Chap 7 Image Segmentation. Edge-Based Segmentation The edge information is used to determine...

Chap 7

Image Segmentation

Image Segmentation

Edge-Based Segmentation The edge information is used to determine

boundaries of objects Pixel-based direct classification methods

The estimation methods derived from the histogram statistics of the image are used

Region-based methods Pixels are analyzed directly for a region growing

process based on a pre-defined similarity criterion

Edge-based image segmentation -Edge Detection Operations

The gradient magnitude and directional information can be obtained by convolving the edge detection masks with the image

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LaplaciangradientorderSecond


Laplacian is very sensitive to noise It is beneficial to apply a smoothing

filter(Gaussian) first before taking a Laplacian of the image

Laplacian of Gaussian (LOG) The image obtained by convolving the LOG mask

with the original image is analyzed for zero-crossing to detect edges.

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[Gonzalez]


LOG operation

[Gonzalez]Threshold = 0Threshold = 0

SobelSobel

LoGLoG

Edge-based image segmentation -Boundary Tracking

The closed regions can be formed by the edge-linking procedures following the edge-detection

The neighborhood bj+1 of edge point bj can be found to be a boundary point if it satisfied the following conditions.

31

21

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1

2mod Tbb

Tbebe

Tbe

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jj

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j

Where e is the magnitude and the edge orientation

Edge-based image segmentationHough Transform

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spaceparameter space -

Haugh Transform

[Gonzalez]

Edge-based image segmentationHough Transform

O

Gradient e

r

p

Figure 7.2. A model of the object shape to be detected in the image using Hough transform. The vector r connects the Centroid and a tangent point p. The magnitude and angle of the vector r are stored in the R-table at a location indexed by the gradient of the tangent point p.

Gray-Level Thesholding

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Optimal Global Thresholding

The parametric distribution based methods can be applied to the histogram of an image Assume the histogram has two Gaussian distributions

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For image segmentation, the objective is to find an optimal threshold T that minimizes the overall probability of error in pixel classification.

The process requires the parameterization of the probability density distributions to find the 1, 1, and 2, 2

The optimal global threshold T can be determined by finding a general solution the minimizes the equation (7.14) with mixture distribution in Equation (7.15)

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(7.14) )()()( 2112 TEPTEPTE

Optimal Global Thresholding (7.14) thus satisfies the following quadratic expression

If the variances of both classes can be assumed to be equal to 2

)/ln(2

)(2

0

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The determined threshold

Pixel Classification Through Clustering

The approach is useful when images with pixels representing a feature vector consisting of multiple parameters of interest are to be segmented. Parameters: Gray value, RGB components, contrast

and local texture measurements for each pixel Clustering may produce disjoint regions with holes or

regions with a single pixel Post-processing algorithm is usually applied region growing, pixel connectivity or rule-based

algorithm


k-means Clustering

Partition d-dimentional data into k clusters The objective function providing the desired properties

of the distribution of feature vectors of clusters

The algorithm is quite sensitive to the initial cluster assignment and the choice of the distance measure. (See the following algorithm for more)


k-means Clustering

Algorithm1. Select the number of clusters k with initial cluster centroids vi ;

i=1,2,…,k2. Partition the input data points into k clusters by assigning

each data point xj to the closest cluster centroid vi .(ex. Euclidean distance)

3. Compute a cluster assignment matrix U representing the partition of the data points with the binary membership value of the jth data point to the ith cluster such that U=[uij], where

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k-means Clustering

Algorithm (cont.)4. Re-computer the centroids using the membership values

as

5. If cluster centroids does not change, stop; otherwise go to step 2

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The k-means clustering method optimizes the sum-of-squared-error based objective function Jw(U,v)

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Fuzzy k-means Clustering

Utilize an adaptable membership value (uij) that can be updated

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Region Growing

Examine pixels in the neighborhood based on a pre-defined similarity criterion

The neighborhood pixels with similar properties are merged to form closed regions for segmentation

Guarantee the segmented regions of connected pixels

Region Growing

Two criteria A similarity criterion that defines the basis for

inclusion of pixels in the growth of the region A stopping criterion stops the growth of the region

1. Determine the seed for each region ( the pixels with gray 255 are selected to be seeds in this case)

2. Choose criteria for region growing• The absolute gray-level difference

between any pixels and seed• The detected pixel had to be 8-

connected to at least one pixel in that region

[Gonzalez]

Region Growing

Center Pixel

Pixels satisfying the similarity

criterion Pixels not satisfying similarity criterion

3x3 neighborhood, 100% (9/9) growth satisfied

5x5 neighborhood , 56%(9/16) growth satisfied

7x7 neighborhood, %(6/24) growth satisfied

Segmented region

1st iteration

2nd iteration

3rd iteration

The stopping criterion : the minimum percentage is 30%

An image is partitioned into a large number of potential homogeneous regions Be examined for homogeneity of pre-defined

property as gray values, contrast, texture… Two neighborhood regions with similar property are

merged

Region -Merging

Case 1 Case 2

Region -Splitting

Examine the heterogeneity of a predefined property of the entire region in terms of its distribution and the mean, variance, minimum and maximum values. Heterogeneous region R is split into two or more

regions R1, R2, …, Rn

Rule-based splitting and quad-tree based splitting

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Region -Splitting

An image with quad region-splitting process

R1

R3

R21

R23

R22

R24

R41

R43

R42

R44

R

R1 R2 R3 R4

R21 R22 R23 R24 R41 R42 R43 R44

Quad-tree structure

Recent advances in segmentation

The large variability in anatomical structures needs of a reliable, accurate, and diagnostically useful segmentation

Model-based estimation methods Rule-based methods

Estimation-Model Based Adaptive Segmentation

-Multi-level Adaptive Segmentation (MAS)

Define Classes

Determination of Model Parameters

(From a set ofmanually segmented and labeled images)

Classification of ImagePixels Using Model

Signatures

Identification of TissueClasses and Pathology

Formation ofa New ClassNecessary?

All pixelsclassified?

ParameterRelaxation

Yes

Yes

No

No

Image Segmentation Using Neural Networks

-Neural Network

Neural Network do not require underlying class probability distribution for accurate classification

The decision boundaries for pixel classification are adapted through an iterative training process (supervised neural network)

Neural Network paradigms Backpropagation, Radial Basis Function, and Self-

Organizing Feature Maps


-Neural Network

Important works for neural network Selection of a useful set of features as input data for

classification Selection of the training examples The structure of the network

Artificial Neural Network: Backpropagation

Input Features

Output Results

Hidden Layer Neurons

Output Layer Neurons

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Layer 1

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Layer L


-Backpropropagation Neural Network

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Architectural graph

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Signal-flow graph

Layer k Layer k+1



Least Mean Squared(LMS) error algorithm for training the neural network

1. Assign random weights in the range of [-1,+1] to all weights wij

k

2. For each classified pattern pair {y(0),yL} in the train set, do the following steps:

a. Compute the output values of each neural element using the current weight matrix.

b. Find the error e(k) between the computed output vector and the desired out vector.

c. Adjust the weight matrix using the change dW(k) computed as dW(k) =e(k)[y(k-1)] for all layers =1,…,L, where is the learning rate that can be set between 0 and 1.

3. Repeat Step 2 for all classified pattern pairs in the training set until the error vector for each training example is sufficiently low or zero.

RBF Network

RBF Unit 1

RBF Unit 2

RBF Unit n

Input ImageSliding

Image Window

Output

Linear Combiner

RBF Layer

RBF Network

The final output of the network f(x)

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RBF Network

Within the basic topology, the system can be represented in state-space from y=(F,w), where F is the matrix of activation functions

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NN

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.........

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The weight van be calculated from the Moore-Penrose inverse

where is a small value such as 0.1

RBF NN Based Segmentation

Chap 7 Image Segmentation. Edge-Based Segmentation The edge information is used to determine...

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