z

P P

1818

Optimal ThresholdingOptimal Thresholding

If the form of the If the form of the

densities is densities is

known or known or

assumed in assumed in

terms of terms of

minimum error minimum error

determining an determining an

optimal optimal

threshold for threshold for

segmenting the segmenting the

image is image is

possiblepossible

1 1 2 2

1 2 1

p z P p z P p z

P P

1919

Optimal ThresholdingOptimal Thresholding

Probability of erroneouslyProbability of erroneously

2020

Optimal ThresholdingOptimal Thresholding

Minimum errorMinimum error Differentiating ( Differentiating ( 微分微分 ) E(T) with respect to T (usi) E(T) with respect to T (usi

ng Leibnizrsquos rule) and equating the result to 0ng Leibnizrsquos rule) and equating the result to 0

find T which makesfind T which makes

2121

Optimal ThresholdingOptimal Thresholding

Minimum errorMinimum error

Specially if Specially if PP11 = P = P22 then the optimum then the optimum

threshold is where the curve pthreshold is where the curve p11(z) and (z) and

pp22(z) intersect(z) intersect

2222

Optimal ThresholdingOptimal Thresholding

For exampleFor example

Let Let PDF=Gaussian densityPDF=Gaussian density p p11(z) and (z) and

pp22(z)(z)

where μwhere μ11 and σ and σ1122 are the mean and are the mean and

variance of the Gaussian density of one variance of the Gaussian density of one

objectobject

μμ22 and σ and σ2222 are the mean and variance of are the mean and variance of

the Gaussian density of the other objectthe Gaussian density of the other object

2323

Optimal ThresholdingOptimal Thresholding

Quadratic equation (Quadratic equation (二次方程二次方程 ))

2424

Problems of ThresholdingProblems of Thresholding

Original imageOriginal image Thresholded imageThresholded image

2525

Problems of ThresholdingProblems of Thresholding

(a)(a) Exact threshold Exact threshold

segmentationsegmentation

(b)(b) Threshold too lowThreshold too low

(c)(c) Threshold too Threshold too

highhigh

2626

72 Point Detection72 Point Detection

a point has been detected at the a point has been detected at the

location on which the mark is location on which the mark is

centered ifcentered if

|R|geT|R|geT

where T is a nonnegative thresholdwhere T is a nonnegative threshold

R is the sum of products of the R is the sum of products of the

coefficients with the gray levels contained coefficients with the gray levels contained

in the region encompassed by the markin the region encompassed by the mark

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

2727

72 Point Detection72 Point Detection

Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

2828

ExampleExample

2929

73 Line Detection73 Line Detection

Horizontal mask will result with max Horizontal mask will result with max

response when a line passed through the response when a line passed through the

middle row of the mask with a constant middle row of the mask with a constant

backgroundbackground

the similar idea is used with other masksthe similar idea is used with other masks

Note the preferred direction of each mask Note the preferred direction of each mask

is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

than other possible directionsthan other possible directions

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

3030

Idea 1 of Line DetectionIdea 1 of Line Detection

Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

if at a certain point in the imageif at a certain point in the image

|Ri||Ri|gtgt|Rj||Rj|

for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

3131

Idea 2 of Line DetectionIdea 2 of Line Detection

Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

3232

ExampleExample

3333

74 Edge-based 74 Edge-based SegmentationSegmentation

Edge-based segmentations rely on edges found in an image by edge detecting operators

these edges mark image locations of discontinuities in gray level

Edge detection is the most common approach for detecting meaningful discontinuities in gray level

There are a large group of methods based on information about edges in the image

3434

What is edgeWhat is edge

Edge is where change occurs Change is measured by derivative in 1D

―Biggest change derivative has maximum magnitude

Or 2nd derivative is zero we discuss approaches for implementing

―first-order derivative (Gradient operator)

―second-order derivative (Laplacian operator)

―we have introduced both derivatives in chapter 3

―Here we will talk only about their properties for edge detection

3535

What is edgeWhat is edge

In other wordsIn other words an edge is a set of an edge is a set of

connected pixelsconnected pixels

that lie on the boundary between two that lie on the boundary between two

regions with relatively distinct gray-level regions with relatively distinct gray-level

propertiesproperties

Note edge vs boundaryNote edge vs boundary

―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

―whereas a region boundary owing to whereas a region boundary owing to

the way it is defined is a more global the way it is defined is a more global

ideaidea

3636

Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

because of because of

optics optics

sampling sampling

image image

acquisition acquisition

imperfectionimperfection

3737

Thick and Thin EdgeThick and Thin Edge

The slope of the ramp is inversely The slope of the ramp is inversely

proportional to the degree of blurring in the proportional to the degree of blurring in the

edgeedge

Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

pathpath

Instead an edge point now is any point Instead an edge point now is any point

contained in the ramp and contained in the ramp and an edge would an edge would

then be a set of such points that are then be a set of such points that are

connectedconnected

The thickness is determined by the length of the The thickness is determined by the length of the

rampramp

The length is determined by the slope which is in The length is determined by the slope which is in

turn determined by the degree of blurringturn determined by the degree of blurring

Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

edges tend to be thinedges tend to be thin

3838

First and Second derivatives (First and Second derivatives ( 导数导数 ))

the signs of the the signs of the

derivatives would be derivatives would be

reversed for an edge reversed for an edge

that transitions from that transitions from

light to darklight to dark

First First derivatderivatee

SeconSecond d derivatderivatee

Gray-Gray-level level profileprofile

3939

Second derivativesSecond derivatives

an undesirable featurean undesirable feature

produces 2 values for every edge in an produces 2 values for every edge in an

imageimage

zero-crossing propertyzero-crossing property

an imaginary straight line joining the an imaginary straight line joining the

extreme positive and negative values of extreme positive and negative values of

the second derivative would cross zero the second derivative would cross zero

near the midpoint of the edgenear the midpoint of the edge

quite useful for locating the centers of quite useful for locating the centers of

thick edgesthick edges

4040

Basic idea of edge detectionBasic idea of edge detection

A profile is defined perpendicularly to A profile is defined perpendicularly to

the edge direction and the results are the edge direction and the results are

interpretedinterpreted

The magnitude of the first derivative is The magnitude of the first derivative is

used to detect an edge (if a point is on a used to detect an edge (if a point is on a

ramp)ramp)

The sign of the second derivative can The sign of the second derivative can

determine whether an edge pixel is on the determine whether an edge pixel is on the

dark or light side of an edgedark or light side of an edge

4141

Review of First DerivateReview of First Derivate

Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

Sobel operators 3Sobel operators 333

6 5 8 5x yG z z G z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

9 5 8 6x yG z z G z z 1 0 0 1

0 1 1 0

1 0 0 1

0 1 1 0

7 8 9 1 2 3

3 6 9 1 4 7

2 2

2 2

x

y

G z z z z z z

G z z z z z z

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

x yf G G

4242

Edge direction and strengthEdge direction and strength

Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

2 2x yf G G

4343

Gradient MasksGradient Masks

1 0 0 1

0 1 1 0

Roberts

1 0 0 1

0 1 1 0

Roberts

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

4444

Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

1515

Basic Adaptive ThresholdingBasic Adaptive Thresholding

subdivide original image into small areassubdivide original image into small areas

utilize a different threshold to segment each utilize a different threshold to segment each subimagessubimages

since the threshold used for each pixel depesince the threshold used for each pixel depends on the location of the pixel in terms of tnds on the location of the pixel in terms of the subimages this type of thresholding is ahe subimages this type of thresholding is adaptivedaptive

See ldquoAdaptive_thresholdmrdquoSee ldquoAdaptive_thresholdmrdquo

1616

Multilevel ThresholdingMultilevel Thresholding

a point (xy) belongs toa point (xy) belongs to an object class if Tan object class if T11 ltlt f(xy) le T f(xy) le T22

another object class if f(xy) another object class if f(xy) gtgt T T22

to background if f(xy) le Tto background if f(xy) le T11

1717

The histogram of an image containing two pThe histogram of an image containing two principal brightness regions can be considererincipal brightness regions can be considered an estimate of the brightness probability d an estimate of the brightness probability density function p(z)density function p(z) p(z) is the sum (or mixture) of two unimodal (p(z) is the sum (or mixture) of two unimodal ( 单单峰的峰的 ) densities (one for light one for dark region) densities (one for light one for dark regions)s)

1 1 2 2

1 2 1

p z P p z P p z

P P

1818

Optimal ThresholdingOptimal Thresholding

If the form of the If the form of the

densities is densities is

known or known or

assumed in assumed in

terms of terms of

minimum error minimum error

determining an determining an

optimal optimal

threshold for threshold for

segmenting the segmenting the

image is image is

possiblepossible

1 1 2 2

1 2 1

p z P p z P p z

P P

1919

Optimal ThresholdingOptimal Thresholding

Probability of erroneouslyProbability of erroneously

2020

Optimal ThresholdingOptimal Thresholding

Minimum errorMinimum error Differentiating ( Differentiating ( 微分微分 ) E(T) with respect to T (usi) E(T) with respect to T (usi

ng Leibnizrsquos rule) and equating the result to 0ng Leibnizrsquos rule) and equating the result to 0

find T which makesfind T which makes

2121

Optimal ThresholdingOptimal Thresholding

Minimum errorMinimum error

Specially if Specially if PP11 = P = P22 then the optimum then the optimum

threshold is where the curve pthreshold is where the curve p11(z) and (z) and

pp22(z) intersect(z) intersect

2222

Optimal ThresholdingOptimal Thresholding

For exampleFor example

Let Let PDF=Gaussian densityPDF=Gaussian density p p11(z) and (z) and

pp22(z)(z)

where μwhere μ11 and σ and σ1122 are the mean and are the mean and

variance of the Gaussian density of one variance of the Gaussian density of one

objectobject

μμ22 and σ and σ2222 are the mean and variance of are the mean and variance of

the Gaussian density of the other objectthe Gaussian density of the other object

2323

Optimal ThresholdingOptimal Thresholding

Quadratic equation (Quadratic equation (二次方程二次方程 ))

2424

Problems of ThresholdingProblems of Thresholding

Original imageOriginal image Thresholded imageThresholded image

2525

Problems of ThresholdingProblems of Thresholding

(a)(a) Exact threshold Exact threshold

segmentationsegmentation

(b)(b) Threshold too lowThreshold too low

(c)(c) Threshold too Threshold too

highhigh

2626

72 Point Detection72 Point Detection

a point has been detected at the a point has been detected at the

location on which the mark is location on which the mark is

centered ifcentered if

|R|geT|R|geT

where T is a nonnegative thresholdwhere T is a nonnegative threshold

R is the sum of products of the R is the sum of products of the

coefficients with the gray levels contained coefficients with the gray levels contained

in the region encompassed by the markin the region encompassed by the mark

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

2727

72 Point Detection72 Point Detection

Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

2828

ExampleExample

2929

73 Line Detection73 Line Detection

Horizontal mask will result with max Horizontal mask will result with max

response when a line passed through the response when a line passed through the

middle row of the mask with a constant middle row of the mask with a constant

backgroundbackground

the similar idea is used with other masksthe similar idea is used with other masks

Note the preferred direction of each mask Note the preferred direction of each mask

is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

than other possible directionsthan other possible directions

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

3030

Idea 1 of Line DetectionIdea 1 of Line Detection

Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

if at a certain point in the imageif at a certain point in the image

|Ri||Ri|gtgt|Rj||Rj|

for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

3131

Idea 2 of Line DetectionIdea 2 of Line Detection

Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

3232

ExampleExample

3333

74 Edge-based 74 Edge-based SegmentationSegmentation

Edge-based segmentations rely on edges found in an image by edge detecting operators

these edges mark image locations of discontinuities in gray level

Edge detection is the most common approach for detecting meaningful discontinuities in gray level

There are a large group of methods based on information about edges in the image

3434

What is edgeWhat is edge

Edge is where change occurs Change is measured by derivative in 1D

―Biggest change derivative has maximum magnitude

Or 2nd derivative is zero we discuss approaches for implementing

―first-order derivative (Gradient operator)

―second-order derivative (Laplacian operator)

―we have introduced both derivatives in chapter 3

―Here we will talk only about their properties for edge detection

3535

What is edgeWhat is edge

In other wordsIn other words an edge is a set of an edge is a set of

connected pixelsconnected pixels

that lie on the boundary between two that lie on the boundary between two

regions with relatively distinct gray-level regions with relatively distinct gray-level

propertiesproperties

Note edge vs boundaryNote edge vs boundary

―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

―whereas a region boundary owing to whereas a region boundary owing to

the way it is defined is a more global the way it is defined is a more global

ideaidea

3636

Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

because of because of

optics optics

sampling sampling

image image

acquisition acquisition

imperfectionimperfection

3737

Thick and Thin EdgeThick and Thin Edge

The slope of the ramp is inversely The slope of the ramp is inversely

proportional to the degree of blurring in the proportional to the degree of blurring in the

edgeedge

Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

pathpath

Instead an edge point now is any point Instead an edge point now is any point

contained in the ramp and contained in the ramp and an edge would an edge would

then be a set of such points that are then be a set of such points that are

connectedconnected

The thickness is determined by the length of the The thickness is determined by the length of the

rampramp

The length is determined by the slope which is in The length is determined by the slope which is in

turn determined by the degree of blurringturn determined by the degree of blurring

Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

edges tend to be thinedges tend to be thin

3838

First and Second derivatives (First and Second derivatives ( 导数导数 ))

the signs of the the signs of the

derivatives would be derivatives would be

reversed for an edge reversed for an edge

that transitions from that transitions from

light to darklight to dark

First First derivatderivatee

SeconSecond d derivatderivatee

Gray-Gray-level level profileprofile

3939

Second derivativesSecond derivatives

an undesirable featurean undesirable feature

produces 2 values for every edge in an produces 2 values for every edge in an

imageimage

zero-crossing propertyzero-crossing property

an imaginary straight line joining the an imaginary straight line joining the

extreme positive and negative values of extreme positive and negative values of

the second derivative would cross zero the second derivative would cross zero

near the midpoint of the edgenear the midpoint of the edge

quite useful for locating the centers of quite useful for locating the centers of

thick edgesthick edges

4040

Basic idea of edge detectionBasic idea of edge detection

A profile is defined perpendicularly to A profile is defined perpendicularly to

the edge direction and the results are the edge direction and the results are

interpretedinterpreted

The magnitude of the first derivative is The magnitude of the first derivative is

used to detect an edge (if a point is on a used to detect an edge (if a point is on a

ramp)ramp)

The sign of the second derivative can The sign of the second derivative can

determine whether an edge pixel is on the determine whether an edge pixel is on the

dark or light side of an edgedark or light side of an edge

4141

Review of First DerivateReview of First Derivate

Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

Sobel operators 3Sobel operators 333

6 5 8 5x yG z z G z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

9 5 8 6x yG z z G z z 1 0 0 1

0 1 1 0

1 0 0 1

0 1 1 0

7 8 9 1 2 3

3 6 9 1 4 7

2 2

2 2

x

y

G z z z z z z

G z z z z z z

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

x yf G G

4242

Edge direction and strengthEdge direction and strength

Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

2 2x yf G G

4343

Gradient MasksGradient Masks

1 0 0 1

0 1 1 0

Roberts

1 0 0 1

0 1 1 0

Roberts

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

4444

Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
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• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

1616

Multilevel ThresholdingMultilevel Thresholding

a point (xy) belongs toa point (xy) belongs to an object class if Tan object class if T11 ltlt f(xy) le T f(xy) le T22

another object class if f(xy) another object class if f(xy) gtgt T T22

to background if f(xy) le Tto background if f(xy) le T11

1717

The histogram of an image containing two pThe histogram of an image containing two principal brightness regions can be considererincipal brightness regions can be considered an estimate of the brightness probability d an estimate of the brightness probability density function p(z)density function p(z) p(z) is the sum (or mixture) of two unimodal (p(z) is the sum (or mixture) of two unimodal ( 单单峰的峰的 ) densities (one for light one for dark region) densities (one for light one for dark regions)s)

1 1 2 2

1 2 1

p z P p z P p z

P P

1818

Optimal ThresholdingOptimal Thresholding

If the form of the If the form of the

densities is densities is

known or known or

assumed in assumed in

terms of terms of

minimum error minimum error

determining an determining an

optimal optimal

threshold for threshold for

segmenting the segmenting the

image is image is

possiblepossible

1 1 2 2

1 2 1

p z P p z P p z

P P

1919

Optimal ThresholdingOptimal Thresholding

Probability of erroneouslyProbability of erroneously

2020

Optimal ThresholdingOptimal Thresholding

Minimum errorMinimum error Differentiating ( Differentiating ( 微分微分 ) E(T) with respect to T (usi) E(T) with respect to T (usi

ng Leibnizrsquos rule) and equating the result to 0ng Leibnizrsquos rule) and equating the result to 0

find T which makesfind T which makes

2121

Optimal ThresholdingOptimal Thresholding

Minimum errorMinimum error

Specially if Specially if PP11 = P = P22 then the optimum then the optimum

threshold is where the curve pthreshold is where the curve p11(z) and (z) and

pp22(z) intersect(z) intersect

2222

Optimal ThresholdingOptimal Thresholding

For exampleFor example

Let Let PDF=Gaussian densityPDF=Gaussian density p p11(z) and (z) and

pp22(z)(z)

where μwhere μ11 and σ and σ1122 are the mean and are the mean and

variance of the Gaussian density of one variance of the Gaussian density of one

objectobject

μμ22 and σ and σ2222 are the mean and variance of are the mean and variance of

the Gaussian density of the other objectthe Gaussian density of the other object

2323

Optimal ThresholdingOptimal Thresholding

Quadratic equation (Quadratic equation (二次方程二次方程 ))

2424

Problems of ThresholdingProblems of Thresholding

Original imageOriginal image Thresholded imageThresholded image

2525

Problems of ThresholdingProblems of Thresholding

(a)(a) Exact threshold Exact threshold

segmentationsegmentation

(b)(b) Threshold too lowThreshold too low

(c)(c) Threshold too Threshold too

highhigh

2626

72 Point Detection72 Point Detection

a point has been detected at the a point has been detected at the

location on which the mark is location on which the mark is

centered ifcentered if

|R|geT|R|geT

where T is a nonnegative thresholdwhere T is a nonnegative threshold

R is the sum of products of the R is the sum of products of the

coefficients with the gray levels contained coefficients with the gray levels contained

in the region encompassed by the markin the region encompassed by the mark

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

2727

72 Point Detection72 Point Detection

Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

2828

ExampleExample

2929

73 Line Detection73 Line Detection

Horizontal mask will result with max Horizontal mask will result with max

response when a line passed through the response when a line passed through the

middle row of the mask with a constant middle row of the mask with a constant

backgroundbackground

the similar idea is used with other masksthe similar idea is used with other masks

Note the preferred direction of each mask Note the preferred direction of each mask

is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

than other possible directionsthan other possible directions

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

3030

Idea 1 of Line DetectionIdea 1 of Line Detection

Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

if at a certain point in the imageif at a certain point in the image

|Ri||Ri|gtgt|Rj||Rj|

for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

3131

Idea 2 of Line DetectionIdea 2 of Line Detection

Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

3232

ExampleExample

3333

74 Edge-based 74 Edge-based SegmentationSegmentation

Edge-based segmentations rely on edges found in an image by edge detecting operators

these edges mark image locations of discontinuities in gray level

Edge detection is the most common approach for detecting meaningful discontinuities in gray level

There are a large group of methods based on information about edges in the image

3434

What is edgeWhat is edge

Edge is where change occurs Change is measured by derivative in 1D

―Biggest change derivative has maximum magnitude

Or 2nd derivative is zero we discuss approaches for implementing

―first-order derivative (Gradient operator)

―second-order derivative (Laplacian operator)

―we have introduced both derivatives in chapter 3

―Here we will talk only about their properties for edge detection

3535

What is edgeWhat is edge

In other wordsIn other words an edge is a set of an edge is a set of

connected pixelsconnected pixels

that lie on the boundary between two that lie on the boundary between two

regions with relatively distinct gray-level regions with relatively distinct gray-level

propertiesproperties

Note edge vs boundaryNote edge vs boundary

―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

―whereas a region boundary owing to whereas a region boundary owing to

the way it is defined is a more global the way it is defined is a more global

ideaidea

3636

Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

because of because of

optics optics

sampling sampling

image image

acquisition acquisition

imperfectionimperfection

3737

Thick and Thin EdgeThick and Thin Edge

The slope of the ramp is inversely The slope of the ramp is inversely

proportional to the degree of blurring in the proportional to the degree of blurring in the

edgeedge

Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

pathpath

Instead an edge point now is any point Instead an edge point now is any point

contained in the ramp and contained in the ramp and an edge would an edge would

then be a set of such points that are then be a set of such points that are

connectedconnected

The thickness is determined by the length of the The thickness is determined by the length of the

rampramp

The length is determined by the slope which is in The length is determined by the slope which is in

turn determined by the degree of blurringturn determined by the degree of blurring

Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

edges tend to be thinedges tend to be thin

3838

First and Second derivatives (First and Second derivatives ( 导数导数 ))

the signs of the the signs of the

derivatives would be derivatives would be

reversed for an edge reversed for an edge

that transitions from that transitions from

light to darklight to dark

First First derivatderivatee

SeconSecond d derivatderivatee

Gray-Gray-level level profileprofile

3939

Second derivativesSecond derivatives

an undesirable featurean undesirable feature

produces 2 values for every edge in an produces 2 values for every edge in an

imageimage

zero-crossing propertyzero-crossing property

an imaginary straight line joining the an imaginary straight line joining the

extreme positive and negative values of extreme positive and negative values of

the second derivative would cross zero the second derivative would cross zero

near the midpoint of the edgenear the midpoint of the edge

quite useful for locating the centers of quite useful for locating the centers of

thick edgesthick edges

4040

Basic idea of edge detectionBasic idea of edge detection

A profile is defined perpendicularly to A profile is defined perpendicularly to

the edge direction and the results are the edge direction and the results are

interpretedinterpreted

The magnitude of the first derivative is The magnitude of the first derivative is

used to detect an edge (if a point is on a used to detect an edge (if a point is on a

ramp)ramp)

The sign of the second derivative can The sign of the second derivative can

determine whether an edge pixel is on the determine whether an edge pixel is on the

dark or light side of an edgedark or light side of an edge

4141

Review of First DerivateReview of First Derivate

Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

Sobel operators 3Sobel operators 333

6 5 8 5x yG z z G z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

9 5 8 6x yG z z G z z 1 0 0 1

0 1 1 0

1 0 0 1

0 1 1 0

7 8 9 1 2 3

3 6 9 1 4 7

2 2

2 2

x

y

G z z z z z z

G z z z z z z

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

x yf G G

4242

Edge direction and strengthEdge direction and strength

Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

2 2x yf G G

4343

Gradient MasksGradient Masks

1 0 0 1

0 1 1 0

Roberts

1 0 0 1

0 1 1 0

Roberts

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

4444

Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
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• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
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• Slide 31
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• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

1717

The histogram of an image containing two pThe histogram of an image containing two principal brightness regions can be considererincipal brightness regions can be considered an estimate of the brightness probability d an estimate of the brightness probability density function p(z)density function p(z) p(z) is the sum (or mixture) of two unimodal (p(z) is the sum (or mixture) of two unimodal ( 单单峰的峰的 ) densities (one for light one for dark region) densities (one for light one for dark regions)s)

1 1 2 2

1 2 1

p z P p z P p z

P P

1818

Optimal ThresholdingOptimal Thresholding

If the form of the If the form of the

densities is densities is

known or known or

assumed in assumed in

terms of terms of

minimum error minimum error

determining an determining an

optimal optimal

threshold for threshold for

segmenting the segmenting the

image is image is

possiblepossible

1 1 2 2

1 2 1

p z P p z P p z

P P

1919

Optimal ThresholdingOptimal Thresholding

Probability of erroneouslyProbability of erroneously

2020

Optimal ThresholdingOptimal Thresholding

Minimum errorMinimum error Differentiating ( Differentiating ( 微分微分 ) E(T) with respect to T (usi) E(T) with respect to T (usi

ng Leibnizrsquos rule) and equating the result to 0ng Leibnizrsquos rule) and equating the result to 0

find T which makesfind T which makes

2121

Optimal ThresholdingOptimal Thresholding

Minimum errorMinimum error

Specially if Specially if PP11 = P = P22 then the optimum then the optimum

threshold is where the curve pthreshold is where the curve p11(z) and (z) and

pp22(z) intersect(z) intersect

2222

Optimal ThresholdingOptimal Thresholding

For exampleFor example

Let Let PDF=Gaussian densityPDF=Gaussian density p p11(z) and (z) and

pp22(z)(z)

where μwhere μ11 and σ and σ1122 are the mean and are the mean and

variance of the Gaussian density of one variance of the Gaussian density of one

objectobject

μμ22 and σ and σ2222 are the mean and variance of are the mean and variance of

the Gaussian density of the other objectthe Gaussian density of the other object

2323

Optimal ThresholdingOptimal Thresholding

Quadratic equation (Quadratic equation (二次方程二次方程 ))

2424

Problems of ThresholdingProblems of Thresholding

Original imageOriginal image Thresholded imageThresholded image

2525

Problems of ThresholdingProblems of Thresholding

(a)(a) Exact threshold Exact threshold

segmentationsegmentation

(b)(b) Threshold too lowThreshold too low

(c)(c) Threshold too Threshold too

highhigh

2626

72 Point Detection72 Point Detection

a point has been detected at the a point has been detected at the

location on which the mark is location on which the mark is

centered ifcentered if

|R|geT|R|geT

where T is a nonnegative thresholdwhere T is a nonnegative threshold

R is the sum of products of the R is the sum of products of the

coefficients with the gray levels contained coefficients with the gray levels contained

in the region encompassed by the markin the region encompassed by the mark

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

2727

72 Point Detection72 Point Detection

Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

2828

ExampleExample

2929

73 Line Detection73 Line Detection

Horizontal mask will result with max Horizontal mask will result with max

response when a line passed through the response when a line passed through the

middle row of the mask with a constant middle row of the mask with a constant

backgroundbackground

the similar idea is used with other masksthe similar idea is used with other masks

Note the preferred direction of each mask Note the preferred direction of each mask

is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

than other possible directionsthan other possible directions

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

3030

Idea 1 of Line DetectionIdea 1 of Line Detection

Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

if at a certain point in the imageif at a certain point in the image

|Ri||Ri|gtgt|Rj||Rj|

for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

3131

Idea 2 of Line DetectionIdea 2 of Line Detection

Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

3232

ExampleExample

3333

74 Edge-based 74 Edge-based SegmentationSegmentation

Edge-based segmentations rely on edges found in an image by edge detecting operators

these edges mark image locations of discontinuities in gray level

Edge detection is the most common approach for detecting meaningful discontinuities in gray level

There are a large group of methods based on information about edges in the image

3434

What is edgeWhat is edge

Edge is where change occurs Change is measured by derivative in 1D

―Biggest change derivative has maximum magnitude

Or 2nd derivative is zero we discuss approaches for implementing

―first-order derivative (Gradient operator)

―second-order derivative (Laplacian operator)

―we have introduced both derivatives in chapter 3

―Here we will talk only about their properties for edge detection

3535

What is edgeWhat is edge

In other wordsIn other words an edge is a set of an edge is a set of

connected pixelsconnected pixels

that lie on the boundary between two that lie on the boundary between two

regions with relatively distinct gray-level regions with relatively distinct gray-level

propertiesproperties

Note edge vs boundaryNote edge vs boundary

―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

―whereas a region boundary owing to whereas a region boundary owing to

the way it is defined is a more global the way it is defined is a more global

ideaidea

3636

Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

because of because of

optics optics

sampling sampling

image image

acquisition acquisition

imperfectionimperfection

3737

Thick and Thin EdgeThick and Thin Edge

The slope of the ramp is inversely The slope of the ramp is inversely

proportional to the degree of blurring in the proportional to the degree of blurring in the

edgeedge

Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

pathpath

Instead an edge point now is any point Instead an edge point now is any point

contained in the ramp and contained in the ramp and an edge would an edge would

then be a set of such points that are then be a set of such points that are

connectedconnected

The thickness is determined by the length of the The thickness is determined by the length of the

rampramp

The length is determined by the slope which is in The length is determined by the slope which is in

turn determined by the degree of blurringturn determined by the degree of blurring

Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

edges tend to be thinedges tend to be thin

3838

First and Second derivatives (First and Second derivatives ( 导数导数 ))

the signs of the the signs of the

derivatives would be derivatives would be

reversed for an edge reversed for an edge

that transitions from that transitions from

light to darklight to dark

First First derivatderivatee

SeconSecond d derivatderivatee

Gray-Gray-level level profileprofile

3939

Second derivativesSecond derivatives

an undesirable featurean undesirable feature

produces 2 values for every edge in an produces 2 values for every edge in an

imageimage

zero-crossing propertyzero-crossing property

an imaginary straight line joining the an imaginary straight line joining the

extreme positive and negative values of extreme positive and negative values of

the second derivative would cross zero the second derivative would cross zero

near the midpoint of the edgenear the midpoint of the edge

quite useful for locating the centers of quite useful for locating the centers of

thick edgesthick edges

4040

Basic idea of edge detectionBasic idea of edge detection

A profile is defined perpendicularly to A profile is defined perpendicularly to

the edge direction and the results are the edge direction and the results are

interpretedinterpreted

The magnitude of the first derivative is The magnitude of the first derivative is

used to detect an edge (if a point is on a used to detect an edge (if a point is on a

ramp)ramp)

The sign of the second derivative can The sign of the second derivative can

determine whether an edge pixel is on the determine whether an edge pixel is on the

dark or light side of an edgedark or light side of an edge

4141

Review of First DerivateReview of First Derivate

Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

Sobel operators 3Sobel operators 333

6 5 8 5x yG z z G z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

9 5 8 6x yG z z G z z 1 0 0 1

0 1 1 0

1 0 0 1

0 1 1 0

7 8 9 1 2 3

3 6 9 1 4 7

2 2

2 2

x

y

G z z z z z z

G z z z z z z

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

x yf G G

4242

Edge direction and strengthEdge direction and strength

Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

2 2x yf G G

4343

Gradient MasksGradient Masks

1 0 0 1

0 1 1 0

Roberts

1 0 0 1

0 1 1 0

Roberts

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

4444

Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

1818

Optimal ThresholdingOptimal Thresholding

If the form of the If the form of the

densities is densities is

known or known or

assumed in assumed in

terms of terms of

minimum error minimum error

determining an determining an

optimal optimal

threshold for threshold for

segmenting the segmenting the

image is image is

possiblepossible

1 1 2 2

1 2 1

p z P p z P p z

P P

1919

Optimal ThresholdingOptimal Thresholding

Probability of erroneouslyProbability of erroneously

2020

Optimal ThresholdingOptimal Thresholding

Minimum errorMinimum error Differentiating ( Differentiating ( 微分微分 ) E(T) with respect to T (usi) E(T) with respect to T (usi

ng Leibnizrsquos rule) and equating the result to 0ng Leibnizrsquos rule) and equating the result to 0

find T which makesfind T which makes

2121

Optimal ThresholdingOptimal Thresholding

Minimum errorMinimum error

Specially if Specially if PP11 = P = P22 then the optimum then the optimum

threshold is where the curve pthreshold is where the curve p11(z) and (z) and

pp22(z) intersect(z) intersect

2222

Optimal ThresholdingOptimal Thresholding

For exampleFor example

Let Let PDF=Gaussian densityPDF=Gaussian density p p11(z) and (z) and

pp22(z)(z)

where μwhere μ11 and σ and σ1122 are the mean and are the mean and

variance of the Gaussian density of one variance of the Gaussian density of one

objectobject

μμ22 and σ and σ2222 are the mean and variance of are the mean and variance of

the Gaussian density of the other objectthe Gaussian density of the other object

2323

Optimal ThresholdingOptimal Thresholding

Quadratic equation (Quadratic equation (二次方程二次方程 ))

2424

Problems of ThresholdingProblems of Thresholding

Original imageOriginal image Thresholded imageThresholded image

2525

Problems of ThresholdingProblems of Thresholding

(a)(a) Exact threshold Exact threshold

segmentationsegmentation

(b)(b) Threshold too lowThreshold too low

(c)(c) Threshold too Threshold too

highhigh

2626

72 Point Detection72 Point Detection

a point has been detected at the a point has been detected at the

location on which the mark is location on which the mark is

centered ifcentered if

|R|geT|R|geT

where T is a nonnegative thresholdwhere T is a nonnegative threshold

R is the sum of products of the R is the sum of products of the

coefficients with the gray levels contained coefficients with the gray levels contained

in the region encompassed by the markin the region encompassed by the mark

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

2727

72 Point Detection72 Point Detection

Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

2828

ExampleExample

2929

73 Line Detection73 Line Detection

Horizontal mask will result with max Horizontal mask will result with max

response when a line passed through the response when a line passed through the

middle row of the mask with a constant middle row of the mask with a constant

backgroundbackground

the similar idea is used with other masksthe similar idea is used with other masks

Note the preferred direction of each mask Note the preferred direction of each mask

is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

than other possible directionsthan other possible directions

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

3030

Idea 1 of Line DetectionIdea 1 of Line Detection

Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

if at a certain point in the imageif at a certain point in the image

|Ri||Ri|gtgt|Rj||Rj|

for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

3131

Idea 2 of Line DetectionIdea 2 of Line Detection

Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

3232

ExampleExample

3333

74 Edge-based 74 Edge-based SegmentationSegmentation

Edge-based segmentations rely on edges found in an image by edge detecting operators

these edges mark image locations of discontinuities in gray level

Edge detection is the most common approach for detecting meaningful discontinuities in gray level

There are a large group of methods based on information about edges in the image

3434

What is edgeWhat is edge

Edge is where change occurs Change is measured by derivative in 1D

―Biggest change derivative has maximum magnitude

Or 2nd derivative is zero we discuss approaches for implementing

―first-order derivative (Gradient operator)

―second-order derivative (Laplacian operator)

―we have introduced both derivatives in chapter 3

―Here we will talk only about their properties for edge detection

3535

What is edgeWhat is edge

In other wordsIn other words an edge is a set of an edge is a set of

connected pixelsconnected pixels

that lie on the boundary between two that lie on the boundary between two

regions with relatively distinct gray-level regions with relatively distinct gray-level

propertiesproperties

Note edge vs boundaryNote edge vs boundary

―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

―whereas a region boundary owing to whereas a region boundary owing to

the way it is defined is a more global the way it is defined is a more global

ideaidea

3636

Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

because of because of

optics optics

sampling sampling

image image

acquisition acquisition

imperfectionimperfection

3737

Thick and Thin EdgeThick and Thin Edge

The slope of the ramp is inversely The slope of the ramp is inversely

proportional to the degree of blurring in the proportional to the degree of blurring in the

edgeedge

Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

pathpath

Instead an edge point now is any point Instead an edge point now is any point

contained in the ramp and contained in the ramp and an edge would an edge would

then be a set of such points that are then be a set of such points that are

connectedconnected

The thickness is determined by the length of the The thickness is determined by the length of the

rampramp

The length is determined by the slope which is in The length is determined by the slope which is in

turn determined by the degree of blurringturn determined by the degree of blurring

Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

edges tend to be thinedges tend to be thin

3838

First and Second derivatives (First and Second derivatives ( 导数导数 ))

the signs of the the signs of the

derivatives would be derivatives would be

reversed for an edge reversed for an edge

that transitions from that transitions from

light to darklight to dark

First First derivatderivatee

SeconSecond d derivatderivatee

Gray-Gray-level level profileprofile

3939

Second derivativesSecond derivatives

an undesirable featurean undesirable feature

produces 2 values for every edge in an produces 2 values for every edge in an

imageimage

zero-crossing propertyzero-crossing property

an imaginary straight line joining the an imaginary straight line joining the

extreme positive and negative values of extreme positive and negative values of

the second derivative would cross zero the second derivative would cross zero

near the midpoint of the edgenear the midpoint of the edge

quite useful for locating the centers of quite useful for locating the centers of

thick edgesthick edges

4040

Basic idea of edge detectionBasic idea of edge detection

A profile is defined perpendicularly to A profile is defined perpendicularly to

the edge direction and the results are the edge direction and the results are

interpretedinterpreted

The magnitude of the first derivative is The magnitude of the first derivative is

used to detect an edge (if a point is on a used to detect an edge (if a point is on a

ramp)ramp)

The sign of the second derivative can The sign of the second derivative can

determine whether an edge pixel is on the determine whether an edge pixel is on the

dark or light side of an edgedark or light side of an edge

4141

Review of First DerivateReview of First Derivate

Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

Sobel operators 3Sobel operators 333

6 5 8 5x yG z z G z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

9 5 8 6x yG z z G z z 1 0 0 1

0 1 1 0

1 0 0 1

0 1 1 0

7 8 9 1 2 3

3 6 9 1 4 7

2 2

2 2

x

y

G z z z z z z

G z z z z z z

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

x yf G G

4242

Edge direction and strengthEdge direction and strength

Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

2 2x yf G G

4343

Gradient MasksGradient Masks

1 0 0 1

0 1 1 0

Roberts

1 0 0 1

0 1 1 0

Roberts

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

4444

Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

1919

Optimal ThresholdingOptimal Thresholding

Probability of erroneouslyProbability of erroneously

2020

Optimal ThresholdingOptimal Thresholding

Minimum errorMinimum error Differentiating ( Differentiating ( 微分微分 ) E(T) with respect to T (usi) E(T) with respect to T (usi

ng Leibnizrsquos rule) and equating the result to 0ng Leibnizrsquos rule) and equating the result to 0

find T which makesfind T which makes

2121

Optimal ThresholdingOptimal Thresholding

Minimum errorMinimum error

Specially if Specially if PP11 = P = P22 then the optimum then the optimum

threshold is where the curve pthreshold is where the curve p11(z) and (z) and

pp22(z) intersect(z) intersect

2222

Optimal ThresholdingOptimal Thresholding

For exampleFor example

Let Let PDF=Gaussian densityPDF=Gaussian density p p11(z) and (z) and

pp22(z)(z)

where μwhere μ11 and σ and σ1122 are the mean and are the mean and

variance of the Gaussian density of one variance of the Gaussian density of one

objectobject

μμ22 and σ and σ2222 are the mean and variance of are the mean and variance of

the Gaussian density of the other objectthe Gaussian density of the other object

2323

Optimal ThresholdingOptimal Thresholding

Quadratic equation (Quadratic equation (二次方程二次方程 ))

2424

Problems of ThresholdingProblems of Thresholding

Original imageOriginal image Thresholded imageThresholded image

2525

Problems of ThresholdingProblems of Thresholding

(a)(a) Exact threshold Exact threshold

segmentationsegmentation

(b)(b) Threshold too lowThreshold too low

(c)(c) Threshold too Threshold too

highhigh

2626

72 Point Detection72 Point Detection

a point has been detected at the a point has been detected at the

location on which the mark is location on which the mark is

centered ifcentered if

|R|geT|R|geT

where T is a nonnegative thresholdwhere T is a nonnegative threshold

R is the sum of products of the R is the sum of products of the

coefficients with the gray levels contained coefficients with the gray levels contained

in the region encompassed by the markin the region encompassed by the mark

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

2727

72 Point Detection72 Point Detection

Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

2828

ExampleExample

2929

73 Line Detection73 Line Detection

Horizontal mask will result with max Horizontal mask will result with max

response when a line passed through the response when a line passed through the

middle row of the mask with a constant middle row of the mask with a constant

backgroundbackground

the similar idea is used with other masksthe similar idea is used with other masks

Note the preferred direction of each mask Note the preferred direction of each mask

is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

than other possible directionsthan other possible directions

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

3030

Idea 1 of Line DetectionIdea 1 of Line Detection

Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

if at a certain point in the imageif at a certain point in the image

|Ri||Ri|gtgt|Rj||Rj|

for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

3131

Idea 2 of Line DetectionIdea 2 of Line Detection

Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

3232

ExampleExample

3333

74 Edge-based 74 Edge-based SegmentationSegmentation

Edge-based segmentations rely on edges found in an image by edge detecting operators

these edges mark image locations of discontinuities in gray level

Edge detection is the most common approach for detecting meaningful discontinuities in gray level

There are a large group of methods based on information about edges in the image

3434

What is edgeWhat is edge

Edge is where change occurs Change is measured by derivative in 1D

―Biggest change derivative has maximum magnitude

Or 2nd derivative is zero we discuss approaches for implementing

―first-order derivative (Gradient operator)

―second-order derivative (Laplacian operator)

―we have introduced both derivatives in chapter 3

―Here we will talk only about their properties for edge detection

3535

What is edgeWhat is edge

In other wordsIn other words an edge is a set of an edge is a set of

connected pixelsconnected pixels

that lie on the boundary between two that lie on the boundary between two

regions with relatively distinct gray-level regions with relatively distinct gray-level

propertiesproperties

Note edge vs boundaryNote edge vs boundary

―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

―whereas a region boundary owing to whereas a region boundary owing to

the way it is defined is a more global the way it is defined is a more global

ideaidea

3636

Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

because of because of

optics optics

sampling sampling

image image

acquisition acquisition

imperfectionimperfection

3737

Thick and Thin EdgeThick and Thin Edge

The slope of the ramp is inversely The slope of the ramp is inversely

proportional to the degree of blurring in the proportional to the degree of blurring in the

edgeedge

Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

pathpath

Instead an edge point now is any point Instead an edge point now is any point

contained in the ramp and contained in the ramp and an edge would an edge would

then be a set of such points that are then be a set of such points that are

connectedconnected

The thickness is determined by the length of the The thickness is determined by the length of the

rampramp

The length is determined by the slope which is in The length is determined by the slope which is in

turn determined by the degree of blurringturn determined by the degree of blurring

Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

edges tend to be thinedges tend to be thin

3838

First and Second derivatives (First and Second derivatives ( 导数导数 ))

the signs of the the signs of the

derivatives would be derivatives would be

reversed for an edge reversed for an edge

that transitions from that transitions from

light to darklight to dark

First First derivatderivatee

SeconSecond d derivatderivatee

Gray-Gray-level level profileprofile

3939

Second derivativesSecond derivatives

an undesirable featurean undesirable feature

produces 2 values for every edge in an produces 2 values for every edge in an

imageimage

zero-crossing propertyzero-crossing property

an imaginary straight line joining the an imaginary straight line joining the

extreme positive and negative values of extreme positive and negative values of

the second derivative would cross zero the second derivative would cross zero

near the midpoint of the edgenear the midpoint of the edge

quite useful for locating the centers of quite useful for locating the centers of

thick edgesthick edges

4040

Basic idea of edge detectionBasic idea of edge detection

A profile is defined perpendicularly to A profile is defined perpendicularly to

the edge direction and the results are the edge direction and the results are

interpretedinterpreted

The magnitude of the first derivative is The magnitude of the first derivative is

used to detect an edge (if a point is on a used to detect an edge (if a point is on a

ramp)ramp)

The sign of the second derivative can The sign of the second derivative can

determine whether an edge pixel is on the determine whether an edge pixel is on the

dark or light side of an edgedark or light side of an edge

4141

Review of First DerivateReview of First Derivate

Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

Sobel operators 3Sobel operators 333

6 5 8 5x yG z z G z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

9 5 8 6x yG z z G z z 1 0 0 1

0 1 1 0

1 0 0 1

0 1 1 0

7 8 9 1 2 3

3 6 9 1 4 7

2 2

2 2

x

y

G z z z z z z

G z z z z z z

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

x yf G G

4242

Edge direction and strengthEdge direction and strength

Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

2 2x yf G G

4343

Gradient MasksGradient Masks

1 0 0 1

0 1 1 0

Roberts

1 0 0 1

0 1 1 0

Roberts

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

4444

Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

2020

Optimal ThresholdingOptimal Thresholding

Minimum errorMinimum error Differentiating ( Differentiating ( 微分微分 ) E(T) with respect to T (usi) E(T) with respect to T (usi

ng Leibnizrsquos rule) and equating the result to 0ng Leibnizrsquos rule) and equating the result to 0

find T which makesfind T which makes

2121

Optimal ThresholdingOptimal Thresholding

Minimum errorMinimum error

Specially if Specially if PP11 = P = P22 then the optimum then the optimum

threshold is where the curve pthreshold is where the curve p11(z) and (z) and

pp22(z) intersect(z) intersect

2222

Optimal ThresholdingOptimal Thresholding

For exampleFor example

Let Let PDF=Gaussian densityPDF=Gaussian density p p11(z) and (z) and

pp22(z)(z)

where μwhere μ11 and σ and σ1122 are the mean and are the mean and

variance of the Gaussian density of one variance of the Gaussian density of one

objectobject

μμ22 and σ and σ2222 are the mean and variance of are the mean and variance of

the Gaussian density of the other objectthe Gaussian density of the other object

2323

Optimal ThresholdingOptimal Thresholding

Quadratic equation (Quadratic equation (二次方程二次方程 ))

2424

Problems of ThresholdingProblems of Thresholding

Original imageOriginal image Thresholded imageThresholded image

2525

Problems of ThresholdingProblems of Thresholding

(a)(a) Exact threshold Exact threshold

segmentationsegmentation

(b)(b) Threshold too lowThreshold too low

(c)(c) Threshold too Threshold too

highhigh

2626

72 Point Detection72 Point Detection

a point has been detected at the a point has been detected at the

location on which the mark is location on which the mark is

centered ifcentered if

|R|geT|R|geT

where T is a nonnegative thresholdwhere T is a nonnegative threshold

R is the sum of products of the R is the sum of products of the

coefficients with the gray levels contained coefficients with the gray levels contained

in the region encompassed by the markin the region encompassed by the mark

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

2727

72 Point Detection72 Point Detection

Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

2828

ExampleExample

2929

73 Line Detection73 Line Detection

Horizontal mask will result with max Horizontal mask will result with max

response when a line passed through the response when a line passed through the

middle row of the mask with a constant middle row of the mask with a constant

backgroundbackground

the similar idea is used with other masksthe similar idea is used with other masks

Note the preferred direction of each mask Note the preferred direction of each mask

is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

than other possible directionsthan other possible directions

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

3030

Idea 1 of Line DetectionIdea 1 of Line Detection

Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

if at a certain point in the imageif at a certain point in the image

|Ri||Ri|gtgt|Rj||Rj|

for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

3131

Idea 2 of Line DetectionIdea 2 of Line Detection

Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

3232

ExampleExample

3333

74 Edge-based 74 Edge-based SegmentationSegmentation

Edge-based segmentations rely on edges found in an image by edge detecting operators

these edges mark image locations of discontinuities in gray level

Edge detection is the most common approach for detecting meaningful discontinuities in gray level

There are a large group of methods based on information about edges in the image

3434

What is edgeWhat is edge

Edge is where change occurs Change is measured by derivative in 1D

―Biggest change derivative has maximum magnitude

Or 2nd derivative is zero we discuss approaches for implementing

―first-order derivative (Gradient operator)

―second-order derivative (Laplacian operator)

―we have introduced both derivatives in chapter 3

―Here we will talk only about their properties for edge detection

3535

What is edgeWhat is edge

In other wordsIn other words an edge is a set of an edge is a set of

connected pixelsconnected pixels

that lie on the boundary between two that lie on the boundary between two

regions with relatively distinct gray-level regions with relatively distinct gray-level

propertiesproperties

Note edge vs boundaryNote edge vs boundary

―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

―whereas a region boundary owing to whereas a region boundary owing to

the way it is defined is a more global the way it is defined is a more global

ideaidea

3636

Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

because of because of

optics optics

sampling sampling

image image

acquisition acquisition

imperfectionimperfection

3737

Thick and Thin EdgeThick and Thin Edge

The slope of the ramp is inversely The slope of the ramp is inversely

proportional to the degree of blurring in the proportional to the degree of blurring in the

edgeedge

Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

pathpath

Instead an edge point now is any point Instead an edge point now is any point

contained in the ramp and contained in the ramp and an edge would an edge would

then be a set of such points that are then be a set of such points that are

connectedconnected

The thickness is determined by the length of the The thickness is determined by the length of the

rampramp

The length is determined by the slope which is in The length is determined by the slope which is in

turn determined by the degree of blurringturn determined by the degree of blurring

Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

edges tend to be thinedges tend to be thin

3838

First and Second derivatives (First and Second derivatives ( 导数导数 ))

the signs of the the signs of the

derivatives would be derivatives would be

reversed for an edge reversed for an edge

that transitions from that transitions from

light to darklight to dark

First First derivatderivatee

SeconSecond d derivatderivatee

Gray-Gray-level level profileprofile

3939

Second derivativesSecond derivatives

an undesirable featurean undesirable feature

produces 2 values for every edge in an produces 2 values for every edge in an

imageimage

zero-crossing propertyzero-crossing property

an imaginary straight line joining the an imaginary straight line joining the

extreme positive and negative values of extreme positive and negative values of

the second derivative would cross zero the second derivative would cross zero

near the midpoint of the edgenear the midpoint of the edge

quite useful for locating the centers of quite useful for locating the centers of

thick edgesthick edges

4040

Basic idea of edge detectionBasic idea of edge detection

A profile is defined perpendicularly to A profile is defined perpendicularly to

the edge direction and the results are the edge direction and the results are

interpretedinterpreted

The magnitude of the first derivative is The magnitude of the first derivative is

used to detect an edge (if a point is on a used to detect an edge (if a point is on a

ramp)ramp)

The sign of the second derivative can The sign of the second derivative can

determine whether an edge pixel is on the determine whether an edge pixel is on the

dark or light side of an edgedark or light side of an edge

4141

Review of First DerivateReview of First Derivate

Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

Sobel operators 3Sobel operators 333

6 5 8 5x yG z z G z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

9 5 8 6x yG z z G z z 1 0 0 1

0 1 1 0

1 0 0 1

0 1 1 0

7 8 9 1 2 3

3 6 9 1 4 7

2 2

2 2

x

y

G z z z z z z

G z z z z z z

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

x yf G G

4242

Edge direction and strengthEdge direction and strength

Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

2 2x yf G G

4343

Gradient MasksGradient Masks

1 0 0 1

0 1 1 0

Roberts

1 0 0 1

0 1 1 0

Roberts

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

4444

Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

2121

Optimal ThresholdingOptimal Thresholding

Minimum errorMinimum error

Specially if Specially if PP11 = P = P22 then the optimum then the optimum

threshold is where the curve pthreshold is where the curve p11(z) and (z) and

pp22(z) intersect(z) intersect

2222

Optimal ThresholdingOptimal Thresholding

For exampleFor example

Let Let PDF=Gaussian densityPDF=Gaussian density p p11(z) and (z) and

pp22(z)(z)

where μwhere μ11 and σ and σ1122 are the mean and are the mean and

variance of the Gaussian density of one variance of the Gaussian density of one

objectobject

μμ22 and σ and σ2222 are the mean and variance of are the mean and variance of

the Gaussian density of the other objectthe Gaussian density of the other object

2323

Optimal ThresholdingOptimal Thresholding

Quadratic equation (Quadratic equation (二次方程二次方程 ))

2424

Problems of ThresholdingProblems of Thresholding

Original imageOriginal image Thresholded imageThresholded image

2525

Problems of ThresholdingProblems of Thresholding

(a)(a) Exact threshold Exact threshold

segmentationsegmentation

(b)(b) Threshold too lowThreshold too low

(c)(c) Threshold too Threshold too

highhigh

2626

72 Point Detection72 Point Detection

a point has been detected at the a point has been detected at the

location on which the mark is location on which the mark is

centered ifcentered if

|R|geT|R|geT

where T is a nonnegative thresholdwhere T is a nonnegative threshold

R is the sum of products of the R is the sum of products of the

coefficients with the gray levels contained coefficients with the gray levels contained

in the region encompassed by the markin the region encompassed by the mark

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

2727

72 Point Detection72 Point Detection

Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

2828

ExampleExample

2929

73 Line Detection73 Line Detection

Horizontal mask will result with max Horizontal mask will result with max

response when a line passed through the response when a line passed through the

middle row of the mask with a constant middle row of the mask with a constant

backgroundbackground

the similar idea is used with other masksthe similar idea is used with other masks

Note the preferred direction of each mask Note the preferred direction of each mask

is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

than other possible directionsthan other possible directions

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

3030

Idea 1 of Line DetectionIdea 1 of Line Detection

Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

if at a certain point in the imageif at a certain point in the image

|Ri||Ri|gtgt|Rj||Rj|

for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

3131

Idea 2 of Line DetectionIdea 2 of Line Detection

Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

3232

ExampleExample

3333

74 Edge-based 74 Edge-based SegmentationSegmentation

Edge-based segmentations rely on edges found in an image by edge detecting operators

these edges mark image locations of discontinuities in gray level

Edge detection is the most common approach for detecting meaningful discontinuities in gray level

There are a large group of methods based on information about edges in the image

3434

What is edgeWhat is edge

Edge is where change occurs Change is measured by derivative in 1D

―Biggest change derivative has maximum magnitude

Or 2nd derivative is zero we discuss approaches for implementing

―first-order derivative (Gradient operator)

―second-order derivative (Laplacian operator)

―we have introduced both derivatives in chapter 3

―Here we will talk only about their properties for edge detection

3535

What is edgeWhat is edge

In other wordsIn other words an edge is a set of an edge is a set of

connected pixelsconnected pixels

that lie on the boundary between two that lie on the boundary between two

regions with relatively distinct gray-level regions with relatively distinct gray-level

propertiesproperties

Note edge vs boundaryNote edge vs boundary

―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

―whereas a region boundary owing to whereas a region boundary owing to

the way it is defined is a more global the way it is defined is a more global

ideaidea

3636

Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

because of because of

optics optics

sampling sampling

image image

acquisition acquisition

imperfectionimperfection

3737

Thick and Thin EdgeThick and Thin Edge

The slope of the ramp is inversely The slope of the ramp is inversely

proportional to the degree of blurring in the proportional to the degree of blurring in the

edgeedge

Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

pathpath

Instead an edge point now is any point Instead an edge point now is any point

contained in the ramp and contained in the ramp and an edge would an edge would

then be a set of such points that are then be a set of such points that are

connectedconnected

The thickness is determined by the length of the The thickness is determined by the length of the

rampramp

The length is determined by the slope which is in The length is determined by the slope which is in

turn determined by the degree of blurringturn determined by the degree of blurring

Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

edges tend to be thinedges tend to be thin

3838

First and Second derivatives (First and Second derivatives ( 导数导数 ))

the signs of the the signs of the

derivatives would be derivatives would be

reversed for an edge reversed for an edge

that transitions from that transitions from

light to darklight to dark

First First derivatderivatee

SeconSecond d derivatderivatee

Gray-Gray-level level profileprofile

3939

Second derivativesSecond derivatives

an undesirable featurean undesirable feature

produces 2 values for every edge in an produces 2 values for every edge in an

imageimage

zero-crossing propertyzero-crossing property

an imaginary straight line joining the an imaginary straight line joining the

extreme positive and negative values of extreme positive and negative values of

the second derivative would cross zero the second derivative would cross zero

near the midpoint of the edgenear the midpoint of the edge

quite useful for locating the centers of quite useful for locating the centers of

thick edgesthick edges

4040

Basic idea of edge detectionBasic idea of edge detection

A profile is defined perpendicularly to A profile is defined perpendicularly to

the edge direction and the results are the edge direction and the results are

interpretedinterpreted

The magnitude of the first derivative is The magnitude of the first derivative is

used to detect an edge (if a point is on a used to detect an edge (if a point is on a

ramp)ramp)

The sign of the second derivative can The sign of the second derivative can

determine whether an edge pixel is on the determine whether an edge pixel is on the

dark or light side of an edgedark or light side of an edge

4141

Review of First DerivateReview of First Derivate

Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

Sobel operators 3Sobel operators 333

6 5 8 5x yG z z G z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

9 5 8 6x yG z z G z z 1 0 0 1

0 1 1 0

1 0 0 1

0 1 1 0

7 8 9 1 2 3

3 6 9 1 4 7

2 2

2 2

x

y

G z z z z z z

G z z z z z z

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

x yf G G

4242

Edge direction and strengthEdge direction and strength

Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

2 2x yf G G

4343

Gradient MasksGradient Masks

1 0 0 1

0 1 1 0

Roberts

1 0 0 1

0 1 1 0

Roberts

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

4444

Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

2222

Optimal ThresholdingOptimal Thresholding

For exampleFor example

Let Let PDF=Gaussian densityPDF=Gaussian density p p11(z) and (z) and

pp22(z)(z)

where μwhere μ11 and σ and σ1122 are the mean and are the mean and

variance of the Gaussian density of one variance of the Gaussian density of one

objectobject

μμ22 and σ and σ2222 are the mean and variance of are the mean and variance of

the Gaussian density of the other objectthe Gaussian density of the other object

2323

Optimal ThresholdingOptimal Thresholding

Quadratic equation (Quadratic equation (二次方程二次方程 ))

2424

Problems of ThresholdingProblems of Thresholding

Original imageOriginal image Thresholded imageThresholded image

2525

Problems of ThresholdingProblems of Thresholding

(a)(a) Exact threshold Exact threshold

segmentationsegmentation

(b)(b) Threshold too lowThreshold too low

(c)(c) Threshold too Threshold too

highhigh

2626

72 Point Detection72 Point Detection

a point has been detected at the a point has been detected at the

location on which the mark is location on which the mark is

centered ifcentered if

|R|geT|R|geT

where T is a nonnegative thresholdwhere T is a nonnegative threshold

R is the sum of products of the R is the sum of products of the

coefficients with the gray levels contained coefficients with the gray levels contained

in the region encompassed by the markin the region encompassed by the mark

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

2727

72 Point Detection72 Point Detection

Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

2828

ExampleExample

2929

73 Line Detection73 Line Detection

Horizontal mask will result with max Horizontal mask will result with max

response when a line passed through the response when a line passed through the

middle row of the mask with a constant middle row of the mask with a constant

backgroundbackground

the similar idea is used with other masksthe similar idea is used with other masks

Note the preferred direction of each mask Note the preferred direction of each mask

is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

than other possible directionsthan other possible directions

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

3030

Idea 1 of Line DetectionIdea 1 of Line Detection

Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

if at a certain point in the imageif at a certain point in the image

|Ri||Ri|gtgt|Rj||Rj|

for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

3131

Idea 2 of Line DetectionIdea 2 of Line Detection

Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

3232

ExampleExample

3333

74 Edge-based 74 Edge-based SegmentationSegmentation

Edge-based segmentations rely on edges found in an image by edge detecting operators

these edges mark image locations of discontinuities in gray level

Edge detection is the most common approach for detecting meaningful discontinuities in gray level

There are a large group of methods based on information about edges in the image

3434

What is edgeWhat is edge

Edge is where change occurs Change is measured by derivative in 1D

―Biggest change derivative has maximum magnitude

Or 2nd derivative is zero we discuss approaches for implementing

―first-order derivative (Gradient operator)

―second-order derivative (Laplacian operator)

―we have introduced both derivatives in chapter 3

―Here we will talk only about their properties for edge detection

3535

What is edgeWhat is edge

In other wordsIn other words an edge is a set of an edge is a set of

connected pixelsconnected pixels

that lie on the boundary between two that lie on the boundary between two

regions with relatively distinct gray-level regions with relatively distinct gray-level

propertiesproperties

Note edge vs boundaryNote edge vs boundary

―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

―whereas a region boundary owing to whereas a region boundary owing to

the way it is defined is a more global the way it is defined is a more global

ideaidea

3636

Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

because of because of

optics optics

sampling sampling

image image

acquisition acquisition

imperfectionimperfection

3737

Thick and Thin EdgeThick and Thin Edge

The slope of the ramp is inversely The slope of the ramp is inversely

proportional to the degree of blurring in the proportional to the degree of blurring in the

edgeedge

Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

pathpath

Instead an edge point now is any point Instead an edge point now is any point

contained in the ramp and contained in the ramp and an edge would an edge would

then be a set of such points that are then be a set of such points that are

connectedconnected

The thickness is determined by the length of the The thickness is determined by the length of the

rampramp

The length is determined by the slope which is in The length is determined by the slope which is in

turn determined by the degree of blurringturn determined by the degree of blurring

Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

edges tend to be thinedges tend to be thin

3838

First and Second derivatives (First and Second derivatives ( 导数导数 ))

the signs of the the signs of the

derivatives would be derivatives would be

reversed for an edge reversed for an edge

that transitions from that transitions from

light to darklight to dark

First First derivatderivatee

SeconSecond d derivatderivatee

Gray-Gray-level level profileprofile

3939

Second derivativesSecond derivatives

an undesirable featurean undesirable feature

produces 2 values for every edge in an produces 2 values for every edge in an

imageimage

zero-crossing propertyzero-crossing property

an imaginary straight line joining the an imaginary straight line joining the

extreme positive and negative values of extreme positive and negative values of

the second derivative would cross zero the second derivative would cross zero

near the midpoint of the edgenear the midpoint of the edge

quite useful for locating the centers of quite useful for locating the centers of

thick edgesthick edges

4040

Basic idea of edge detectionBasic idea of edge detection

A profile is defined perpendicularly to A profile is defined perpendicularly to

the edge direction and the results are the edge direction and the results are

interpretedinterpreted

The magnitude of the first derivative is The magnitude of the first derivative is

used to detect an edge (if a point is on a used to detect an edge (if a point is on a

ramp)ramp)

The sign of the second derivative can The sign of the second derivative can

determine whether an edge pixel is on the determine whether an edge pixel is on the

dark or light side of an edgedark or light side of an edge

4141

Review of First DerivateReview of First Derivate

Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

Sobel operators 3Sobel operators 333

6 5 8 5x yG z z G z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

9 5 8 6x yG z z G z z 1 0 0 1

0 1 1 0

1 0 0 1

0 1 1 0

7 8 9 1 2 3

3 6 9 1 4 7

2 2

2 2

x

y

G z z z z z z

G z z z z z z

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

x yf G G

4242

Edge direction and strengthEdge direction and strength

Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

2 2x yf G G

4343

Gradient MasksGradient Masks

1 0 0 1

0 1 1 0

Roberts

1 0 0 1

0 1 1 0

Roberts

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

4444

Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

2323

Optimal ThresholdingOptimal Thresholding

Quadratic equation (Quadratic equation (二次方程二次方程 ))

2424

Problems of ThresholdingProblems of Thresholding

Original imageOriginal image Thresholded imageThresholded image

2525

Problems of ThresholdingProblems of Thresholding

(a)(a) Exact threshold Exact threshold

segmentationsegmentation

(b)(b) Threshold too lowThreshold too low

(c)(c) Threshold too Threshold too

highhigh

2626

72 Point Detection72 Point Detection

a point has been detected at the a point has been detected at the

location on which the mark is location on which the mark is

centered ifcentered if

|R|geT|R|geT

where T is a nonnegative thresholdwhere T is a nonnegative threshold

R is the sum of products of the R is the sum of products of the

coefficients with the gray levels contained coefficients with the gray levels contained

in the region encompassed by the markin the region encompassed by the mark

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

2727

72 Point Detection72 Point Detection

Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

2828

ExampleExample

2929

73 Line Detection73 Line Detection

Horizontal mask will result with max Horizontal mask will result with max

response when a line passed through the response when a line passed through the

middle row of the mask with a constant middle row of the mask with a constant

backgroundbackground

the similar idea is used with other masksthe similar idea is used with other masks

Note the preferred direction of each mask Note the preferred direction of each mask

is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

than other possible directionsthan other possible directions

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

3030

Idea 1 of Line DetectionIdea 1 of Line Detection

Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

if at a certain point in the imageif at a certain point in the image

|Ri||Ri|gtgt|Rj||Rj|

for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

3131

Idea 2 of Line DetectionIdea 2 of Line Detection

Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

3232

ExampleExample

3333

74 Edge-based 74 Edge-based SegmentationSegmentation

Edge-based segmentations rely on edges found in an image by edge detecting operators

these edges mark image locations of discontinuities in gray level

Edge detection is the most common approach for detecting meaningful discontinuities in gray level

There are a large group of methods based on information about edges in the image

3434

What is edgeWhat is edge

Edge is where change occurs Change is measured by derivative in 1D

―Biggest change derivative has maximum magnitude

Or 2nd derivative is zero we discuss approaches for implementing

―first-order derivative (Gradient operator)

―second-order derivative (Laplacian operator)

―we have introduced both derivatives in chapter 3

―Here we will talk only about their properties for edge detection

3535

What is edgeWhat is edge

In other wordsIn other words an edge is a set of an edge is a set of

connected pixelsconnected pixels

that lie on the boundary between two that lie on the boundary between two

regions with relatively distinct gray-level regions with relatively distinct gray-level

propertiesproperties

Note edge vs boundaryNote edge vs boundary

―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

―whereas a region boundary owing to whereas a region boundary owing to

the way it is defined is a more global the way it is defined is a more global

ideaidea

3636

Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

because of because of

optics optics

sampling sampling

image image

acquisition acquisition

imperfectionimperfection

3737

Thick and Thin EdgeThick and Thin Edge

The slope of the ramp is inversely The slope of the ramp is inversely

proportional to the degree of blurring in the proportional to the degree of blurring in the

edgeedge

Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

pathpath

Instead an edge point now is any point Instead an edge point now is any point

contained in the ramp and contained in the ramp and an edge would an edge would

then be a set of such points that are then be a set of such points that are

connectedconnected

The thickness is determined by the length of the The thickness is determined by the length of the

rampramp

The length is determined by the slope which is in The length is determined by the slope which is in

turn determined by the degree of blurringturn determined by the degree of blurring

Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

edges tend to be thinedges tend to be thin

3838

First and Second derivatives (First and Second derivatives ( 导数导数 ))

the signs of the the signs of the

derivatives would be derivatives would be

reversed for an edge reversed for an edge

that transitions from that transitions from

light to darklight to dark

First First derivatderivatee

SeconSecond d derivatderivatee

Gray-Gray-level level profileprofile

3939

Second derivativesSecond derivatives

an undesirable featurean undesirable feature

produces 2 values for every edge in an produces 2 values for every edge in an

imageimage

zero-crossing propertyzero-crossing property

an imaginary straight line joining the an imaginary straight line joining the

extreme positive and negative values of extreme positive and negative values of

the second derivative would cross zero the second derivative would cross zero

near the midpoint of the edgenear the midpoint of the edge

quite useful for locating the centers of quite useful for locating the centers of

thick edgesthick edges

4040

Basic idea of edge detectionBasic idea of edge detection

A profile is defined perpendicularly to A profile is defined perpendicularly to

the edge direction and the results are the edge direction and the results are

interpretedinterpreted

The magnitude of the first derivative is The magnitude of the first derivative is

used to detect an edge (if a point is on a used to detect an edge (if a point is on a

ramp)ramp)

The sign of the second derivative can The sign of the second derivative can

determine whether an edge pixel is on the determine whether an edge pixel is on the

dark or light side of an edgedark or light side of an edge

4141

Review of First DerivateReview of First Derivate

Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

Sobel operators 3Sobel operators 333

6 5 8 5x yG z z G z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

9 5 8 6x yG z z G z z 1 0 0 1

0 1 1 0

1 0 0 1

0 1 1 0

7 8 9 1 2 3

3 6 9 1 4 7

2 2

2 2

x

y

G z z z z z z

G z z z z z z

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

x yf G G

4242

Edge direction and strengthEdge direction and strength

Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

2 2x yf G G

4343

Gradient MasksGradient Masks

1 0 0 1

0 1 1 0

Roberts

1 0 0 1

0 1 1 0

Roberts

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

4444

Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

2424

Problems of ThresholdingProblems of Thresholding

Original imageOriginal image Thresholded imageThresholded image

2525

Problems of ThresholdingProblems of Thresholding

(a)(a) Exact threshold Exact threshold

segmentationsegmentation

(b)(b) Threshold too lowThreshold too low

(c)(c) Threshold too Threshold too

highhigh

2626

72 Point Detection72 Point Detection

a point has been detected at the a point has been detected at the

location on which the mark is location on which the mark is

centered ifcentered if

|R|geT|R|geT

where T is a nonnegative thresholdwhere T is a nonnegative threshold

R is the sum of products of the R is the sum of products of the

coefficients with the gray levels contained coefficients with the gray levels contained

in the region encompassed by the markin the region encompassed by the mark

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

2727

72 Point Detection72 Point Detection

Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

2828

ExampleExample

2929

73 Line Detection73 Line Detection

Horizontal mask will result with max Horizontal mask will result with max

response when a line passed through the response when a line passed through the

middle row of the mask with a constant middle row of the mask with a constant

backgroundbackground

the similar idea is used with other masksthe similar idea is used with other masks

Note the preferred direction of each mask Note the preferred direction of each mask

is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

than other possible directionsthan other possible directions

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

3030

Idea 1 of Line DetectionIdea 1 of Line Detection

Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

if at a certain point in the imageif at a certain point in the image

|Ri||Ri|gtgt|Rj||Rj|

for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

3131

Idea 2 of Line DetectionIdea 2 of Line Detection

Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

3232

ExampleExample

3333

74 Edge-based 74 Edge-based SegmentationSegmentation

Edge-based segmentations rely on edges found in an image by edge detecting operators

these edges mark image locations of discontinuities in gray level

Edge detection is the most common approach for detecting meaningful discontinuities in gray level

There are a large group of methods based on information about edges in the image

3434

What is edgeWhat is edge

Edge is where change occurs Change is measured by derivative in 1D

―Biggest change derivative has maximum magnitude

Or 2nd derivative is zero we discuss approaches for implementing

―first-order derivative (Gradient operator)

―second-order derivative (Laplacian operator)

―we have introduced both derivatives in chapter 3

―Here we will talk only about their properties for edge detection

3535

What is edgeWhat is edge

In other wordsIn other words an edge is a set of an edge is a set of

connected pixelsconnected pixels

that lie on the boundary between two that lie on the boundary between two

regions with relatively distinct gray-level regions with relatively distinct gray-level

propertiesproperties

Note edge vs boundaryNote edge vs boundary

―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

―whereas a region boundary owing to whereas a region boundary owing to

the way it is defined is a more global the way it is defined is a more global

ideaidea

3636

Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

because of because of

optics optics

sampling sampling

image image

acquisition acquisition

imperfectionimperfection

3737

Thick and Thin EdgeThick and Thin Edge

The slope of the ramp is inversely The slope of the ramp is inversely

proportional to the degree of blurring in the proportional to the degree of blurring in the

edgeedge

Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

pathpath

Instead an edge point now is any point Instead an edge point now is any point

contained in the ramp and contained in the ramp and an edge would an edge would

then be a set of such points that are then be a set of such points that are

connectedconnected

The thickness is determined by the length of the The thickness is determined by the length of the

rampramp

The length is determined by the slope which is in The length is determined by the slope which is in

turn determined by the degree of blurringturn determined by the degree of blurring

Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

edges tend to be thinedges tend to be thin

3838

First and Second derivatives (First and Second derivatives ( 导数导数 ))

the signs of the the signs of the

derivatives would be derivatives would be

reversed for an edge reversed for an edge

that transitions from that transitions from

light to darklight to dark

First First derivatderivatee

SeconSecond d derivatderivatee

Gray-Gray-level level profileprofile

3939

Second derivativesSecond derivatives

an undesirable featurean undesirable feature

produces 2 values for every edge in an produces 2 values for every edge in an

imageimage

zero-crossing propertyzero-crossing property

an imaginary straight line joining the an imaginary straight line joining the

extreme positive and negative values of extreme positive and negative values of

the second derivative would cross zero the second derivative would cross zero

near the midpoint of the edgenear the midpoint of the edge

quite useful for locating the centers of quite useful for locating the centers of

thick edgesthick edges

4040

Basic idea of edge detectionBasic idea of edge detection

A profile is defined perpendicularly to A profile is defined perpendicularly to

the edge direction and the results are the edge direction and the results are

interpretedinterpreted

The magnitude of the first derivative is The magnitude of the first derivative is

used to detect an edge (if a point is on a used to detect an edge (if a point is on a

ramp)ramp)

The sign of the second derivative can The sign of the second derivative can

determine whether an edge pixel is on the determine whether an edge pixel is on the

dark or light side of an edgedark or light side of an edge

4141

Review of First DerivateReview of First Derivate

Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

Sobel operators 3Sobel operators 333

6 5 8 5x yG z z G z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

9 5 8 6x yG z z G z z 1 0 0 1

0 1 1 0

1 0 0 1

0 1 1 0

7 8 9 1 2 3

3 6 9 1 4 7

2 2

2 2

x

y

G z z z z z z

G z z z z z z

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

x yf G G

4242

Edge direction and strengthEdge direction and strength

Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

2 2x yf G G

4343

Gradient MasksGradient Masks

1 0 0 1

0 1 1 0

Roberts

1 0 0 1

0 1 1 0

Roberts

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

4444

Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

2525

Problems of ThresholdingProblems of Thresholding

(a)(a) Exact threshold Exact threshold

segmentationsegmentation

(b)(b) Threshold too lowThreshold too low

(c)(c) Threshold too Threshold too

highhigh

2626

72 Point Detection72 Point Detection

a point has been detected at the a point has been detected at the

location on which the mark is location on which the mark is

centered ifcentered if

|R|geT|R|geT

where T is a nonnegative thresholdwhere T is a nonnegative threshold

R is the sum of products of the R is the sum of products of the

coefficients with the gray levels contained coefficients with the gray levels contained

in the region encompassed by the markin the region encompassed by the mark

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

2727

72 Point Detection72 Point Detection

Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

2828

ExampleExample

2929

73 Line Detection73 Line Detection

Horizontal mask will result with max Horizontal mask will result with max

response when a line passed through the response when a line passed through the

middle row of the mask with a constant middle row of the mask with a constant

backgroundbackground

the similar idea is used with other masksthe similar idea is used with other masks

Note the preferred direction of each mask Note the preferred direction of each mask

is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

than other possible directionsthan other possible directions

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

3030

Idea 1 of Line DetectionIdea 1 of Line Detection

Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

if at a certain point in the imageif at a certain point in the image

|Ri||Ri|gtgt|Rj||Rj|

for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

3131

Idea 2 of Line DetectionIdea 2 of Line Detection

Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

3232

ExampleExample

3333

74 Edge-based 74 Edge-based SegmentationSegmentation

Edge-based segmentations rely on edges found in an image by edge detecting operators

these edges mark image locations of discontinuities in gray level

Edge detection is the most common approach for detecting meaningful discontinuities in gray level

There are a large group of methods based on information about edges in the image

3434

What is edgeWhat is edge

Edge is where change occurs Change is measured by derivative in 1D

―Biggest change derivative has maximum magnitude

Or 2nd derivative is zero we discuss approaches for implementing

―first-order derivative (Gradient operator)

―second-order derivative (Laplacian operator)

―we have introduced both derivatives in chapter 3

―Here we will talk only about their properties for edge detection

3535

What is edgeWhat is edge

In other wordsIn other words an edge is a set of an edge is a set of

connected pixelsconnected pixels

that lie on the boundary between two that lie on the boundary between two

regions with relatively distinct gray-level regions with relatively distinct gray-level

propertiesproperties

Note edge vs boundaryNote edge vs boundary

―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

―whereas a region boundary owing to whereas a region boundary owing to

the way it is defined is a more global the way it is defined is a more global

ideaidea

3636

Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

because of because of

optics optics

sampling sampling

image image

acquisition acquisition

imperfectionimperfection

3737

Thick and Thin EdgeThick and Thin Edge

The slope of the ramp is inversely The slope of the ramp is inversely

proportional to the degree of blurring in the proportional to the degree of blurring in the

edgeedge

Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

pathpath

Instead an edge point now is any point Instead an edge point now is any point

contained in the ramp and contained in the ramp and an edge would an edge would

then be a set of such points that are then be a set of such points that are

connectedconnected

The thickness is determined by the length of the The thickness is determined by the length of the

rampramp

The length is determined by the slope which is in The length is determined by the slope which is in

turn determined by the degree of blurringturn determined by the degree of blurring

Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

edges tend to be thinedges tend to be thin

3838

First and Second derivatives (First and Second derivatives ( 导数导数 ))

the signs of the the signs of the

derivatives would be derivatives would be

reversed for an edge reversed for an edge

that transitions from that transitions from

light to darklight to dark

First First derivatderivatee

SeconSecond d derivatderivatee

Gray-Gray-level level profileprofile

3939

Second derivativesSecond derivatives

an undesirable featurean undesirable feature

produces 2 values for every edge in an produces 2 values for every edge in an

imageimage

zero-crossing propertyzero-crossing property

an imaginary straight line joining the an imaginary straight line joining the

extreme positive and negative values of extreme positive and negative values of

the second derivative would cross zero the second derivative would cross zero

near the midpoint of the edgenear the midpoint of the edge

quite useful for locating the centers of quite useful for locating the centers of

thick edgesthick edges

4040

Basic idea of edge detectionBasic idea of edge detection

A profile is defined perpendicularly to A profile is defined perpendicularly to

the edge direction and the results are the edge direction and the results are

interpretedinterpreted

The magnitude of the first derivative is The magnitude of the first derivative is

used to detect an edge (if a point is on a used to detect an edge (if a point is on a

ramp)ramp)

The sign of the second derivative can The sign of the second derivative can

determine whether an edge pixel is on the determine whether an edge pixel is on the

dark or light side of an edgedark or light side of an edge

4141

Review of First DerivateReview of First Derivate

Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

Sobel operators 3Sobel operators 333

6 5 8 5x yG z z G z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

9 5 8 6x yG z z G z z 1 0 0 1

0 1 1 0

1 0 0 1

0 1 1 0

7 8 9 1 2 3

3 6 9 1 4 7

2 2

2 2

x

y

G z z z z z z

G z z z z z z

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

x yf G G

4242

Edge direction and strengthEdge direction and strength

Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

2 2x yf G G

4343

Gradient MasksGradient Masks

1 0 0 1

0 1 1 0

Roberts

1 0 0 1

0 1 1 0

Roberts

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

4444

Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
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• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

2626

72 Point Detection72 Point Detection

a point has been detected at the a point has been detected at the

location on which the mark is location on which the mark is

centered ifcentered if

|R|geT|R|geT

where T is a nonnegative thresholdwhere T is a nonnegative threshold

R is the sum of products of the R is the sum of products of the

coefficients with the gray levels contained coefficients with the gray levels contained

in the region encompassed by the markin the region encompassed by the mark

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

2727

72 Point Detection72 Point Detection

Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

2828

ExampleExample

2929

73 Line Detection73 Line Detection

Horizontal mask will result with max Horizontal mask will result with max

response when a line passed through the response when a line passed through the

middle row of the mask with a constant middle row of the mask with a constant

backgroundbackground

the similar idea is used with other masksthe similar idea is used with other masks

Note the preferred direction of each mask Note the preferred direction of each mask

is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

than other possible directionsthan other possible directions

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

3030

Idea 1 of Line DetectionIdea 1 of Line Detection

Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

if at a certain point in the imageif at a certain point in the image

|Ri||Ri|gtgt|Rj||Rj|

for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

3131

Idea 2 of Line DetectionIdea 2 of Line Detection

Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

3232

ExampleExample

3333

74 Edge-based 74 Edge-based SegmentationSegmentation

Edge-based segmentations rely on edges found in an image by edge detecting operators

these edges mark image locations of discontinuities in gray level

Edge detection is the most common approach for detecting meaningful discontinuities in gray level

There are a large group of methods based on information about edges in the image

3434

What is edgeWhat is edge

Edge is where change occurs Change is measured by derivative in 1D

―Biggest change derivative has maximum magnitude

Or 2nd derivative is zero we discuss approaches for implementing

―first-order derivative (Gradient operator)

―second-order derivative (Laplacian operator)

―we have introduced both derivatives in chapter 3

―Here we will talk only about their properties for edge detection

3535

What is edgeWhat is edge

In other wordsIn other words an edge is a set of an edge is a set of

connected pixelsconnected pixels

that lie on the boundary between two that lie on the boundary between two

regions with relatively distinct gray-level regions with relatively distinct gray-level

propertiesproperties

Note edge vs boundaryNote edge vs boundary

―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

―whereas a region boundary owing to whereas a region boundary owing to

the way it is defined is a more global the way it is defined is a more global

ideaidea

3636

Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

because of because of

optics optics

sampling sampling

image image

acquisition acquisition

imperfectionimperfection

3737

Thick and Thin EdgeThick and Thin Edge

The slope of the ramp is inversely The slope of the ramp is inversely

proportional to the degree of blurring in the proportional to the degree of blurring in the

edgeedge

Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

pathpath

Instead an edge point now is any point Instead an edge point now is any point

contained in the ramp and contained in the ramp and an edge would an edge would

then be a set of such points that are then be a set of such points that are

connectedconnected

The thickness is determined by the length of the The thickness is determined by the length of the

rampramp

The length is determined by the slope which is in The length is determined by the slope which is in

turn determined by the degree of blurringturn determined by the degree of blurring

Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

edges tend to be thinedges tend to be thin

3838

First and Second derivatives (First and Second derivatives ( 导数导数 ))

the signs of the the signs of the

derivatives would be derivatives would be

reversed for an edge reversed for an edge

that transitions from that transitions from

light to darklight to dark

First First derivatderivatee

SeconSecond d derivatderivatee

Gray-Gray-level level profileprofile

3939

Second derivativesSecond derivatives

an undesirable featurean undesirable feature

produces 2 values for every edge in an produces 2 values for every edge in an

imageimage

zero-crossing propertyzero-crossing property

an imaginary straight line joining the an imaginary straight line joining the

extreme positive and negative values of extreme positive and negative values of

the second derivative would cross zero the second derivative would cross zero

near the midpoint of the edgenear the midpoint of the edge

quite useful for locating the centers of quite useful for locating the centers of

thick edgesthick edges

4040

Basic idea of edge detectionBasic idea of edge detection

A profile is defined perpendicularly to A profile is defined perpendicularly to

the edge direction and the results are the edge direction and the results are

interpretedinterpreted

The magnitude of the first derivative is The magnitude of the first derivative is

used to detect an edge (if a point is on a used to detect an edge (if a point is on a

ramp)ramp)

The sign of the second derivative can The sign of the second derivative can

determine whether an edge pixel is on the determine whether an edge pixel is on the

dark or light side of an edgedark or light side of an edge

4141

Review of First DerivateReview of First Derivate

Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

Sobel operators 3Sobel operators 333

6 5 8 5x yG z z G z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

9 5 8 6x yG z z G z z 1 0 0 1

0 1 1 0

1 0 0 1

0 1 1 0

7 8 9 1 2 3

3 6 9 1 4 7

2 2

2 2

x

y

G z z z z z z

G z z z z z z

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

x yf G G

4242

Edge direction and strengthEdge direction and strength

Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

2 2x yf G G

4343

Gradient MasksGradient Masks

1 0 0 1

0 1 1 0

Roberts

1 0 0 1

0 1 1 0

Roberts

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

4444

Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
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• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

2727

72 Point Detection72 Point Detection

Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

2828

ExampleExample

2929

73 Line Detection73 Line Detection

Horizontal mask will result with max Horizontal mask will result with max

response when a line passed through the response when a line passed through the

middle row of the mask with a constant middle row of the mask with a constant

backgroundbackground

the similar idea is used with other masksthe similar idea is used with other masks

Note the preferred direction of each mask Note the preferred direction of each mask

is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

than other possible directionsthan other possible directions

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

3030

Idea 1 of Line DetectionIdea 1 of Line Detection

Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

if at a certain point in the imageif at a certain point in the image

|Ri||Ri|gtgt|Rj||Rj|

for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

3131

Idea 2 of Line DetectionIdea 2 of Line Detection

Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

3232

ExampleExample

3333

74 Edge-based 74 Edge-based SegmentationSegmentation

Edge-based segmentations rely on edges found in an image by edge detecting operators

these edges mark image locations of discontinuities in gray level

Edge detection is the most common approach for detecting meaningful discontinuities in gray level

There are a large group of methods based on information about edges in the image

3434

What is edgeWhat is edge

Edge is where change occurs Change is measured by derivative in 1D

―Biggest change derivative has maximum magnitude

Or 2nd derivative is zero we discuss approaches for implementing

―first-order derivative (Gradient operator)

―second-order derivative (Laplacian operator)

―we have introduced both derivatives in chapter 3

―Here we will talk only about their properties for edge detection

3535

What is edgeWhat is edge

In other wordsIn other words an edge is a set of an edge is a set of

connected pixelsconnected pixels

that lie on the boundary between two that lie on the boundary between two

regions with relatively distinct gray-level regions with relatively distinct gray-level

propertiesproperties

Note edge vs boundaryNote edge vs boundary

―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

―whereas a region boundary owing to whereas a region boundary owing to

the way it is defined is a more global the way it is defined is a more global

ideaidea

3636

Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

because of because of

optics optics

sampling sampling

image image

acquisition acquisition

imperfectionimperfection

3737

Thick and Thin EdgeThick and Thin Edge

The slope of the ramp is inversely The slope of the ramp is inversely

proportional to the degree of blurring in the proportional to the degree of blurring in the

edgeedge

Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

pathpath

Instead an edge point now is any point Instead an edge point now is any point

contained in the ramp and contained in the ramp and an edge would an edge would

then be a set of such points that are then be a set of such points that are

connectedconnected

The thickness is determined by the length of the The thickness is determined by the length of the

rampramp

The length is determined by the slope which is in The length is determined by the slope which is in

turn determined by the degree of blurringturn determined by the degree of blurring

Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

edges tend to be thinedges tend to be thin

3838

First and Second derivatives (First and Second derivatives ( 导数导数 ))

the signs of the the signs of the

derivatives would be derivatives would be

reversed for an edge reversed for an edge

that transitions from that transitions from

light to darklight to dark

First First derivatderivatee

SeconSecond d derivatderivatee

Gray-Gray-level level profileprofile

3939

Second derivativesSecond derivatives

an undesirable featurean undesirable feature

produces 2 values for every edge in an produces 2 values for every edge in an

imageimage

zero-crossing propertyzero-crossing property

an imaginary straight line joining the an imaginary straight line joining the

extreme positive and negative values of extreme positive and negative values of

the second derivative would cross zero the second derivative would cross zero

near the midpoint of the edgenear the midpoint of the edge

quite useful for locating the centers of quite useful for locating the centers of

thick edgesthick edges

4040

Basic idea of edge detectionBasic idea of edge detection

A profile is defined perpendicularly to A profile is defined perpendicularly to

the edge direction and the results are the edge direction and the results are

interpretedinterpreted

The magnitude of the first derivative is The magnitude of the first derivative is

used to detect an edge (if a point is on a used to detect an edge (if a point is on a

ramp)ramp)

The sign of the second derivative can The sign of the second derivative can

determine whether an edge pixel is on the determine whether an edge pixel is on the

dark or light side of an edgedark or light side of an edge

4141

Review of First DerivateReview of First Derivate

Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

Sobel operators 3Sobel operators 333

6 5 8 5x yG z z G z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

9 5 8 6x yG z z G z z 1 0 0 1

0 1 1 0

1 0 0 1

0 1 1 0

7 8 9 1 2 3

3 6 9 1 4 7

2 2

2 2

x

y

G z z z z z z

G z z z z z z

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

x yf G G

4242

Edge direction and strengthEdge direction and strength

Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

2 2x yf G G

4343

Gradient MasksGradient Masks

1 0 0 1

0 1 1 0

Roberts

1 0 0 1

0 1 1 0

Roberts

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

4444

Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

2828

ExampleExample

2929

73 Line Detection73 Line Detection

Horizontal mask will result with max Horizontal mask will result with max

response when a line passed through the response when a line passed through the

middle row of the mask with a constant middle row of the mask with a constant

backgroundbackground

the similar idea is used with other masksthe similar idea is used with other masks

Note the preferred direction of each mask Note the preferred direction of each mask

is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

than other possible directionsthan other possible directions

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

3030

Idea 1 of Line DetectionIdea 1 of Line Detection

Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

if at a certain point in the imageif at a certain point in the image

|Ri||Ri|gtgt|Rj||Rj|

for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

3131

Idea 2 of Line DetectionIdea 2 of Line Detection

Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

3232

ExampleExample

3333

74 Edge-based 74 Edge-based SegmentationSegmentation

Edge-based segmentations rely on edges found in an image by edge detecting operators

these edges mark image locations of discontinuities in gray level

Edge detection is the most common approach for detecting meaningful discontinuities in gray level

There are a large group of methods based on information about edges in the image

3434

What is edgeWhat is edge

Edge is where change occurs Change is measured by derivative in 1D

―Biggest change derivative has maximum magnitude

Or 2nd derivative is zero we discuss approaches for implementing

―first-order derivative (Gradient operator)

―second-order derivative (Laplacian operator)

―we have introduced both derivatives in chapter 3

―Here we will talk only about their properties for edge detection

3535

What is edgeWhat is edge

In other wordsIn other words an edge is a set of an edge is a set of

connected pixelsconnected pixels

that lie on the boundary between two that lie on the boundary between two

regions with relatively distinct gray-level regions with relatively distinct gray-level

propertiesproperties

Note edge vs boundaryNote edge vs boundary

―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

―whereas a region boundary owing to whereas a region boundary owing to

the way it is defined is a more global the way it is defined is a more global

ideaidea

3636

Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

because of because of

optics optics

sampling sampling

image image

acquisition acquisition

imperfectionimperfection

3737

Thick and Thin EdgeThick and Thin Edge

The slope of the ramp is inversely The slope of the ramp is inversely

proportional to the degree of blurring in the proportional to the degree of blurring in the

edgeedge

Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

pathpath

Instead an edge point now is any point Instead an edge point now is any point

contained in the ramp and contained in the ramp and an edge would an edge would

then be a set of such points that are then be a set of such points that are

connectedconnected

The thickness is determined by the length of the The thickness is determined by the length of the

rampramp

The length is determined by the slope which is in The length is determined by the slope which is in

turn determined by the degree of blurringturn determined by the degree of blurring

Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

edges tend to be thinedges tend to be thin

3838

First and Second derivatives (First and Second derivatives ( 导数导数 ))

the signs of the the signs of the

derivatives would be derivatives would be

reversed for an edge reversed for an edge

that transitions from that transitions from

light to darklight to dark

First First derivatderivatee

SeconSecond d derivatderivatee

Gray-Gray-level level profileprofile

3939

Second derivativesSecond derivatives

an undesirable featurean undesirable feature

produces 2 values for every edge in an produces 2 values for every edge in an

imageimage

zero-crossing propertyzero-crossing property

an imaginary straight line joining the an imaginary straight line joining the

extreme positive and negative values of extreme positive and negative values of

the second derivative would cross zero the second derivative would cross zero

near the midpoint of the edgenear the midpoint of the edge

quite useful for locating the centers of quite useful for locating the centers of

thick edgesthick edges

4040

Basic idea of edge detectionBasic idea of edge detection

A profile is defined perpendicularly to A profile is defined perpendicularly to

the edge direction and the results are the edge direction and the results are

interpretedinterpreted

The magnitude of the first derivative is The magnitude of the first derivative is

used to detect an edge (if a point is on a used to detect an edge (if a point is on a

ramp)ramp)

The sign of the second derivative can The sign of the second derivative can

determine whether an edge pixel is on the determine whether an edge pixel is on the

dark or light side of an edgedark or light side of an edge

4141

Review of First DerivateReview of First Derivate

Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

Sobel operators 3Sobel operators 333

6 5 8 5x yG z z G z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

9 5 8 6x yG z z G z z 1 0 0 1

0 1 1 0

1 0 0 1

0 1 1 0

7 8 9 1 2 3

3 6 9 1 4 7

2 2

2 2

x

y

G z z z z z z

G z z z z z z

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

x yf G G

4242

Edge direction and strengthEdge direction and strength

Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

2 2x yf G G

4343

Gradient MasksGradient Masks

1 0 0 1

0 1 1 0

Roberts

1 0 0 1

0 1 1 0

Roberts

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

4444

Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

2929

73 Line Detection73 Line Detection

Horizontal mask will result with max Horizontal mask will result with max

response when a line passed through the response when a line passed through the

middle row of the mask with a constant middle row of the mask with a constant

backgroundbackground

the similar idea is used with other masksthe similar idea is used with other masks

Note the preferred direction of each mask Note the preferred direction of each mask

is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

than other possible directionsthan other possible directions

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

3030

Idea 1 of Line DetectionIdea 1 of Line Detection

Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

if at a certain point in the imageif at a certain point in the image

|Ri||Ri|gtgt|Rj||Rj|

for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

3131

Idea 2 of Line DetectionIdea 2 of Line Detection

Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

3232

ExampleExample

3333

74 Edge-based 74 Edge-based SegmentationSegmentation

Edge-based segmentations rely on edges found in an image by edge detecting operators

these edges mark image locations of discontinuities in gray level

Edge detection is the most common approach for detecting meaningful discontinuities in gray level

There are a large group of methods based on information about edges in the image

3434

What is edgeWhat is edge

Edge is where change occurs Change is measured by derivative in 1D

―Biggest change derivative has maximum magnitude

Or 2nd derivative is zero we discuss approaches for implementing

―first-order derivative (Gradient operator)

―second-order derivative (Laplacian operator)

―we have introduced both derivatives in chapter 3

―Here we will talk only about their properties for edge detection

3535

What is edgeWhat is edge

In other wordsIn other words an edge is a set of an edge is a set of

connected pixelsconnected pixels

that lie on the boundary between two that lie on the boundary between two

regions with relatively distinct gray-level regions with relatively distinct gray-level

propertiesproperties

Note edge vs boundaryNote edge vs boundary

―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

―whereas a region boundary owing to whereas a region boundary owing to

the way it is defined is a more global the way it is defined is a more global

ideaidea

3636

Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

because of because of

optics optics

sampling sampling

image image

acquisition acquisition

imperfectionimperfection

3737

Thick and Thin EdgeThick and Thin Edge

The slope of the ramp is inversely The slope of the ramp is inversely

proportional to the degree of blurring in the proportional to the degree of blurring in the

edgeedge

Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

pathpath

Instead an edge point now is any point Instead an edge point now is any point

contained in the ramp and contained in the ramp and an edge would an edge would

then be a set of such points that are then be a set of such points that are

connectedconnected

The thickness is determined by the length of the The thickness is determined by the length of the

rampramp

The length is determined by the slope which is in The length is determined by the slope which is in

turn determined by the degree of blurringturn determined by the degree of blurring

Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

edges tend to be thinedges tend to be thin

3838

First and Second derivatives (First and Second derivatives ( 导数导数 ))

the signs of the the signs of the

derivatives would be derivatives would be

reversed for an edge reversed for an edge

that transitions from that transitions from

light to darklight to dark

First First derivatderivatee

SeconSecond d derivatderivatee

Gray-Gray-level level profileprofile

3939

Second derivativesSecond derivatives

an undesirable featurean undesirable feature

produces 2 values for every edge in an produces 2 values for every edge in an

imageimage

zero-crossing propertyzero-crossing property

an imaginary straight line joining the an imaginary straight line joining the

extreme positive and negative values of extreme positive and negative values of

the second derivative would cross zero the second derivative would cross zero

near the midpoint of the edgenear the midpoint of the edge

quite useful for locating the centers of quite useful for locating the centers of

thick edgesthick edges

4040

Basic idea of edge detectionBasic idea of edge detection

A profile is defined perpendicularly to A profile is defined perpendicularly to

the edge direction and the results are the edge direction and the results are

interpretedinterpreted

The magnitude of the first derivative is The magnitude of the first derivative is

used to detect an edge (if a point is on a used to detect an edge (if a point is on a

ramp)ramp)

The sign of the second derivative can The sign of the second derivative can

determine whether an edge pixel is on the determine whether an edge pixel is on the

dark or light side of an edgedark or light side of an edge

4141

Review of First DerivateReview of First Derivate

Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

Sobel operators 3Sobel operators 333

6 5 8 5x yG z z G z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

9 5 8 6x yG z z G z z 1 0 0 1

0 1 1 0

1 0 0 1

0 1 1 0

7 8 9 1 2 3

3 6 9 1 4 7

2 2

2 2

x

y

G z z z z z z

G z z z z z z

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

x yf G G

4242

Edge direction and strengthEdge direction and strength

Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

2 2x yf G G

4343

Gradient MasksGradient Masks

1 0 0 1

0 1 1 0

Roberts

1 0 0 1

0 1 1 0

Roberts

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

4444

Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

3030

Idea 1 of Line DetectionIdea 1 of Line Detection

Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

if at a certain point in the imageif at a certain point in the image

|Ri||Ri|gtgt|Rj||Rj|

for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

3131

Idea 2 of Line DetectionIdea 2 of Line Detection

Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

3232

ExampleExample

3333

74 Edge-based 74 Edge-based SegmentationSegmentation

Edge-based segmentations rely on edges found in an image by edge detecting operators

these edges mark image locations of discontinuities in gray level

Edge detection is the most common approach for detecting meaningful discontinuities in gray level

There are a large group of methods based on information about edges in the image

3434

What is edgeWhat is edge

Edge is where change occurs Change is measured by derivative in 1D

―Biggest change derivative has maximum magnitude

Or 2nd derivative is zero we discuss approaches for implementing

―first-order derivative (Gradient operator)

―second-order derivative (Laplacian operator)

―we have introduced both derivatives in chapter 3

―Here we will talk only about their properties for edge detection

3535

What is edgeWhat is edge

In other wordsIn other words an edge is a set of an edge is a set of

connected pixelsconnected pixels

that lie on the boundary between two that lie on the boundary between two

regions with relatively distinct gray-level regions with relatively distinct gray-level

propertiesproperties

Note edge vs boundaryNote edge vs boundary

―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

―whereas a region boundary owing to whereas a region boundary owing to

the way it is defined is a more global the way it is defined is a more global

ideaidea

3636

Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

because of because of

optics optics

sampling sampling

image image

acquisition acquisition

imperfectionimperfection

3737

Thick and Thin EdgeThick and Thin Edge

The slope of the ramp is inversely The slope of the ramp is inversely

proportional to the degree of blurring in the proportional to the degree of blurring in the

edgeedge

Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

pathpath

Instead an edge point now is any point Instead an edge point now is any point

contained in the ramp and contained in the ramp and an edge would an edge would

then be a set of such points that are then be a set of such points that are

connectedconnected

The thickness is determined by the length of the The thickness is determined by the length of the

rampramp

The length is determined by the slope which is in The length is determined by the slope which is in

turn determined by the degree of blurringturn determined by the degree of blurring

Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

edges tend to be thinedges tend to be thin

3838

First and Second derivatives (First and Second derivatives ( 导数导数 ))

the signs of the the signs of the

derivatives would be derivatives would be

reversed for an edge reversed for an edge

that transitions from that transitions from

light to darklight to dark

First First derivatderivatee

SeconSecond d derivatderivatee

Gray-Gray-level level profileprofile

3939

Second derivativesSecond derivatives

an undesirable featurean undesirable feature

produces 2 values for every edge in an produces 2 values for every edge in an

imageimage

zero-crossing propertyzero-crossing property

an imaginary straight line joining the an imaginary straight line joining the

extreme positive and negative values of extreme positive and negative values of

the second derivative would cross zero the second derivative would cross zero

near the midpoint of the edgenear the midpoint of the edge

quite useful for locating the centers of quite useful for locating the centers of

thick edgesthick edges

4040

Basic idea of edge detectionBasic idea of edge detection

A profile is defined perpendicularly to A profile is defined perpendicularly to

the edge direction and the results are the edge direction and the results are

interpretedinterpreted

The magnitude of the first derivative is The magnitude of the first derivative is

used to detect an edge (if a point is on a used to detect an edge (if a point is on a

ramp)ramp)

The sign of the second derivative can The sign of the second derivative can

determine whether an edge pixel is on the determine whether an edge pixel is on the

dark or light side of an edgedark or light side of an edge

4141

Review of First DerivateReview of First Derivate

Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

Sobel operators 3Sobel operators 333

6 5 8 5x yG z z G z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

9 5 8 6x yG z z G z z 1 0 0 1

0 1 1 0

1 0 0 1

0 1 1 0

7 8 9 1 2 3

3 6 9 1 4 7

2 2

2 2

x

y

G z z z z z z

G z z z z z z

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

x yf G G

4242

Edge direction and strengthEdge direction and strength

Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

2 2x yf G G

4343

Gradient MasksGradient Masks

1 0 0 1

0 1 1 0

Roberts

1 0 0 1

0 1 1 0

Roberts

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

4444

Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

3131

Idea 2 of Line DetectionIdea 2 of Line Detection

Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

3232

ExampleExample

3333

74 Edge-based 74 Edge-based SegmentationSegmentation

Edge-based segmentations rely on edges found in an image by edge detecting operators

these edges mark image locations of discontinuities in gray level

Edge detection is the most common approach for detecting meaningful discontinuities in gray level

There are a large group of methods based on information about edges in the image

3434

What is edgeWhat is edge

Edge is where change occurs Change is measured by derivative in 1D

―Biggest change derivative has maximum magnitude

Or 2nd derivative is zero we discuss approaches for implementing

―first-order derivative (Gradient operator)

―second-order derivative (Laplacian operator)

―we have introduced both derivatives in chapter 3

―Here we will talk only about their properties for edge detection

3535

What is edgeWhat is edge

In other wordsIn other words an edge is a set of an edge is a set of

connected pixelsconnected pixels

that lie on the boundary between two that lie on the boundary between two

regions with relatively distinct gray-level regions with relatively distinct gray-level

propertiesproperties

Note edge vs boundaryNote edge vs boundary

―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

―whereas a region boundary owing to whereas a region boundary owing to

the way it is defined is a more global the way it is defined is a more global

ideaidea

3636

Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

because of because of

optics optics

sampling sampling

image image

acquisition acquisition

imperfectionimperfection

3737

Thick and Thin EdgeThick and Thin Edge

The slope of the ramp is inversely The slope of the ramp is inversely

proportional to the degree of blurring in the proportional to the degree of blurring in the

edgeedge

Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

pathpath

Instead an edge point now is any point Instead an edge point now is any point

contained in the ramp and contained in the ramp and an edge would an edge would

then be a set of such points that are then be a set of such points that are

connectedconnected

The thickness is determined by the length of the The thickness is determined by the length of the

rampramp

The length is determined by the slope which is in The length is determined by the slope which is in

turn determined by the degree of blurringturn determined by the degree of blurring

Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

edges tend to be thinedges tend to be thin

3838

First and Second derivatives (First and Second derivatives ( 导数导数 ))

the signs of the the signs of the

derivatives would be derivatives would be

reversed for an edge reversed for an edge

that transitions from that transitions from

light to darklight to dark

First First derivatderivatee

SeconSecond d derivatderivatee

Gray-Gray-level level profileprofile

3939

Second derivativesSecond derivatives

an undesirable featurean undesirable feature

produces 2 values for every edge in an produces 2 values for every edge in an

imageimage

zero-crossing propertyzero-crossing property

an imaginary straight line joining the an imaginary straight line joining the

extreme positive and negative values of extreme positive and negative values of

the second derivative would cross zero the second derivative would cross zero

near the midpoint of the edgenear the midpoint of the edge

quite useful for locating the centers of quite useful for locating the centers of

thick edgesthick edges

4040

Basic idea of edge detectionBasic idea of edge detection

A profile is defined perpendicularly to A profile is defined perpendicularly to

the edge direction and the results are the edge direction and the results are

interpretedinterpreted

The magnitude of the first derivative is The magnitude of the first derivative is

used to detect an edge (if a point is on a used to detect an edge (if a point is on a

ramp)ramp)

The sign of the second derivative can The sign of the second derivative can

determine whether an edge pixel is on the determine whether an edge pixel is on the

dark or light side of an edgedark or light side of an edge

4141

Review of First DerivateReview of First Derivate

Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

Sobel operators 3Sobel operators 333

6 5 8 5x yG z z G z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

9 5 8 6x yG z z G z z 1 0 0 1

0 1 1 0

1 0 0 1

0 1 1 0

7 8 9 1 2 3

3 6 9 1 4 7

2 2

2 2

x

y

G z z z z z z

G z z z z z z

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

x yf G G

4242

Edge direction and strengthEdge direction and strength

Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

2 2x yf G G

4343

Gradient MasksGradient Masks

1 0 0 1

0 1 1 0

Roberts

1 0 0 1

0 1 1 0

Roberts

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

4444

Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

3232

ExampleExample

3333

74 Edge-based 74 Edge-based SegmentationSegmentation

Edge-based segmentations rely on edges found in an image by edge detecting operators

these edges mark image locations of discontinuities in gray level

Edge detection is the most common approach for detecting meaningful discontinuities in gray level

There are a large group of methods based on information about edges in the image

3434

What is edgeWhat is edge

Edge is where change occurs Change is measured by derivative in 1D

―Biggest change derivative has maximum magnitude

Or 2nd derivative is zero we discuss approaches for implementing

―first-order derivative (Gradient operator)

―second-order derivative (Laplacian operator)

―we have introduced both derivatives in chapter 3

―Here we will talk only about their properties for edge detection

3535

What is edgeWhat is edge

In other wordsIn other words an edge is a set of an edge is a set of

connected pixelsconnected pixels

that lie on the boundary between two that lie on the boundary between two

regions with relatively distinct gray-level regions with relatively distinct gray-level

propertiesproperties

Note edge vs boundaryNote edge vs boundary

―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

―whereas a region boundary owing to whereas a region boundary owing to

the way it is defined is a more global the way it is defined is a more global

ideaidea

3636

Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

because of because of

optics optics

sampling sampling

image image

acquisition acquisition

imperfectionimperfection

3737

Thick and Thin EdgeThick and Thin Edge

The slope of the ramp is inversely The slope of the ramp is inversely

proportional to the degree of blurring in the proportional to the degree of blurring in the

edgeedge

Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

pathpath

Instead an edge point now is any point Instead an edge point now is any point

contained in the ramp and contained in the ramp and an edge would an edge would

then be a set of such points that are then be a set of such points that are

connectedconnected

The thickness is determined by the length of the The thickness is determined by the length of the

rampramp

The length is determined by the slope which is in The length is determined by the slope which is in

turn determined by the degree of blurringturn determined by the degree of blurring

Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

edges tend to be thinedges tend to be thin

3838

First and Second derivatives (First and Second derivatives ( 导数导数 ))

the signs of the the signs of the

derivatives would be derivatives would be

reversed for an edge reversed for an edge

that transitions from that transitions from

light to darklight to dark

First First derivatderivatee

SeconSecond d derivatderivatee

Gray-Gray-level level profileprofile

3939

Second derivativesSecond derivatives

an undesirable featurean undesirable feature

produces 2 values for every edge in an produces 2 values for every edge in an

imageimage

zero-crossing propertyzero-crossing property

an imaginary straight line joining the an imaginary straight line joining the

extreme positive and negative values of extreme positive and negative values of

the second derivative would cross zero the second derivative would cross zero

near the midpoint of the edgenear the midpoint of the edge

quite useful for locating the centers of quite useful for locating the centers of

thick edgesthick edges

4040

Basic idea of edge detectionBasic idea of edge detection

A profile is defined perpendicularly to A profile is defined perpendicularly to

the edge direction and the results are the edge direction and the results are

interpretedinterpreted

The magnitude of the first derivative is The magnitude of the first derivative is

used to detect an edge (if a point is on a used to detect an edge (if a point is on a

ramp)ramp)

The sign of the second derivative can The sign of the second derivative can

determine whether an edge pixel is on the determine whether an edge pixel is on the

dark or light side of an edgedark or light side of an edge

4141

Review of First DerivateReview of First Derivate

Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

Sobel operators 3Sobel operators 333

6 5 8 5x yG z z G z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

9 5 8 6x yG z z G z z 1 0 0 1

0 1 1 0

1 0 0 1

0 1 1 0

7 8 9 1 2 3

3 6 9 1 4 7

2 2

2 2

x

y

G z z z z z z

G z z z z z z

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

x yf G G

4242

Edge direction and strengthEdge direction and strength

Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

2 2x yf G G

4343

Gradient MasksGradient Masks

1 0 0 1

0 1 1 0

Roberts

1 0 0 1

0 1 1 0

Roberts

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

4444

Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

3333

74 Edge-based 74 Edge-based SegmentationSegmentation

Edge-based segmentations rely on edges found in an image by edge detecting operators

these edges mark image locations of discontinuities in gray level

Edge detection is the most common approach for detecting meaningful discontinuities in gray level

There are a large group of methods based on information about edges in the image

3434

What is edgeWhat is edge

Edge is where change occurs Change is measured by derivative in 1D

―Biggest change derivative has maximum magnitude

Or 2nd derivative is zero we discuss approaches for implementing

―first-order derivative (Gradient operator)

―second-order derivative (Laplacian operator)

―we have introduced both derivatives in chapter 3

―Here we will talk only about their properties for edge detection

3535

What is edgeWhat is edge

In other wordsIn other words an edge is a set of an edge is a set of

connected pixelsconnected pixels

that lie on the boundary between two that lie on the boundary between two

regions with relatively distinct gray-level regions with relatively distinct gray-level

propertiesproperties

Note edge vs boundaryNote edge vs boundary

―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

―whereas a region boundary owing to whereas a region boundary owing to

the way it is defined is a more global the way it is defined is a more global

ideaidea

3636

Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

because of because of

optics optics

sampling sampling

image image

acquisition acquisition

imperfectionimperfection

3737

Thick and Thin EdgeThick and Thin Edge

The slope of the ramp is inversely The slope of the ramp is inversely

proportional to the degree of blurring in the proportional to the degree of blurring in the

edgeedge

Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

pathpath

Instead an edge point now is any point Instead an edge point now is any point

contained in the ramp and contained in the ramp and an edge would an edge would

then be a set of such points that are then be a set of such points that are

connectedconnected

The thickness is determined by the length of the The thickness is determined by the length of the

rampramp

The length is determined by the slope which is in The length is determined by the slope which is in

turn determined by the degree of blurringturn determined by the degree of blurring

Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

edges tend to be thinedges tend to be thin

3838

First and Second derivatives (First and Second derivatives ( 导数导数 ))

the signs of the the signs of the

derivatives would be derivatives would be

reversed for an edge reversed for an edge

that transitions from that transitions from

light to darklight to dark

First First derivatderivatee

SeconSecond d derivatderivatee

Gray-Gray-level level profileprofile

3939

Second derivativesSecond derivatives

an undesirable featurean undesirable feature

produces 2 values for every edge in an produces 2 values for every edge in an

imageimage

zero-crossing propertyzero-crossing property

an imaginary straight line joining the an imaginary straight line joining the

extreme positive and negative values of extreme positive and negative values of

the second derivative would cross zero the second derivative would cross zero

near the midpoint of the edgenear the midpoint of the edge

quite useful for locating the centers of quite useful for locating the centers of

thick edgesthick edges

4040

Basic idea of edge detectionBasic idea of edge detection

A profile is defined perpendicularly to A profile is defined perpendicularly to

the edge direction and the results are the edge direction and the results are

interpretedinterpreted

The magnitude of the first derivative is The magnitude of the first derivative is

used to detect an edge (if a point is on a used to detect an edge (if a point is on a

ramp)ramp)

The sign of the second derivative can The sign of the second derivative can

determine whether an edge pixel is on the determine whether an edge pixel is on the

dark or light side of an edgedark or light side of an edge

4141

Review of First DerivateReview of First Derivate

Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

Sobel operators 3Sobel operators 333

6 5 8 5x yG z z G z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

9 5 8 6x yG z z G z z 1 0 0 1

0 1 1 0

1 0 0 1

0 1 1 0

7 8 9 1 2 3

3 6 9 1 4 7

2 2

2 2

x

y

G z z z z z z

G z z z z z z

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

x yf G G

4242

Edge direction and strengthEdge direction and strength

Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

2 2x yf G G

4343

Gradient MasksGradient Masks

1 0 0 1

0 1 1 0

Roberts

1 0 0 1

0 1 1 0

Roberts

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

4444

Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

3434

What is edgeWhat is edge

Edge is where change occurs Change is measured by derivative in 1D

―Biggest change derivative has maximum magnitude

Or 2nd derivative is zero we discuss approaches for implementing

―first-order derivative (Gradient operator)

―second-order derivative (Laplacian operator)

―we have introduced both derivatives in chapter 3

―Here we will talk only about their properties for edge detection

3535

What is edgeWhat is edge

In other wordsIn other words an edge is a set of an edge is a set of

connected pixelsconnected pixels

that lie on the boundary between two that lie on the boundary between two

regions with relatively distinct gray-level regions with relatively distinct gray-level

propertiesproperties

Note edge vs boundaryNote edge vs boundary

―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

―whereas a region boundary owing to whereas a region boundary owing to

the way it is defined is a more global the way it is defined is a more global

ideaidea

3636

Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

because of because of

optics optics

sampling sampling

image image

acquisition acquisition

imperfectionimperfection

3737

Thick and Thin EdgeThick and Thin Edge

The slope of the ramp is inversely The slope of the ramp is inversely

proportional to the degree of blurring in the proportional to the degree of blurring in the

edgeedge

Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

pathpath

Instead an edge point now is any point Instead an edge point now is any point

contained in the ramp and contained in the ramp and an edge would an edge would

then be a set of such points that are then be a set of such points that are

connectedconnected

The thickness is determined by the length of the The thickness is determined by the length of the

rampramp

The length is determined by the slope which is in The length is determined by the slope which is in

turn determined by the degree of blurringturn determined by the degree of blurring

Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

edges tend to be thinedges tend to be thin

3838

First and Second derivatives (First and Second derivatives ( 导数导数 ))

the signs of the the signs of the

derivatives would be derivatives would be

reversed for an edge reversed for an edge

that transitions from that transitions from

light to darklight to dark

First First derivatderivatee

SeconSecond d derivatderivatee

Gray-Gray-level level profileprofile

3939

Second derivativesSecond derivatives

an undesirable featurean undesirable feature

produces 2 values for every edge in an produces 2 values for every edge in an

imageimage

zero-crossing propertyzero-crossing property

an imaginary straight line joining the an imaginary straight line joining the

extreme positive and negative values of extreme positive and negative values of

the second derivative would cross zero the second derivative would cross zero

near the midpoint of the edgenear the midpoint of the edge

quite useful for locating the centers of quite useful for locating the centers of

thick edgesthick edges

4040

Basic idea of edge detectionBasic idea of edge detection

A profile is defined perpendicularly to A profile is defined perpendicularly to

the edge direction and the results are the edge direction and the results are

interpretedinterpreted

The magnitude of the first derivative is The magnitude of the first derivative is

used to detect an edge (if a point is on a used to detect an edge (if a point is on a

ramp)ramp)

The sign of the second derivative can The sign of the second derivative can

determine whether an edge pixel is on the determine whether an edge pixel is on the

dark or light side of an edgedark or light side of an edge

4141

Review of First DerivateReview of First Derivate

Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

Sobel operators 3Sobel operators 333

6 5 8 5x yG z z G z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

9 5 8 6x yG z z G z z 1 0 0 1

0 1 1 0

1 0 0 1

0 1 1 0

7 8 9 1 2 3

3 6 9 1 4 7

2 2

2 2

x

y

G z z z z z z

G z z z z z z

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

x yf G G

4242

Edge direction and strengthEdge direction and strength

Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

2 2x yf G G

4343

Gradient MasksGradient Masks

1 0 0 1

0 1 1 0

Roberts

1 0 0 1

0 1 1 0

Roberts

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

4444

Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

3535

What is edgeWhat is edge

In other wordsIn other words an edge is a set of an edge is a set of

connected pixelsconnected pixels

that lie on the boundary between two that lie on the boundary between two

regions with relatively distinct gray-level regions with relatively distinct gray-level

propertiesproperties

Note edge vs boundaryNote edge vs boundary

―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

―whereas a region boundary owing to whereas a region boundary owing to

the way it is defined is a more global the way it is defined is a more global

ideaidea

3636

Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

because of because of

optics optics

sampling sampling

image image

acquisition acquisition

imperfectionimperfection

3737

Thick and Thin EdgeThick and Thin Edge

The slope of the ramp is inversely The slope of the ramp is inversely

proportional to the degree of blurring in the proportional to the degree of blurring in the

edgeedge

Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

pathpath

Instead an edge point now is any point Instead an edge point now is any point

contained in the ramp and contained in the ramp and an edge would an edge would

then be a set of such points that are then be a set of such points that are

connectedconnected

The thickness is determined by the length of the The thickness is determined by the length of the

rampramp

The length is determined by the slope which is in The length is determined by the slope which is in

turn determined by the degree of blurringturn determined by the degree of blurring

Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

edges tend to be thinedges tend to be thin

3838

First and Second derivatives (First and Second derivatives ( 导数导数 ))

the signs of the the signs of the

derivatives would be derivatives would be

reversed for an edge reversed for an edge

that transitions from that transitions from

light to darklight to dark

First First derivatderivatee

SeconSecond d derivatderivatee

Gray-Gray-level level profileprofile

3939

Second derivativesSecond derivatives

an undesirable featurean undesirable feature

produces 2 values for every edge in an produces 2 values for every edge in an

imageimage

zero-crossing propertyzero-crossing property

an imaginary straight line joining the an imaginary straight line joining the

extreme positive and negative values of extreme positive and negative values of

the second derivative would cross zero the second derivative would cross zero

near the midpoint of the edgenear the midpoint of the edge

quite useful for locating the centers of quite useful for locating the centers of

thick edgesthick edges

4040

Basic idea of edge detectionBasic idea of edge detection

A profile is defined perpendicularly to A profile is defined perpendicularly to

the edge direction and the results are the edge direction and the results are

interpretedinterpreted

The magnitude of the first derivative is The magnitude of the first derivative is

used to detect an edge (if a point is on a used to detect an edge (if a point is on a

ramp)ramp)

The sign of the second derivative can The sign of the second derivative can

determine whether an edge pixel is on the determine whether an edge pixel is on the

dark or light side of an edgedark or light side of an edge

4141

Review of First DerivateReview of First Derivate

Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

Sobel operators 3Sobel operators 333

6 5 8 5x yG z z G z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

9 5 8 6x yG z z G z z 1 0 0 1

0 1 1 0

1 0 0 1

0 1 1 0

7 8 9 1 2 3

3 6 9 1 4 7

2 2

2 2

x

y

G z z z z z z

G z z z z z z

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

x yf G G

4242

Edge direction and strengthEdge direction and strength

Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

2 2x yf G G

4343

Gradient MasksGradient Masks

1 0 0 1

0 1 1 0

Roberts

1 0 0 1

0 1 1 0

Roberts

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

4444

Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

3636

Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

because of because of

optics optics

sampling sampling

image image

acquisition acquisition

imperfectionimperfection

3737

Thick and Thin EdgeThick and Thin Edge

The slope of the ramp is inversely The slope of the ramp is inversely

proportional to the degree of blurring in the proportional to the degree of blurring in the

edgeedge

Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

pathpath

Instead an edge point now is any point Instead an edge point now is any point

contained in the ramp and contained in the ramp and an edge would an edge would

then be a set of such points that are then be a set of such points that are

connectedconnected

The thickness is determined by the length of the The thickness is determined by the length of the

rampramp

The length is determined by the slope which is in The length is determined by the slope which is in

turn determined by the degree of blurringturn determined by the degree of blurring

Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

edges tend to be thinedges tend to be thin

3838

First and Second derivatives (First and Second derivatives ( 导数导数 ))

the signs of the the signs of the

derivatives would be derivatives would be

reversed for an edge reversed for an edge

that transitions from that transitions from

light to darklight to dark

First First derivatderivatee

SeconSecond d derivatderivatee

Gray-Gray-level level profileprofile

3939

Second derivativesSecond derivatives

an undesirable featurean undesirable feature

produces 2 values for every edge in an produces 2 values for every edge in an

imageimage

zero-crossing propertyzero-crossing property

an imaginary straight line joining the an imaginary straight line joining the

extreme positive and negative values of extreme positive and negative values of

the second derivative would cross zero the second derivative would cross zero

near the midpoint of the edgenear the midpoint of the edge

quite useful for locating the centers of quite useful for locating the centers of

thick edgesthick edges

4040

Basic idea of edge detectionBasic idea of edge detection

A profile is defined perpendicularly to A profile is defined perpendicularly to

the edge direction and the results are the edge direction and the results are

interpretedinterpreted

The magnitude of the first derivative is The magnitude of the first derivative is

used to detect an edge (if a point is on a used to detect an edge (if a point is on a

ramp)ramp)

The sign of the second derivative can The sign of the second derivative can

determine whether an edge pixel is on the determine whether an edge pixel is on the

dark or light side of an edgedark or light side of an edge

4141

Review of First DerivateReview of First Derivate

Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

Sobel operators 3Sobel operators 333

6 5 8 5x yG z z G z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

9 5 8 6x yG z z G z z 1 0 0 1

0 1 1 0

1 0 0 1

0 1 1 0

7 8 9 1 2 3

3 6 9 1 4 7

2 2

2 2

x

y

G z z z z z z

G z z z z z z

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

x yf G G

4242

Edge direction and strengthEdge direction and strength

Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

2 2x yf G G

4343

Gradient MasksGradient Masks

1 0 0 1

0 1 1 0

Roberts

1 0 0 1

0 1 1 0

Roberts

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

4444

Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

3737

Thick and Thin EdgeThick and Thin Edge

The slope of the ramp is inversely The slope of the ramp is inversely

proportional to the degree of blurring in the proportional to the degree of blurring in the

edgeedge

Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

pathpath

Instead an edge point now is any point Instead an edge point now is any point

contained in the ramp and contained in the ramp and an edge would an edge would

then be a set of such points that are then be a set of such points that are

connectedconnected

The thickness is determined by the length of the The thickness is determined by the length of the

rampramp

The length is determined by the slope which is in The length is determined by the slope which is in

turn determined by the degree of blurringturn determined by the degree of blurring

Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

edges tend to be thinedges tend to be thin

3838

First and Second derivatives (First and Second derivatives ( 导数导数 ))

the signs of the the signs of the

derivatives would be derivatives would be

reversed for an edge reversed for an edge

that transitions from that transitions from

light to darklight to dark

First First derivatderivatee

SeconSecond d derivatderivatee

Gray-Gray-level level profileprofile

3939

Second derivativesSecond derivatives

an undesirable featurean undesirable feature

produces 2 values for every edge in an produces 2 values for every edge in an

imageimage

zero-crossing propertyzero-crossing property

an imaginary straight line joining the an imaginary straight line joining the

extreme positive and negative values of extreme positive and negative values of

the second derivative would cross zero the second derivative would cross zero

near the midpoint of the edgenear the midpoint of the edge

quite useful for locating the centers of quite useful for locating the centers of

thick edgesthick edges

4040

Basic idea of edge detectionBasic idea of edge detection

A profile is defined perpendicularly to A profile is defined perpendicularly to

the edge direction and the results are the edge direction and the results are

interpretedinterpreted

The magnitude of the first derivative is The magnitude of the first derivative is

used to detect an edge (if a point is on a used to detect an edge (if a point is on a

ramp)ramp)

The sign of the second derivative can The sign of the second derivative can

determine whether an edge pixel is on the determine whether an edge pixel is on the

dark or light side of an edgedark or light side of an edge

4141

Review of First DerivateReview of First Derivate

Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

Sobel operators 3Sobel operators 333

6 5 8 5x yG z z G z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

9 5 8 6x yG z z G z z 1 0 0 1

0 1 1 0

1 0 0 1

0 1 1 0

7 8 9 1 2 3

3 6 9 1 4 7

2 2

2 2

x

y

G z z z z z z

G z z z z z z

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

x yf G G

4242

Edge direction and strengthEdge direction and strength

Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

2 2x yf G G

4343

Gradient MasksGradient Masks

1 0 0 1

0 1 1 0

Roberts

1 0 0 1

0 1 1 0

Roberts

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

4444

Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

3838

First and Second derivatives (First and Second derivatives ( 导数导数 ))

the signs of the the signs of the

derivatives would be derivatives would be

reversed for an edge reversed for an edge

that transitions from that transitions from

light to darklight to dark

First First derivatderivatee

SeconSecond d derivatderivatee

Gray-Gray-level level profileprofile

3939

Second derivativesSecond derivatives

an undesirable featurean undesirable feature

produces 2 values for every edge in an produces 2 values for every edge in an

imageimage

zero-crossing propertyzero-crossing property

an imaginary straight line joining the an imaginary straight line joining the

extreme positive and negative values of extreme positive and negative values of

the second derivative would cross zero the second derivative would cross zero

near the midpoint of the edgenear the midpoint of the edge

quite useful for locating the centers of quite useful for locating the centers of

thick edgesthick edges

4040

Basic idea of edge detectionBasic idea of edge detection

A profile is defined perpendicularly to A profile is defined perpendicularly to

the edge direction and the results are the edge direction and the results are

interpretedinterpreted

The magnitude of the first derivative is The magnitude of the first derivative is

used to detect an edge (if a point is on a used to detect an edge (if a point is on a

ramp)ramp)

The sign of the second derivative can The sign of the second derivative can

determine whether an edge pixel is on the determine whether an edge pixel is on the

dark or light side of an edgedark or light side of an edge

4141

Review of First DerivateReview of First Derivate

Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

Sobel operators 3Sobel operators 333

6 5 8 5x yG z z G z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

9 5 8 6x yG z z G z z 1 0 0 1

0 1 1 0

1 0 0 1

0 1 1 0

7 8 9 1 2 3

3 6 9 1 4 7

2 2

2 2

x

y

G z z z z z z

G z z z z z z

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

x yf G G

4242

Edge direction and strengthEdge direction and strength

Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

2 2x yf G G

4343

Gradient MasksGradient Masks

1 0 0 1

0 1 1 0

Roberts

1 0 0 1

0 1 1 0

Roberts

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

4444

Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

3939

Second derivativesSecond derivatives

an undesirable featurean undesirable feature

produces 2 values for every edge in an produces 2 values for every edge in an

imageimage

zero-crossing propertyzero-crossing property

an imaginary straight line joining the an imaginary straight line joining the

extreme positive and negative values of extreme positive and negative values of

the second derivative would cross zero the second derivative would cross zero

near the midpoint of the edgenear the midpoint of the edge

quite useful for locating the centers of quite useful for locating the centers of

thick edgesthick edges

4040

Basic idea of edge detectionBasic idea of edge detection

A profile is defined perpendicularly to A profile is defined perpendicularly to

the edge direction and the results are the edge direction and the results are

interpretedinterpreted

The magnitude of the first derivative is The magnitude of the first derivative is

used to detect an edge (if a point is on a used to detect an edge (if a point is on a

ramp)ramp)

The sign of the second derivative can The sign of the second derivative can

determine whether an edge pixel is on the determine whether an edge pixel is on the

dark or light side of an edgedark or light side of an edge

4141

Review of First DerivateReview of First Derivate

Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

Sobel operators 3Sobel operators 333

6 5 8 5x yG z z G z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

9 5 8 6x yG z z G z z 1 0 0 1

0 1 1 0

1 0 0 1

0 1 1 0

7 8 9 1 2 3

3 6 9 1 4 7

2 2

2 2

x

y

G z z z z z z

G z z z z z z

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

x yf G G

4242

Edge direction and strengthEdge direction and strength

Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

2 2x yf G G

4343

Gradient MasksGradient Masks

1 0 0 1

0 1 1 0

Roberts

1 0 0 1

0 1 1 0

Roberts

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

4444

Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

4040

Basic idea of edge detectionBasic idea of edge detection

A profile is defined perpendicularly to A profile is defined perpendicularly to

the edge direction and the results are the edge direction and the results are

interpretedinterpreted

The magnitude of the first derivative is The magnitude of the first derivative is

used to detect an edge (if a point is on a used to detect an edge (if a point is on a

ramp)ramp)

The sign of the second derivative can The sign of the second derivative can

determine whether an edge pixel is on the determine whether an edge pixel is on the

dark or light side of an edgedark or light side of an edge

4141

Review of First DerivateReview of First Derivate

Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

Sobel operators 3Sobel operators 333

6 5 8 5x yG z z G z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

9 5 8 6x yG z z G z z 1 0 0 1

0 1 1 0

1 0 0 1

0 1 1 0

7 8 9 1 2 3

3 6 9 1 4 7

2 2

2 2

x

y

G z z z z z z

G z z z z z z

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

x yf G G

4242

Edge direction and strengthEdge direction and strength

Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

2 2x yf G G

4343

Gradient MasksGradient Masks

1 0 0 1

0 1 1 0

Roberts

1 0 0 1

0 1 1 0

Roberts

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

4444

Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

4141

Review of First DerivateReview of First Derivate

Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

Sobel operators 3Sobel operators 333

6 5 8 5x yG z z G z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

9 5 8 6x yG z z G z z 1 0 0 1

0 1 1 0

1 0 0 1

0 1 1 0

7 8 9 1 2 3

3 6 9 1 4 7

2 2

2 2

x

y

G z z z z z z

G z z z z z z

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

x yf G G

4242

Edge direction and strengthEdge direction and strength

Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

2 2x yf G G

4343

Gradient MasksGradient Masks

1 0 0 1

0 1 1 0

Roberts

1 0 0 1

0 1 1 0

Roberts

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

4444

Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

4242

Edge direction and strengthEdge direction and strength

Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

2 2x yf G G

4343

Gradient MasksGradient Masks

1 0 0 1

0 1 1 0

Roberts

1 0 0 1

0 1 1 0

Roberts

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

4444

Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

4343

Gradient MasksGradient Masks

1 0 0 1

0 1 1 0

Roberts

1 0 0 1

0 1 1 0

Roberts

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

4444

Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

4444

Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
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• Slide 24
• Slide 25
• Slide 26
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• Slide 29
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• Slide 31
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• Slide 33
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• Slide 35
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• Slide 37
• Slide 38
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• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
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5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

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5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

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5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

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5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

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5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

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6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

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6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

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6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

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