Notes on Image Processing

Edge Detection of images

BY: Mohammed Kamel

Supervisor : Prof: R.M.Farouk

Department of Mathematics & computer

science

what is edge detection?

Edges are places in the image with strong intensity contrast (Or) it ‘difference of intensity between two images (Or) Edges are pixels where image brightness changes abruptly. Example:

(a) (b) Figure 1.1 edge detection: (a) shows [original image] that will be used to find its edge, (b) shows edge Detection of image using [canny edge detection] method.

Why is Edge Detection Useful?

-Important features can be extracted from the edges of an image (e.g., corners, lines). -These features are used by higher-level computer vision algorithms (e.g., recognition). -Lane and directions detection.

Figure 1.2

Figure 1.2: shows Lane and directions detection for navigation

Effect of Illumination (error) reflection:

(a) (b) (c) (d)

Figure 1.3

Figure 1.3: error: (a) original image, (b) shows actual edge detection of (a), (b) shows same

original image but with light source, (d) shows how light source effect on edge of image.

Edge direction: perpendicular to the direction of maximum intensity change (i.e. edge normal)

Edge Descriptors!

Edge direction: perpendicular to the direction of maximum intensity change (i.e. edge normal)

Edge strength: related to the local image contrast along the Normal.

Edge position: the image position at which the edge is Located.

Figure 1.4

Figure 1.4: shows how edge direction and normal represented in image.

Step edge: the image intensity abruptly changes from one value on one side of the Discontinuity to a different value on the opposite side. Is also called (Step Function, hard edge).

Modeling Intensity Changes:

Ramp edge: a step edge where the intensity change is not instantaneous but occur

over a Finite distance.

Modeling Intensity Changes(cont’d)

Within some short distance (i.e., usually generated by lines

Ridge edge: the image intensity abruptly changes value but then returns to the

starting

Value within some short distance (i.e., usually generated by lines)

Roof edge: a ridge edge where the intensity change is not instantaneous but occur over

a finite distance (i.e., usually generated by the intersection of two surfaces).

Main Steps in Edge Detection

(1) Smoothing: suppress as much noise as possible, without destroying true edges. (Or) it

Make pixel nearest as possible to each other to make image more clearly

Is also called [Blur].(2)Enhancement: apply differentiation to enhance the quality of edges (sharpening).and apply

some kind of filter that responds strongly at edges for example Take highest

Value of pixel in image and replace it with all pixel to Increase quality of image.

(3)Thresholding: determine which edge pixels should be discarded as noise and which should be

retain (I.e. threshold edge magnitude).

(4) Localization: determine the exact edge location.

** (Sub-pixel resolution): might be required for some applications to estimate the location of an

edge to better than the spacing between pixels.

Edge Detection Using Derivative

Often, points that lie on an edge are detected by:

(1) Detecting the local maxima or minima of the first derivative.

(2) Detecting the zero-crossings of the second derivative.

Edge Detection Using First Derivative

1D functions (not centered at x)

0

( ) ( )'( ) lim ( 1) ( )h

f x h f xf x f x f xh

(h=1) Mask: [-1 1]

1,0,1Mask M= (Centered at x)

Edge Detection Using Second Derivative

Approximate finding maxima/minima of gradient magnitude by finding places where: 2

( ) 02d f xdx

Can’t always find discrete pixels where the second derivative is ((zero)) look for zero-crossing

instead. (See below).

0

'( ) '( )''( ) lim '( 1) '( )h

f x h f xf x f x f xh

( 2) 2 ( 1) ( )f x f x f x (h=1)

Replace x+1 with x (i.e., centered at x): ''( ) ( 1) 2 ( ) ( 1)f x f x f x f x Mask: [1 -2 1]

-Four cases of zero-crossing: {+,-}, {+, 0,-}, {-, +}, {-, 0, +}

-Slope of zero-crossing {a, -b} is: |a+b|

-To detect “strong” zero-crossing, Threshold the slope.

20 20 20

10 10 10 10 10

10

10 10

0 0 0 0 0 0 0

0 0 0 00

( )f x

'( ) f(x 1) f(x)f x (approximation f'(x) at x+1/2)

''( ) f(x 1) 2f(x) f(x 1)f x (approximation f''(x) at x)

Smoothing

We will try to suppress as much noise as possible, without smoothing away the meaningful

edges. It make pixel nearest as possible to each other to make image more clearly It’s also

known by [Blur].

But it different than zooming because when you zoom an image using pixel replication

and zooming factor is increased, you saw a blurred image. This image also has less details,

but it is not true blurring. Because in zooming, you add new pixels to an image, which

increase the overall number of pixels in an image, whereas in blurring, the number of pixels

of a normal image and a blurred image remains the same.

Smoothing can be achieved by many ways. The common type of filters that are used to

perform blurring: [1] Mean filter (Box filter).

[2]Weighted average filter.

[3]Gaussian filter.

Smoothing(cont’d)

Noise :( unwanted signals)

Its random variation of brightness or color information in image, it’s also undesirable when

image capture that adds spurious and extraneous information.

Effects of noise

Consider a single row or column of the image (Plotting intensity as a function of position gives

a Signal)

Smoothing(cont’d)

f

h

( )h f

( * )h fx

Solution: smooth first

Look for peaks in ( * )h fx

Derivative theorem of convolution

( ) ( )h f h fx x

(i.e., saves one operation)

f

hx

( )h fx

Thresholding:

Determine which edge pixels should be discarded as noise and which should be retained (i.e.

threshold edge magnitude).for example: by looking to image and set any number then make

biggest numbers 1 and smallest numbers 0.

-Reduce number of false edges by applying a threshold T

-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 edges will disappear if T is too high.

- Some edges will disappear due to softening of the edge contrast by shadows.

*If the gradient at a pixel is above ‘High’, declare it an ‘edge pixel’.

*If the gradient at a pixel is below ‘Low’, declare it a ‘non-edge-pixel’.

*If the gradient at a pixel is between ‘Low’ and ‘High’ then declare it an ‘edge pixel’ if and

only if it is Connected to an ‘edge pixel’ directly or via pixels between ‘Low’ and ‘High’.

Thresholding(cont’d)

(a) (b)

(c) Figure 1.5 (d)

Figure 1.5: (a) shows [original image], (b) shows [Fine scale high threshold],

(c) shows [Coarse scale high threshold], (d) shows [coarse scale low threshold].

Edge Detection Using First Derivative (Gradient)

The first derivate of an image can be computed using the gradient

Grad(f)=

fxfy

Gradient Representation

The gradient is a vector which has magnitude and direction:

Magnitude (grad (f)) = Direction (grad (f)) =

2 2f fx y

1tan /f f

y x

*Magnitude: indicates edge strength.

*Direction: indicates edge direction. i.e., perpendicular to edge direction

Approximate Gradient

Approximate gradient using finite differences

0

( , ) ( , )limh

f f x h y f x yx h

0

( , ) ( , )limh

f f x y h f x yy h

( , ) ( , ) ( 1, ) ( , )X

X

f x h y f x yf f x y f x yx h

( , ) ( , )

( , 1) ( , )y

y

f x y h f x yf f x y f x yy h

1Xh

1yh

Cartesian vs pixel-coordinates:

- J corresponds to x- direction.

- I corresponds to y- direction.

( 1, ) ( , )f x y f x y

( , 1) ( , )f x y f x y

( , 1) f(i, j)f f i jx

( , ) ( 1, )f f i j f i jy

Sensitive to horizontal edges!

3 2 3 3

2 3

(x , y ) (x , y )y y

f f

(x, y ) (x, y)f Dy fDy

( , ) f(x, y 1) f x y

3 2 2 3(y = y + Dy, y = y , x = x, Dy = 1)

3 2(x , y )

1 1(x , y )

(𝑥3 , 𝑦3 )

(𝑥2 , 𝑦1 )

Or (Grad in the Y direction)

2 1 1 1

2 1

(x , y ) (x , y )f fx x

(x Dx, y) (x, y)f fDx

( 1, ) f(x, y) f x y

(Grad in the X direction)

2 1 1 2(x = x+Dx, x = x, y =y = y, Dx =1)

Or

Edge in the x-direction Edge in the y-direction

(x+1/2, y)

(X, y+1/2)

fx

fy

We can implement and using following masks:

fx

Good approximation at (x+1/2, y)

Good approximation at (x, y+1/2)

-A different approximation of the gradient

( , ) ( , ) ( 1, 1)f x y f x y f x yx

( , ) ( 1, ) ( , 1)f x y f x y f x y

y

fy

another Approximation

0 1 2

7 3

6 5 4

[i,j]

a a aa aa a a

Consider the arrangement of pixels about the pixel (I, J):

3 x 3 neighborhood:

fx

fy

The partial derivatives can be computed by:

2 3 4 0 7 6(a a a ) (a a a )XM c c 6 5 4 0 1 2(a a a ) (a a a )yM c c

The constant c implies the emphasis given to pixels closer to the center of the mask

Roberts Operatorfx

fy

= f (i, j) - f )i + 1, j + 1)

= f (i + 1, j)- f (i, j + 1)

This approximation can be implemented by the following masks: 1 00 1

0 11 0

(Note: Mx and My are approximations at (i + 1/2, j + 1/2))

Prewitt Operator

XM yMSetting c = 1, we get the Prewitt operator:

are approximation at (i, j)

1 0 11 0 11 0 1

1 1 10 0 01 1 1

Flow chart

Image Blurred Edges in x

111111

11

101101101

Image Blurred Edges in y

11

111000111

111111

𝑀 𝑋=¿ 𝑀 𝑦=¿

AverageSmoothing in x

DerivativeFiltering in x

Results

Results

And

And

AverageSmoothing in y

DerivativeFiltering in y

Setting c = 2, we get the Sobel operator:

Sobel Operator

1 0 12 0 21 0 1

1 2 10 0 01 2 1

𝑀 𝑋=¿ 𝑀 𝑦=¿

Flow chart

Image Blurred Edges in x

112211

11

101202101

Image Blurred Edges in y

11

121000121

121121

AverageSmoothing in x

DerivativeFiltering in x

DerivativeFiltering in y

AverageSmoothing in y

Results

Results

And

And

101202101

121000121

*

*

Idxd

Idyd

22

I

dydI

dxd

Threshold EdgesImage I

Edge Detection Steps Using Gradient

A) Smooth the input image ( ( , ) ( , )*G(x, y))F x y f x y

( , )*M ( , )X XF f x y x y

C) ( , )*M ( , )y yF f x y x yB)

D) Magnitude ( , ) X yx y F F

.

E) Direction 1( , ) tan (F / F )X yx y F) IF magnitude ( , ) Tx y , then possible edge point.

(a) (b) (c) (d) (e)

Figure1.6

Figure 1.6:(a) shows [Original image], (b) shows how image represented using [Sobel 3x3],(c) show [Sobel 3x3

grayscale],(d) shows how image represented using [Prewitt 3x3], and (e) shows [Prewitt 3x3 grayscale] .

Marr Hildreth Edge Detector

The derivative operators presented so far are not very useful because they are very sensitive to

noise. To filter the noise before enhancement, Marr and Hildreth proposed a Gaussian Filter,

combined with the Laplacian for edge detection. Is the Laplacian of Gaussian (LoG)?

The fundamental characteristics of LoG edge detector are:

[1] Smooth image by Gaussian filter:

S = G * I Where: S: is smoothed image. G: is Gaussian filter represented by

2 2

221( , )2

x y

G x y e

I: the original image.

[2] Apply Laplacian 2 2

22 2s s s

x y

Note: [ [ ] is used for gradient or derivative.]is used for laplacian

Marr Hildreth Edge Detector (cont’d)

[3] Deriving the Laplacian of Gaussian (LoG)2 2 2( * ) ( )*IS G I G

2 2

22 2

2223

1 (2 )2

x yx yg e

Used in mechanics, electromagnetics, wave theory, quantum mechanics and Laplace equation

[4] Find zero crossings(a)Scan along each row, record an edge point at the location of zero-crossing.

(b)Repeat above step along each column.

-Four cases of zero-crossing: {+,-}, {+, 0,-}, {-, +}, {-, 0, +}

-Slope of zero-crossing {a, -b} is: |a+b|

-To detect “strong” zero-crossing, and threshold the slope.

The Separability of LoG

Similar to separability of Gaussian filter two-dimensional Gaussian can be separated into 2 one-dimensional Gaussians.

Image G(x) G(y) + I G

Laplacian of Gaussian Filtering

Image

Gxx(x)

Gyy(y)

G(x)

G(y)

+ S2

Canny has shown that the first derivative of the Gaussian closely approximates the operator that optimizes the product of signal-to-noise ratio and localization. (i.e., analysis based on "step-edges" corrupted by "Gaussian noise“)

J. Canny, a Computational Approach to Edge Detection, IEEE Trans. Pattern Analysis and Machine Intelligence, 8:679-714, 1986.

:

canny edge detector

Steps of Canny edge detector:

Xf yf[1]Compute and

( * ) * *G Xf G f G fx x

( * ) * *G yf G f G fy y

Xf yf

G(x, y) is the Gaussian function

2 ( , y)x G x

2 ( , y)y G x

G xG x(x, y) is the derivate of (x, y) with respect to x: (x, y) =

G yG y(x, y) is the derivate of (x, y) with respect to y: (x, y) =

[2]Compute the gradient magnitude and direction: Magnitude (I, j) =2 2

X yf f

[3]Apply non-maxima suppression.

[4]Apply hysteresis thresholding/edge linking.

Canny Edge Detector Steps

1. Noise Reduction: Since edge detection is susceptible to noise in the image, first step is to

Remove the noise in the image with a 5x5 Gaussian filter.

2. Finding Intensity Gradient of the Image: Smoothened image is then filtered with Sobel kernel

in both horizontal and vertical direction to get first derivative in Horizontal

Direction and vertical direction from these two images, we can Find Edge

Gradient and Direction for each pixel as follows:2 2X yG GEdge Gradient= Angle ( ) = 1tan y

X

GG

3. Non-maxima suppression: After getting gradient magnitude and direction, a full scan of

image is done to remove any unwanted pixels which may not

constitute the edge for this, at every pixel, pixel is checked if it is

a local maximum in its neighborhood in the direction of gradient.

shows Point A is on the edge (in vertical direction).

Gradient direction is normal to the edge. Point B and C are in gradient directions. So point A is

checked with point B and C to see if it forms a local maximum. If so, it is considered for next

stage, otherwise, it is suppressed (put to zero). In short, the result you get is a binary image with

“thin edges”.

4. Hysteresis thresholding: (Hysteresis: A lag or momentum factor).

1 if ( , ) for some threshold T

( , )0 otherwise

f x y T

E x y

- Can only select “strong” edges and does not guarantee “continuity”.

- For “maybe” edges, decide on the edge if neighboring pixel is a strong edge.

 

Hysteresis thresholding uses two thresholds:-Low threshold 1t -high threshold ht ht 1t(usually = 2 )

1

0 1

0

( , ) definitely an edge

( , ) mayebe an edge,depends on context

( , ) < definitely not an edge

f x y t

t f x y t

f x y t

(a) (b) (c) (d) (e)

Figure 1.7

(a) shows [Original image], (b) shows how image represented using [Grayscale image], (c) show [Gradient

magnitude], (d) shows how image represented using [Thresholded gradient magnitude], and (e) shows

[non-maxima suppression (thinning)].

Criteria for Optimal Edge Detection

(A) Good detection: Minimize the probability of false positives (i.e., spurious edges).

Minimize the probability of false negatives (i.e., missing real edges).

(B) Good localization: Detected edges must be as close as possible to the true edges.

(C) Single response: Minimize the number of local maxima around the true edge.

Appendix

Sobel+Edge detection

Gaussian Filters

Gaussian blur (smoothing)Gaussian with convolution

Smoothing filter tool

Marr Hildreth Edge

Detector

References[1] J. F. Canny, .A computational approach to edge detection. IEEE Trans.Pattern Anal.

Machine Intel vol. 8, No. 6, pp. 679.698, Nov. 2007.

[2] Classical feature detection: URL: http://www.dai.ed.ac.uk/CVonline/LOCAL_COPIES/OWENS/LECT6/node2.html

[3] Edge detection tutorial URL: http://www.pages.drexel.edu/~weg22/edge.html

[4] Canny edge detection tutorial URL: http://www.pages.drexel.edu/~weg22/can_tut.html

[5] Sobel edge detection algorithm URL: http://www.dai.ed.ac.uk/HIPR2/sobel.htm

[6] Gaussian filtering URL: http://robotics.eecs.berkeley.edu/~mayi/imgproc/gademo.html

[7] URL: http://docs.opencv.org/doc/tutorials/imgproc/imgtrans/canny_detector/canny_detector.html

[19] URL: http://www.mathworks.com/help/coder/examples/edge-detection-on-images.html[20] URL: https://www.youtube.com/watch?v=q6fn-i16h20[21] URL: https://www.youtube.com/watch?v=q6fn-i16h20[22] URL: https://www.youtube.com/watch?v=CuOoz0eLmG8[23] URL: http://northstar-www.dartmouth.edu/doc/idl/html_6.2/Smoothing_an_Image.html[24] URL: http://en.wikipedia.org/wiki/Canny_edge_detector[25] URL: http://en.wikipedia.org/wiki/Edge_detection[26] URL: http://www.mathworks.com/discovery/edge-detection.html

The End