Digital Image Processing Chapter 10 Image Segmentation 13

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Digital Image Processing Chapter 10: Image Segmentation 13 July 2005

Digital Image Processing Chapter 10: Image Segmentation 13 July 2005

Element of Image Analysis Preprocess Image acquisition, restoration, and enhancement Intermediate process Image segmentation

Element of Image Analysis Preprocess Image acquisition, restoration, and enhancement Intermediate process Image segmentation and feature extraction High level process Image interpretation and recognition ISAN-DSP GROUP

Importance of Image Segmentation Image segmentation is used to separate an image into constituent

Importance of Image Segmentation Image segmentation is used to separate an image into constituent parts based on some image attributes. Image segmentation is an important step in image analysis Benefit 1. Image segmentation reduces huge amount of unnecessary data while retaining only importance data for image analysis 2. Image segmentation converts bitmap data into better structured data which is easier to be interpreted

Image Attributes for Image Segmentation 1. Similarity properties of pixels inside the object are

Image Attributes for Image Segmentation 1. Similarity properties of pixels inside the object are used to group pixels into the same set. 2. Discontinuity of pixel properties at the boundary between object and background is used to distinguish between pixels belonging to the object and those of background. Discontinuity: Intensity change at boundary Similarity: Internal pixels share the same intensity

Spatial Filtering Application to Shape Detection v One application of spatial filtering is shape

Spatial Filtering Application to Shape Detection v One application of spatial filtering is shape detection: finding locations of objects with the desired shape. v Unlike frequency selective masks that are designed based on the concept of frequency, shape detection masks are derived from the shapes to be detected themselves. v A mask for shape detection usually contains the shape or a part of the shape to be detected. v The location that is most correlated to the mask is the location where the highest filter response occurs. The shape is most likely to exist there.

Point Detection v We can use Laplacian masks for point detection. v Laplacian masks

Point Detection v We can use Laplacian masks for point detection. v Laplacian masks have the largest coefficient at the center of the mask while neighbor pixels have an opposite sign. -1 -1 -1 0 -1 8 -1 -1 4 -1 -1 0 v This mask will give the high response to the object that has the similar shape as the mask such as isolated points. v Notice that sum of all coefficients of the mask is equal to zero. This is due to the need that the response of the filter must be zero inside a constant intensity area

Point Detection Point detection can be done by applying the thresholding function: Location of

Point Detection Point detection can be done by applying the thresholding function: Location of porosity X-ray image of the turbine blade with porosity Laplacian image After thresholding (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd Edition.

Line Detection v Similar to point detection, line detection can be performed using the

Line Detection v Similar to point detection, line detection can be performed using the mask the has the shape look similar to a part of a line v There are several directions that the line in a digital image can be. v For a simple line detection, 4 directions that are mostly used are Horizontal, +45 degree, vertical and – 45 degree. Line detection masks (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd Edition.

Line Detection Example Binary wire bond mask image Absolute value of result after processing

Line Detection Example Binary wire bond mask image Absolute value of result after processing with -45 line detector Result after thresholding Notice that – 45 degree lines are most sensitive (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd Edition.

Edges Ideal step edge Ideal ramp edge Blurred edge Generally, objects and background have

Edges Ideal step edge Ideal ramp edge Blurred edge Generally, objects and background have different intensities. Therefore, Edges of the objects are the areas where abrupt intensity changes occur. (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd Edition.

Ideal Ramp Edges and its Derivatives Edge Original image (Images from Rafael C. Gonzalez

Ideal Ramp Edges and its Derivatives Edge Original image (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd Edition.

Gray level profile Intensity Smoothed Step Edge and Its Derivatives The 1 st derivative

Gray level profile Intensity Smoothed Step Edge and Its Derivatives The 1 st derivative The 2 nd derivative 1. 2 1 0. 8 0. 6 0. 4 0. 2 0 -0. 2 -1. 5 0. 06 0. 04 0. 02 0 -0. 02 -0. 04 -0. 06 -1. 5 Edge -1 -0. 5 Edge 0 0. 5 1 1. 5 2 2. 5 3 3. 5 Minimum point -1 x 10 -3 5 4 3 2 1 0 -1 -2 -3 -4 -5 -1 Maximum -0. 5 point 0 0. 5 1. 5 2 Zero crossing + -0. 5 1 0 0. 5 1 1. 5 + 2

Derivative Based Edge Detection v From the previous slide, we can conclude that: Local

Derivative Based Edge Detection v From the previous slide, we can conclude that: Local maxima of the absolute of the 1 st derivative and Zero crossing of the 2 nd derivative occur at edges. v Therefore, for detecting edges, we can apply zero crossing detection to the 2 nd derivative image or thresholding the absolute of the 1 st derivative image. v Nevertheless, derivative operator is very sensitive to noise as we will see in the next slide.

Noisy Edges and Derivatives f(x) Derivative operator is a highpass filter and thus enhances

Noisy Edges and Derivatives f(x) Derivative operator is a highpass filter and thus enhances noise. AWGN s = 0. 1 AWGN s = 1. 0 Edge responses are buried by noise. AWGN s = 10 (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd Edition.

Masks for Estimating Partial Derivatives Normally, the mask for estimating partial derivative is antisymmetry

Masks for Estimating Partial Derivatives Normally, the mask for estimating partial derivative is antisymmetry with respect to the orthogonal axis For example, the Sobel mask for computing is anti-symmetry with respect to the y-axis. It has the positive sign on the right side and negative sign on the left side. Notice that sum of all coefficients is equal to zero to make sure that the response of a constant intensity area is zero. (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd Edition.

Masks for Detecting Diagonal Edges The mask for detecting -45 -degree edges is anti-symmetry

Masks for Detecting Diagonal Edges The mask for detecting -45 -degree edges is anti-symmetry with respect to the – 45 -degree lines while the mask for detecting 45 -degree edges is anti-symmetry with respect to the 45 -degree lines. (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd Edition.

Example of Image Gradient (Images from Rafael C. Gonzalez and Richard E. Wood, Digital

Example of Image Gradient (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd Edition.

Example of Image Gradient Note: the original image is smoothed by a 5 x

Example of Image Gradient Note: the original image is smoothed by a 5 x 5 moving average mask first. (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd Edition.

Example of Diagonal Edges Using -45 -degree mask Using 45 -degree mask Note: the

Example of Diagonal Edges Using -45 -degree mask Using 45 -degree mask Note: the original image is smoothed by a 5 x 5 moving average mask first. (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd Edition.

Laplacian Masks The Laplacian masks are used to estimate the Laplacian image: Ideally, the

Laplacian Masks The Laplacian masks are used to estimate the Laplacian image: Ideally, the Laplacian mask must be directional invariant: symmetry in all direction (radially symmetry ). However, for 3 x 3 masks, there are Only 8 possible directions. Hence, we can use the following masks: (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd Edition.

Laplacian Masks For a large scale Laplacian mask, we can use a Laplacian of

Laplacian Masks For a large scale Laplacian mask, we can use a Laplacian of Gaussian (LOG) as a mask: Surface plot of LOG, Looks like a “Mexican hat” Cross section of LOG image 5 x 5 LOG mask (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd Edition.

Example of Laplacian Image The angiogram image (blood vessels) We can compute the Laplacian

Example of Laplacian Image The angiogram image (blood vessels) We can compute the Laplacian image by: . 1 Smooth the image by the Gaussian mask 2. Compute the Laplacian image using the mask 1 2 Threshold of Zero crossing The left image (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd Edition.

Pixel Oriented Image Segmentation: Thresholding �������������� ������ Intensity Thresholding T = 102 ��������� ���

Pixel Oriented Image Segmentation: Thresholding �������������� ������ Intensity Thresholding T = 102 ��������� ��� Histogram ������ multimo �������������� After thresholding

Intensity Thresholding Example (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image

Intensity Thresholding Example (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd Edition.

Automatic Threshold Level Selection The major problem of intensity thresholding is to find a

Automatic Threshold Level Selection The major problem of intensity thresholding is to find a good threshold level Algorithm: effective for bimodal histogram 1. Set initial value of T 2. 3. 4. 5. Repeat step 2

Automatic Threshold Level Selection Example (Images from Rafael C. Gonzalez and Richard E. Wood,

Automatic Threshold Level Selection Example (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd Edition.

Multilevel Intensity Thresholding �������������� Histogram T 1 = 158 T 2 = 196 T

Multilevel Intensity Thresholding �������������� Histogram T 1 = 158 T 2 = 196 T 3 = 228 T 1< P <T 2 T 2< P <T 3 P > T 3

Noise Problem �������������� peak ���� Histogram Image degraded by Gaussian noise (s =12) T

Noise Problem �������������� peak ���� Histogram Image degraded by Gaussian noise (s =12) T 1 = 158 T 2 = 196 T 3 = 228 T 1< P <T 2 T 2< P <T 3 P > T 3

Nonuniform Illumination Problem Reflectance Function r(x, y) Histogram An image can be expressed as

Nonuniform Illumination Problem Reflectance Function r(x, y) Histogram An image can be expressed as i(x, y) = illumination component r(x, y) = reflectance component Illumination Function i(x, y) Image histogram f(x, y) (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd Edition.

Nonuniform Illumination and Global Thresholding Global threshold level Global thresholding of nonuniform illumination image

Nonuniform Illumination and Global Thresholding Global threshold level Global thresholding of nonuniform illumination image can cause huge errors! Histogram Nonuniform illumination image Global thresholding result (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd Edition.

Nonuniform Illumination and Local Thresholding Local thresholding: 1. Divide an image into subimages. 2.

Nonuniform Illumination and Local Thresholding Local thresholding: 1. Divide an image into subimages. 2. Threshold each subimage independently 2. 1 Compute histogram of each subimage and select a suitable threshold value for each subimage 2. 2 threshold each subimage using a threshold value in 2. 1 2. 3 Combine all local thresholding results Error 16 subimages Result of local thresholding

Histogram of Subimages and Local Thresholding If areas of object and background are not

Histogram of Subimages and Local Thresholding If areas of object and background are not balanced, a histogram will be unimodal. If areas of object and background are nearly equal, a histogram will be bimodal (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd Edition.

Example of Nonuniform Illumination Problem ������������� T=0. 4 Error ������ ISAN-DSP GROUP

Example of Nonuniform Illumination Problem ������������� T=0. 4 Error ������ ISAN-DSP GROUP

Optimum Thresholding Background Object p 1(z) = PDF of object pixels p 2(z) =

Optimum Thresholding Background Object p 1(z) = PDF of object pixels p 2(z) = PDF of background pixels Error due to background pixels classified as object pixels is : Error due to object pixels classified as background pixels is: Total error = P 1 = Probability of occurrence of object pixels P 2 = Probability of occurrence of background pixels (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd Edition.

Optimum Thresholding

Optimum Thresholding

Optimum Thresholding (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing,

Optimum Thresholding (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd Edition.

Optimum Thresholding (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing,

Optimum Thresholding (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd Edition.

Pixel Oriented Image Segmentation for Color Images RGB, CMY color models: Thresholding based on

Pixel Oriented Image Segmentation for Color Images RGB, CMY color models: Thresholding based on distance in the color space. HSI color model: Thresholding based on H and S component mainly. (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd Edition.

Color Segmentation in HSI Color Space Color image Hue 1 2 3 4 Saturation

Color Segmentation in HSI Color Space Color image Hue 1 2 3 4 Saturation Intensity (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd Edition.

Color Segmentation in HSI Color Space (cont. ) Binary thresholding of S component with

Color Segmentation in HSI Color Space (cont. ) Binary thresholding of S component with T = 10% 5 Product of 2 and 5 6 Red pixels 7 Histogram of 6 8 Segmentation of (Images red color from Rafaelpixels C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd Edition.

Color Segmentation in HSI Color Space (cont. ) Color image Segmented results of red

Color Segmentation in HSI Color Space (cont. ) Color image Segmented results of red pixels (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd Edition.

Color Segmentation in RGB Vector Space (Images from Rafael C. Gonzalez and Richard E.

Color Segmentation in RGB Vector Space (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd Edition. 1. Each point with (R, G, B) coordinate in the vector space represents one color. 2. Segmentation is based on distance thresholding in a vector space D(u, v) = distance function c. T = color to be segmented. c(x, y) = RGB vector at pixel (x, y). (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd Edition.

Example: Segmentation in RGB Vector Space Color image Reference color c. T to be

Example: Segmentation in RGB Vector Space Color image Reference color c. T to be segmented Results of segmentation in RGB vector space with Threshold value T = 1. 25 times the SD of R, G, B values In the box (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd Edition.

Color Image Segmentation Example (Images from Rafael C. Gonzalez and Richard E. Wood, Digital

Color Image Segmentation Example (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd Edition.

Pixel Oriented Image Segmentation for Multispectral Images ����� pixel oriented image segmentation ���� multispectrum

Pixel Oriented Image Segmentation for Multispectral Images ����� pixel oriented image segmentation ���� multispectrum ������ partition �� Feature spac ������������ Partition Image Domain Boundary ������ Munich ���� Blue 1 0. 5 0 1 0. 2 0. 8 0. 4 0. 6 From www. jpl. nasa. gov/radar/ sircxsar/munch. html 0. 4 0. 8 Red 1 0. 2 Green Feature Space (RGB)

Region Oriented Image Segmentation �������������� ��������� �������������������� Region oriented image segmentation 1. Region Growing

Region Oriented Image Segmentation �������������� ��������� �������������������� Region oriented image segmentation 1. Region Growing 2. Region Splitting and Merging P Connected pixels Pixel P and its neighbors

Region Growing Image Segmentation Example �������� ISAN-DSP GROUP ���� Region growing � �� �����

Region Growing Image Segmentation Example �������� ISAN-DSP GROUP ���� Region growing � �� ����� ��� ���������� ������

Region Growing Image Segmentation Example Histogram X-ray image of defective weld (Images from Rafael

Region Growing Image Segmentation Example Histogram X-ray image of defective weld (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd Edition.

Region Growing Image Segmentation Example (Images from Rafael C. Gonzalez and Richard E. Wood,

Region Growing Image Segmentation Example (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd Edition.

Quadtree for Region Splitting Representation (Images from Rafael C. Gonzalez and Richard E. Wood,

Quadtree for Region Splitting Representation (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd Edition.

Region Splitting Algorithm Region Splitting ISAN-DSP GROUP

Region Splitting Algorithm Region Splitting ISAN-DSP GROUP

Region Splitting Example ���������� Region Spliting ����� standard deviation ��� ����� ISAN-DSP GROUP

Region Splitting Example ���������� Region Spliting ����� standard deviation ��� ����� ISAN-DSP GROUP

Region Merging Algorithm 2. Merging ����������� ��� ISAN-DSP GROUP

Region Merging Algorithm 2. Merging ����������� ��� ISAN-DSP GROUP

Region Merging Example ���������� Region Merging ��������� ISAN-DSP GROUP

Region Merging Example ���������� Region Merging ��������� ISAN-DSP GROUP

Region Splitting and Merging Example (Images from Rafael C. Gonzalez and Richard E. Wood,

Region Splitting and Merging Example (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd Edition.

Gray level profile Intensity Smoothed Step Edge and Its Derivatives The 1 st derivative

Gray level profile Intensity Smoothed Step Edge and Its Derivatives The 1 st derivative The 2 nd derivative 1. 2 1 0. 8 0. 6 0. 4 0. 2 0 -0. 2 -1. 5 0. 06 0. 04 0. 02 0 -0. 02 -0. 04 -0. 06 -1. 5 Edge -1 -0. 5 Edge 0 0. 5 1 1. 5 2 2. 5 3 3. 5 Minimum point -1 x 10 -3 5 4 3 2 1 0 -1 -2 -3 -4 -5 -1 Maximum -0. 5 point 0 0. 5 1. 5 2 Zero crossing + -0. 5 1 0 0. 5 1 1. 5 + 2

Derivative Based Edge Detection v From the previous slide, we can conclude that: Local

Derivative Based Edge Detection v From the previous slide, we can conclude that: Local maxima of the absolute of the 1 st derivative and Zero crossing of the 2 nd derivative occur at edges. v Therefore, for detecting edges, we can apply zero crossing detection to the 2 nd derivative image or thresholding the absolute of the 1 st derivative image. v Nevertheless, derivative operator is very sensitive to noise as we will see in the next slide.

Masks for Estimating Partial Derivatives Normally, the mask for estimating partial derivative is antisymmetry

Masks for Estimating Partial Derivatives Normally, the mask for estimating partial derivative is antisymmetry with respect to the orthogonal axis For example, the Sobel mask for computing is anti-symmetry with respect to the y-axis. It has the positive sign on the right side and negative sign on the left side. Notice that sum of all coefficients is equal to zero to make sure that the response of a constant intensity area is zero. (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd Edition.

Gradient Vector Field Gradient ������� Vector ����������� Gradient Vector Field ������� �� ����

Gradient Vector Field Gradient ������� Vector ����������� Gradient Vector Field ������� �� ����

Gradient Based Image Segmentation ����� |ÑP| Thresholding ��� T ����� Edge map ��� T

Gradient Based Image Segmentation ����� |ÑP| Thresholding ��� T ����� Edge map ��� T �������������� T=60 T=100

Laplacian Based Image Segmentation ����� Ñ 2 P Zero Crossing Detection Edge map Laplacian

Laplacian Based Image Segmentation ����� Ñ 2 P Zero Crossing Detection Edge map Laplacian �������� Zero crossing Detection ����� thresholding ����� T=0 ISAN-DSP GROUP

Gradient VS Laplacian Gradient Laplacian ÑP Ñ 2 P Local Extrema Zero Crossing ��������

Gradient VS Laplacian Gradient Laplacian ÑP Ñ 2 P Local Extrema Zero Crossing �������� of |ÑP| ������� (step edge) �������������� ����������� ��� Thresholding Zero Crossing ������� Detection |ÑP| ����������� �� edge map ������ �� ISAN-DSP GROUP

Boundary Extraction Based on Particle motion in a Vector Field Gradient Vector Field Edge

Boundary Extraction Based on Particle motion in a Vector Field Gradient Vector Field Edge Vector Field (Hamiltonian Gradient Field)

Edge Vector Field Edge vector field ��������� ISAN-DSP GROUP

Edge Vector Field Edge vector field ��������� ISAN-DSP GROUP

Particle Trajectory in an Edge Vector Field ����������� edge vector field ���� a>0 ������������

Particle Trajectory in an Edge Vector Field ����������� edge vector field ���� a>0 ������������ : �� edge vector field ����������� (spiral trajecto

Example: Particle Trajectory in an Edge Vector Field ����� P 0 Edge vector field

Example: Particle Trajectory in an Edge Vector Field ����� P 0 Edge vector field �����

A Normal Compressive Vector Field ������������ vector field ������ compressive) ���������� Normal Compressive Field

A Normal Compressive Vector Field ������������ vector field ������ compressive) ���������� Normal Compressive Field

Example: A Normal Compressive Vector Field ISAN-DSP GROUP

Example: A Normal Compressive Vector Field ISAN-DSP GROUP

Combined Vector Field �������� ISAN-DSP GROUP

Combined Vector Field �������� ISAN-DSP GROUP

Particle Trajectory in a Combined Vector Field ���������� ���������

Particle Trajectory in a Combined Vector Field ���������� ���������

Example: Particle Trajectory in a Combined Vector Field Edge vector field Particle trajectory P

Example: Particle Trajectory in a Combined Vector Field Edge vector field Particle trajectory P 0 Shape distortion at a corner

Boundary Extraction of Multiple Objects 1 -way boundary extraction 2 -way boundary extraction �������������������������

Boundary Extraction of Multiple Objects 1 -way boundary extraction 2 -way boundary extraction ������������������������� ISAN-DSP GROUP

Boundary Extraction Algorithm Based on Particle Motion in a Vector Field . 1����� Mask

Boundary Extraction Algorithm Based on Particle Motion in a Vector Field . 1����� Mask ���������� Gradient ��� Laplacian ������. 2����� edge vector field ��� normal compressive vector f . 3������������ Local maxima ��� Gradient magnitude image. 4������������ . 5 ������������������������� ISAN-DSP GROUP

Boundary Extraction Example (Obtained using multiple scale Gaussian differential masks with smin = 0.

Boundary Extraction Example (Obtained using multiple scale Gaussian differential masks with smin = 0. 5, smax = 2. 0) ������ The whole brain atlas, www. med. harvard. edu/AANLIB/home. html ISAN-DSP GROUP

Boundary Extraction Example: Flaw Detection Original X-ray image Flaws Gradient image Boundary extraction results

Boundary Extraction Example: Flaw Detection Original X-ray image Flaws Gradient image Boundary extraction results (s=1. 0)

Multiscale Boundary Extraction Gradient images and extracted boundaries s 1=0. 5 s 2=4. 77

Multiscale Boundary Extraction Gradient images and extracted boundaries s 1=0. 5 s 2=4. 77 s 3=18. 81

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd Edition.

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd Edition.

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd Edition.

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd Edition.

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd Edition.

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd Edition.

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd Edition.

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd Edition.

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd Edition.

Edge Linking

Edge Linking