Segmentation and Representation

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Ilene Atkinson
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1 Segmentation and Representation

2 Topics Segmentation Texture Based Segmentation Level Set Segmentation Watershed Segmentation Representation Introduction Chain Codes Polygonal Approximations Signatures Boundary Segments Skeletons

3 Introduction Image segmentation is the process of partitioning the digital image into multiple regions that can be associated with the properties of one or more objects

4 Topics Segmentation Texture Based Segmentation Level Set Segmentation Watershed Segmentation Representation Introduction Chain Codes Polygonal Approximations Signatures Boundary Segments Skeletons

5 Texture Texture provides measures of properties such as smoothness, coarseness, and regularity.

6 Texture Based Segmentation

7 Co-Occurance Matrix Let P be a position operator, and A a k x k matrix. a ij shows the number of times that pixels with gray level z i occur at position given by P relative to points with gray level z j. Matrix A is called co-occurance matrix and can provide statistical properties of the texture.

8 Example Assume P is one pixel to the right and one pixel below Gray level values are : 0, 1, and 2 Image data: Co-occurance matrix is:

9 Statistical Moments of Texture Let Matrix C be formed by dividing every element of A by the number of point pairs that satisfy P. The following moments are defined to compare textures:

10 Topics Segmentation Texture Based Segmentation Level Set Segmentation Watershed Segmentation Representation Introduction Chain Codes Polygonal Approximations Signatures Boundary Segments Skeletons

11 Level Set Segmentation Instead of manipulating the contour directly, the contour is embedded as the zero level set of a higher dimensional function called the level-set function y(x, t). The level-set function is evolved under the control of a differential equation. At any time, the evolving contour can be obtained by extracting the zero level-set G((X), t) = {y(x, t) = 0} from the output

12 Zero Set in a Level Set

13 Example

14 Topics Segmentation Texture Based Segmentation Level Set Segmentation Watershed Segmentation Representation Introduction Chain Codes Polygonal Approximations Signatures Boundary Segments Skeletons

15 Watershed Segmentation Algorithm: Convert the gray level image into a topographic image where the height of each point is proportional to its gray level intensity. Punch a hole at each region minimum at let the whole topography be flooded from below. The points where the water from different regions join are boundaries of the regions

16 Watershed Segmentation

17 Example

18 Example

19 Topics Segmentation Texture Based Segmentation Level Set Segmentation Watershed Segmentation Representation Introduction Chain Codes Polygonal Approximations Signatures Boundary Segments Skeletons

20 Representation The result of segmentation should be represented and described in a form suitable for further computer processing. A region can be represented in terms of its external characteristics (boundary). A region can be represented in terms of its internal characteristics.

21 Topics Segmentation Texture Based Segmentation Level Set Segmentation Watershed Segmentation Representation Introduction Chain Codes Polygonal Approximations Signatures Boundary Segments Skeletons

22 Chain Codes Chain codes are generated by following a boundary in a clockwise or counter-clockwise direction and assigning a direction to the segments connecting every pair of pixels. Disadvantage: Can be unacceptably long. Solution: Re-sampling (down sample) the boundary Disadvantage: Is starting point dependent Solution: Normalize the representation string to the smallest integer.

23 Chain Code Directions

24 Sample Chain Code

25 Down Sampling

26 Topics Segmentation Texture Based Segmentation Level Set Segmentation Watershed Segmentation Representation Introduction Chain Codes Polygonal Approximations Signatures Boundary Segments Skeletons

27 Polygonal Approximation A boundary can be represented with arbitrary accuracy by a polygon. The approximation is exact when the number of sides is equal to the number of points in the boundary. Finding a polygonal representation can be very timeconsuming.

28 Minimum Perimeter Polygons

29 Splitting Techniques

30 Topics Segmentation Texture Based Segmentation Level Set Segmentation Watershed Segmentation Representation Introduction Chain Codes Polygonal Approximations Signatures Boundary Segments Skeletons

31 Signature A signature is a 1D representation of a boundary. e.g. Plotting distance to centroid as a function of angle Invariant to translation Disadvantages: Rotation and scaling dependant Defined only for convex regions

32 Signature Example

33 Topics Segmentation Texture Based Segmentation Level Set Segmentation Watershed Segmentation Representation Introduction Chain Codes Polygonal Approximations Signatures Boundary Segments Skeletons

34 Boundary Segments Decomposing a boundary into segments simplifies representation. Convex Hull can be used for decomposition. A new segment can be started whenever a Convex Hull deficiency is entered or exited.

35 Boundary Segments Example

36 Topics Segmentation Texture Based Segmentation Level Set Segmentation Watershed Segmentation Representation Introduction Chain Codes Polygonal Approximations Signatures Boundary Segments Skeletons

37 Skeleton The structural shape of a region can be represented by a graph. The structural graph is obtained by thinning the region and finding the skeleton.

38 Questions?

Torgerson s Classical MDS derivation: 1: Determining Coordinates from Euclidean Distances

Torgerson s Classical MDS derivation: 1: Determining Coordinates from Euclidean Distances It is possible to construct a matrix X of Cartesian coordinates of points in Euclidean space when we know the Euclidean