Subdividing an image into different regions based on some …dip111/wiki.files/DIP3E... · 2011-01-21 · Digital Image Processing, 3rd ed. Gonzalez & Woods Ch t 10 Non-Maxima Suppression

Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.

www.ImageProcessingPlace.com

Gonzalez & Woods

Ch t 10Ch t 10Chapter 10Segmentation

Chapter 10Segmentation

Image Segmentationg gSubdividing an image into different regions based on some criterion/criteria.

© 1992–2008 R. C. Gonzalez & R. E. Woods



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Image Derivativesg1. First order generally produces

thick edges2. Second order has stronger

response to fine detailresponse to fine detail3. Second order derivates may

produce double edges (in ramp and step transition intensities).

4. The sign of the second derivative indicates whether the transition from dark to light or light to darklight or light to dark

)()1(

2

f

xfxfxf


)1()(2)2(2 xfxfxf

xf



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Detection of Isolate PointsThe advantages of the second derivative implies using the Laplacian

22 ff

),1(),(2),2(

),(

2

2

2

2

2

22

yxfyxfyxfxf

yf

xfyxf

),(4)2,(),2(),(

)1,(),(2)2,(

2

2

2

yxfyxfyxfyxf

yxfyxfyxfyf

)1,(),1( yxfyxf

A point at location (x,y) can be detected using its response g(x,y)

R(x y) is the result of applying a filter


otherwise

TyxRyxg

0|),(|1

),(

R(x,y) is the result of applying a filter at the location x,y



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Line Detection Lines are detected by 1. Finding the points of the

lines2. Collect these points into

line.

Applying Laplacian on an image.1. Original2. Laplacian Result3. Absolute value4. Positive only




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Line/Edge Detection g1. The points along the lines do not

usually fall on an ideal line.2. The line may not be continuous.




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Edge ModelsgThe transition from a bright intensity to dark one, have different patterns.




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Edge ModelsgThe existence of noise complicate line detection therefore1. Image smoothing often contribute

to noise reduction.2. Detection of edge points- potential

candidates for edge points.3 d l li i l i3. Edge localization- to select points

which are actually on the lines.




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Image Gradientg

x xf

gfgradf )(

yx

y

ggfmagyxM

xfg

fgradf

22)(),(

)(

x

y

yx

gg

yx

ggfgy

1tan),(

)(),(

Gradient is often expressed as

yx ggyxM ),(


y



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Robert operation

)( 59

f

zzxfgx

p

)( 68 zzyfg y

The simplest approximation of partial

)()( 321987 zzzzzzxfgx

dervatives on 3x3 mask

)()( 741963 zzzzzzyfg y




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Marr-Hildreth Edge DetectorMarr and Hildreh argued1. Intensity changes are not independent

of image scale2 The studden change in intensity show2. The studden change in intensity show

a peak in the first derivative and zero crossing in the second

2

22

2)( yx

eyxG

2

22

2

22

2

2

2

22

2

),(),(),(

),(

yxyx yx

yyxG

xyxGyxG

eyxG

2

22

2

22

22

224

22

24

2

22

22

11

yxyx

eyex

eyy

exx

This expression is often called Laplacian of Gaussian


2

22

24

222 2

yx

eyx

This expression is often called Laplacian of Gaussian (LoG)



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Gonzalez & Woods



Canny Edge Detector1. Low error rate2. Edge point should be well localized3. Single edge point response

Algorithm1. Smooth image with a Gaussian

- optimizes the trade-off between noisefiltering and edge localization

2. Compute the Gradient magnitude using approximations of partial derivatives

3 Thi d b l i i3. Thin edges by applying non-maxima suppression to the gradient magnitude

4. Detect edges by double thresholding




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Gradient

• At each point convolve with

11

11

G

• magnitude and orientation of the Gradient are

1111

xG

11

yG

• magnitude and orientation of the Gradient are computed as

22 ],[],[],[ jiQjiPjiM

A id fl i i i h i f f i

],[],[],[ jQjj

]),[],,[(tan],[ 1 jiPjiQji

© 1992–2008 R. C. Gonzalez & R. E. WoodsCanny Edge Detector 19

• Avoid floating point arithmetic for fast computation



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Ch t 10Ch t 10Non-Maxima Suppression



• Thin edges by keeping large values of GradientThin edges by keeping large values of Gradient– not always at the location of an edge– there are many thick edgesy g

1312100031110000

010012300011231001121200

120102321010023201001230


12010232



Gonzalez & Woods

Ch t 10Ch t 10Non-Maxima Suppression (2)



• Thin the broad ridges in M[i,j] into ridges that are only one i l idpixel wide

• Find local maxima in M[i,j] by suppressing all values along the line of the Gradient that are not peak values of the ridge

1312100331110000

0011231001121200

falseedges

120102321010023231001230

gaps

edges


12010232



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Ch t 10Ch t 10Gradient Orientation



• Reduce angle of Gradient θ[i,j] to one of the 4 sectorsReduce angle of Gradient θ[i,j] to one of the 4 sectors• Check the 3x3 region of each M[i,j]• If the value at the center is not greater than the 2• If the value at the center is not greater than the 2

values along the gradient, then M[i,j] is set to 0




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local31110000maxima

0112120013121000

removed

depends010012300011231001121200

dependson condition

1010023201001230

12010232




Gonzalez & Woods

Ch t 10Ch t 10 30000000 Chapter 10Segmentation


0312000030000000

0000030000021200

false edges

0000023000000300 g

0201003010100030

• The suppressed magnitude image will contain many false edges caused by noise or fine texture


false edges caused by noise or fine texture



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Ch t 10Ch t 10Thresholding



g

• Reduce number of false edges by applying aReduce number of false edges by applying a threshold T

all values below T are changed to 0– all values below T are changed to 0– selecting a good values for T is difficult

some false edges will remain if T is too low– some false edges will remain if T is too low– some edges will disappear if T is too high

d ill di d t ft i f th– some edges will disappear due to softening of the edge contrast by shadows




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Ch t 10Ch t 10Double Thresholding



g

• Apply two thresholds in the suppressed imageApply two thresholds in the suppressed image– T2 = 2T2

– two images in the outputg p– the image from T2 contains fewer edges but has gaps in the

contours h i f h f l d– the image from T1 has many false edges

– combine the results from T1 and T2

link the edges of T into contours until we reach a gap– link the edges of T2 into contours until we reach a gap– link the edge from T2 with edge pixels from a T1 contour

until a T2 edge is found again


2



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Ch t 10Ch t 10T2=2 T =1Chapter 10Segmentation


T2 2 T1=1

0302000030000000

0312000030000000

000003000002020003020000

000003000002120003120000

gapsfilled

0000003000000230

1010003000000230from

T1

02000030 02010030

• A T2 contour has pixels along the green arrowsA T2 contour has pixels along the green arrows• Linking: search in a 3x3 of each pixel and connect the

pixel at the center with the one having greater valueS h i th di ti f th d (di ti f G di t)


• Search in the direction of the edge (direction of Gradient)



Gonzalez & Woods

Ch t 10Ch t 10Line Detection



• Model of a line: two edges with oppositeModel of a line: two edges with opposite polarity in distance less than the size of the smoothing filterg– edge detection filters respond to step edges– they do not provide meaningful response to lines

• Apply nonmaxima suppression on the smoothed output– a line is the derivative of a step the derivative

step of the Canny algorithm is not necessary




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ThresholdingThresholding1. Select initial estimate2. Segment the images using T into two

groups G1 and G2groups G1 and G2 3. Compute the averages of the two

groups m1 and m2.4. Computer a new threshold as

T = (m1+m2)/25. Repeat 2..4 until the difference

between two successive thresholds is less than t.




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Optimal ThresholdingOptimal Thresholding1

1

0

L

iip

Suppose we select a thershold T(k)Suppose we select a thershold T(k)

)(0

1 pkPk

ii

)(1)( 1

1

12 kPpkP

L

kii

The mean intensity in each group is

)(/)()()(

)()(

1)(/)()()(010

110

11

BPAPABPBAP

iiPkP

CiPiCiPCiiPkmk

i

k

i

k

i


)(/)()()( BPAPABPBAP



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Optimal ThresholdingOptimal Thresholding

111

)(1)(/)()()(LLL

iiPCiPiCiPCiiPkm

The mean intensity of the second group is

121

221

22 )()(

)(/)()()(kikiki

iiPkP

CiPiCiPCiiPkm

The accumulative mean up to k level is

The average intensity of the entire image is:

k

ik)( 1Lg

i

iipkm0

)(

0

)(i

iG ipkm

2211 mPmPmG

1

22 )(L

iG pmi2211G

)](1)[()()( 2

12

kPkPkmkPmG

B

0

)(i

iG p


)](1)[( 11 kPkP



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Optimal ThresholdingOptimal Thresholding

2

2B

We can measure goodness of the threshold at level k is

2G

1

0

22 )(L

iiGG pmiWhere

and )( 212222 mPmG

Taking k into account

)()( 2kmkPm

and)1()()()()(

21

122121

222

211

2

PPmPmmmPPmmPmmP G

GGB

2 )(k

The optimum threshold k* is

)](1)[()()()(

11

12

kPkPkmkPmk G

B

2

)()(G

B kk

Where


))((max)( 2

10

*2 kk BLkB



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Otsu’s Algorithm for Optimal ThresholdOtsu s Algorithm for Optimal Threshold1. Compute the normalized histogram (pi) of the input image2. Compute the accumulative sum Pk for k=0, .., L-13 C h l i f k 0 L 13. Compute the accumulative mean mk for k=0, .., L-14. Compute the global intensity mean mG5. Compute the between class variance 6 Obtain the Otus threshold k* as the value of k that maximize

)(2 kB)(2 kB6. Obtain the Otus threshold k* as the value of k that maximize

7. Obtain the separablility measure at k = k*)(B




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Subdividing an image into different regions based on some …dip111/wiki.files/DIP3E... · 2011-01-21 · Digital Image Processing, 3rd ed. Gonzalez & Woods Ch t 10 Non-Maxima Suppression