Random Walks for Image segmentation

1. Leo Grady, Member, IEEE Random Walks for Image segmentation IEEE Transaction on pattern analysis and machine intelligence, November 2006

2. Outline • Introduction • Algorithm • Dirichlet Problem • Behavioral Properties • Result--Demo

3. Introduction • K-way image segmentation • User-defined seeds user-defined labels • How to labels theunseeded pixel ? K objects

4. Introduction • Algorithm is resolving the question: • Given a random walker starting at this location, what is the probability that it first reaches each of the K seed points ? • Probability = ? A random walker first reach each of the K seed points. K-tuple vector Probability? A pixel Seed point

5. Introduction The probability a random walker first reaches a seed point. The solution to the Dirichelet problem with boundary conditions at the locations of the seed points and the seed point in question fixed to unity while the others are set to zero.

6. Introduction • Goal • 1. location of weak (or missing) boundaries. • 2. noise robustness. • 3. ability to identify multiple objects simultaneously. • 4. fast computation (and editing). • 5. avoidance of small/trivial solutions.

7. Algorithm • 1. generating the graph weights • 2. establishing the system of equations to solve the problem. • 3. the practical details of implementation.

8. Defining a graph • Graph G = ( V , E ) • And edge, e, spanning two vertices, vi and vj, is denoted by eij. • The weight of an edge, eij, is denoted by w(eij) or wij. • Degree of a vertex is • Assume this graph is connected and undirected.

9. Edge Weights • Gaussian weighting function The only free parameter gi indicates the image intensity at pixel i

10. Combinatorial Dirichlet Problem • The Dirichlet integral • A harmonic functionis a function that satisfies the Laplace equation • Dirichlet problem - finding a harmonic function subject to its boundary values.

11. Combinatorial Dirichlet Problem(cont.) • Combinatorial Laplacian matrix Where Lij is indexed by vertices vi and vj. • The m x nedge-node incidence matrix as • Incidence matrix is indexed by edge eij and node vk.

12. Combinatorial Dirichlet Problem(cont.) • A combinatorial formulation of the Dirichlet integral • C is the mxmconstitutive matrix (the diagonal matrix with the weights of each edge along the diagonal).

13. Combinatorial Dirichlet Problem(cont.) • Partition the vertices into two sets, • VM (marked/seed nodes) • VU (unseeded nodes) • Finding the critical point yields

14. Combinatorial Dirichlet Problem(cont.) • The probability (potential) assumed at node, vi, for each label, s, by . • Define the set of labels for the seed points as a function • Define the vector for each label, s, at node as

15. Solving the combinatorial Dirichlet problem • For one label • For all labels • X has K columns taken by each and M has colums given by each

16. Equivalences between random walks and electrical circuits • Three fundamental equations of circuit theory. • These three equations may be combined into the linear system • It is equivalent to with f = 0.

17. Algorithm Summary • Using , map the image intensities to edge weights in the lattice. • Obtain a set, VM, of marked (labeled) pixels with K labels, either interactively or automatically. • Solve outright for the potentials or solve for each label except the final one, f. Set • Obtain a final segmentation by assigning to each node, vi, the label corresponding to

18. Overview of segmentation computation

19. Analogies • Assigns an unseeded pixel to a label, given a weighted graph: • If a random walker leaving the pixel is most likely to first reach a bearing label s, assign the pixel to label s. • If the seeds are alternately replaced by grounds/unit voltage sources, assign the pixel to the label for which its seeds being “on” produces the greatest electrical potential. • Assign the pixel to the label for which its seeds have the largest effective conductance. • If a 2-tree is drawn randomly from the graph, assign the pixel to the label for which the pixel is most likely to remain connected to.

20. Effective Conductance Unit voltage • Effective conductance • Dirichlet integral equals the effective conductance between nodes labeled “1” (“on”) and those labeled “0” (“off”). • x is intended to include both xM and xU equals i j i j Current flow

21. Effective Conductance (cont.) • Effective conductance between two nodes, vi, vj is given by • where T is a set of edges defining a connected tree • the sum is over all possible trees in the graph • Where TT(i, j) is used to represent the set of edges defining a 2-tree, such that node vi is in one component and vj is in another.

22. 2-tree • A 2-tree is defined to be a tree with one edge removed. • vi and vj are indifferent components and vt is in the same component as vj. • Note that

23. 2-tree (cont.) • Then, the following expressions are equivalent • The segmentation is computed from the potentials by assigning the pixel to the label for which it has greatest potential (probability).

24. Behavioral Properties • Weak Boundary detection • Noise robustness • Assignment of ambiguous regions

25. Weak Boundaries

26. Weak Boundaries - Comparison

27. Noise Robustness

28. Ambiguous Unseeded Regions

29. Demo Videos • http://www.cns.bu.edu/~lgrady/Random_Walker_Image_Segmentation.html • Brain • Lung tumor • Aorta-3D

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37. Definition of a harmonic function • Any real function u(x, y) with continuous second partial derivatives which satisfies Laplace's equation, is called a harmonic function. Back Reference from Mathworld: http://mathworld.wolfram.com/HarmonicFunction.html