1 / 37

Random Walks for Image segmentation

Leo Grady, Member, IEEE. Random Walks for Image segmentation. IEEE Transaction on pattern analysis and machine intelligence, November 2006. Outline. Introduction Algorithm Dirichlet Problem Behavioral Properties Result--Demo. Introduction. K-way image segmentation

tyne
Télécharger la présentation

Random Walks for Image segmentation

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

  30. References • L. Grady and G. Funka-Lea, “Multi-Label Image Segmentation for Medical Applications Based on Graph-Theoretic Electrical Potentials,” Proc. Workshop Computer Vision and Math. Methods in Medical and Biomedical Image Analysis, pp. 230-245, May 2004. • R. Wallace, P.-W. Ong and E. Schwartz, “Space Variant Image Processing,” Int"l J. Computer Vision, vol. 13, no. 1, pp. 71-90, Sept. 1994. • L. Grady, “Space-Variant Computer Vision: A Graph-Theoretic Approach,” PhD dissertation, Boston Univ., 2004. • S. Kakutani, “Markov Processes and the Dirichlet Problem,” Proc. Japanese Academy, vol. 21, pp. 227-233, 1945. • P. Doyle and L. Snell, Random Walks and Electric Networks, no. 22, Washington, D.C.: Math. Assoc. of Am., 1984. • R. Courant and D. Hilbert, Methods of Math. Physics, vol. 2. John Wiley and Sons, 1989. • R. Hersh and R.J. Griego, “Brownian Motion and Potential Theory,” Scientific Am., vol. 220, pp. 67-74, 1969. • F.H. BraninJr. “The Algebraic-Topological Basis for Network Analogies and the Vector Calculus,” Proc. Conf. Generalized Networks, pp. 453-491, Apr. 1966. • P. Tetali, “Random Walks and the Effective Resistance of Networks,” J. Theoretical Probability, vol. 4, no. 1, pp. 101-109, 1991. • N. Biggs, “Algebraic Potential Theory on Graphs,” Bull. London Math. Soc., vol. 29, pp. 641-682, 1997.

  31. Z. Tu and S.-C. Zhu, “Image Segmentation by Data-Driven Markov Chain Monte Carlo,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 24, no. 5, pp. 657-673, May 2002. S.C. Zhu, Y. Wu and D. Mumford, “Filters, Random Field and Maximum Entropy (FRAME),” Int"l J. Computer Vision, vol. 27, no. 2, pp. 1-20, Mar./Apr. 1998. P. Perona and J. Malik, “Scale-Space and Edge Detection Using Anisotropic Diffusion,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 12, no. 7, pp. 629-639, July 1990. J. Shi and J. Malik, “Normalized Cuts and Image Segmentation,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 22, no. 8, pp. 888-905, Aug. 2000. E. Mortensen and W. Barrett, “Interactive Segmentation with Intelligent Scissors,” Graphical Models in Image Processing, vol. 60, no. 5, pp. 349-384, 1998. J.A. Sethian, Level Set Methods and Fast Marching Methods. Cambridge Univ. Press, 1999. M. Kass, A. Witkin and D. Terzopoulos, “Snakes: Active Contour Models,” Int"l J. Computer Vision, vol. 1, no. 4, pp. 321-331, 1987. Y. Boykov and M.-P. Jolly, “Interactive Graph Cuts for Optimal Boundary and Region Segmentation of Objects in N-D Images,” Proc. Int"l Conf. Computer Vision, pp. 105-112, 2001. Y. Boykov and V. Kolmogorov, “An Experimental Comparison of Min-Cut/Max-Flow Algorithms for Energy Minimization in Vision,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 26, no. 9, pp. 1124-1137, Sept. 2004. Y. Boykov, O. Veksler and R. Zabih, “A New Algorithm for Energy Minimization with Discontinuities,” Proc. Second Int"l Workshop Energy Minimization Methods in Computer Vision and Pattern Recognition, pp. 205-220, July 1999.

  32. L. Grady, “Multilabel Random Walker Image Segmentation Using Prior Models,” Proc. 2005 IEEE CS Conf. Computer Vision and Pattern Recognition, pp. 763-770, June 2005. H. Lombaert, Y. Sun, L. Grady and C. Xu, “A Multilevel Banded Graph Cuts Method for Fast Image Segmentation,” Proc. Int"l Conf. Computer Vision 2005, vol. 1, pp. 259-265, Oct. 2005. Y. Li, J. Sun, C. Tang and H. Shum, “Lazy Snapping,” Proc. ACM SIGGRAPH Conf., pp. 303-308, Apr. 2004. A. Blake, C. Rother, M. Brown, P. Perez and P. Torr, “Interactive Image Segmentation Using an Adaptic GMMRF Model,” Proc. European Conf. Computer Vision, pp. 428-441, May 2004. C. Rother, V. Kolmogorov and A. Blake, “`GrabCut"—Interactive Foreground Extraction Using Iterated Graph Cuts,” ACM Trans. Graphics, vol. 23, no. 3, pp. 309-314, 2004. C. Zahn, “Graph Theoretical Methods for Detecting and Describing Gestalt Clusters,” IEEE Trans. Computers, vol. 20, pp. 68-86, 1971. Z. Wu and R. Leahy, “An Optimal Graph Theoretic Approach to Data Clustering: Theory and Its Application to Image Segmentation,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 11, pp. 1101-1113, 1993. S. Sarkar and P. Soundararajan, “Supervised Learning of Large Perceptual Organization: Graph Spectral Partitioning and Learning Automata,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 22, no. 5, pp. 504-525, May 2000. P. Perona and W. Freeman, “A Factorization Approach to Grouping,” Proc. Fifth European Conf. Computer Vision, pp. 655-670, June 1998. L. Grady and E.L. Schwartz, “Isoperimetric Graph Partitioning for Image Segmentation,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 28, no. 3, pp. 469-475, Mar. 2006.

  33. A. Barbu and S.-C. Zhu, “Graph Partition by Swendsen-Wang Cuts,” Proc. Ninth IEEE Conf. Computer Vision, vol. 1, pp. 320-327, Oct. 2003. L. Grady and E. Schwartz, “Anisotropic Interpolation on Graphs: The Combinatorial Dirichlet Problem,” Technical Report CAS/CNS-TR-03-014, Dept. of Cognitive and Neural Systems, Boston Univ., 2003. A. Levin, D. Lischinski and Y. Weiss, “Colorization Using Optimization,” Proc. ACM SIGGRAPH Conf., pp. 689-694, Aug. 2004. X. Zhu, Z. Ghahramani and J. Lafferty, “Semi-Supervised Learning Using Gaussian Fields and Harmonic Functions,” Proc. 20th Int"l Conf. Machine Learning, pp. 912-919, 2003. V.B. Eckmann, “HarmonischeFunktionen und Randwertaufgaben in einemKomplex,” Commentarii Math. Helvetici, vol. 17, pp. 240-255, 1945. U.R. Kodres, “Geometrical Positioning of Circuit Elements in a Computer,” Proc. 1959 AIEE Fall General Meeting, Oct. 1959. W.T. Tutte, “How to Draw a Graph,” Proc. London Math. Soc., vol. 13, no. 3, pp. 743-768, 1963. H. Wechsler and M. Kidode, “A Random Walk Procedure for Texture Discrimination,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 1, no. 3, pp. 272-280, 1979. L. Gorelick, M. Galun, E. Sharon, R. Basri and A. Brandt, “Shape Representation and Classification Using the Poisson Equation,” Proc. 2004 IEEE CS Conf. Computer Vision and Pattern Recognition (CVPR "04), vol. 2, pp. 61-67, July 2004. L. Grady and E.L. Schwartz, “Isoperimetric Partitioning: A New Algorithm for Graph Partitioning,” SIAM J. Scientific Computing, vol. 27, no. 6, pp. 1844-1866, June 2006.

  34. D. Harel and Y. Koren, “On Clustering Using Random Walks,” Proc. 21st Conf. Foundations of Software Technology and Theoretical Computer Science, vol. 2245, pp. 18-41, 2001. L. Yen, D. Vanvyve, F. Wouters, F. Fouss, M. Verleysen and M. Saerens, “Clustering Using a Random-Walk Based Distance Measure,” Proc. 13th Symp. Artificial Neural Networks, pp. 317-324, 2005. M.E.J. Newman, “A Measure of Betweenness Centrality Based on Random Walks,” Social Networks, vol. 27, no. 1, pp. 39-54, Jan. 2005. F. Harary, Graph Theory. Addison-Wesley, 1994. M.J. Black, G. Sapiro, D.H. Marimont and D. Heeger, “Robust Anisotropic Diffusion,” IEEE Trans. Image Processing, vol. 7, no. 3, pp. 421-432, Mar. 1998. X. Zhu, J. Lafferty and Z. Ghahramani, “Combining Active Learning and Semi-Supervised Learning Using Gaussian Fields and Harmonic Functions,” Proc. ICML 2003 Workshop Continuum from Labeled to Unlabeled Data in Machine Learning and Data Mining, pp. 58-65, 2003. J. Dodziuk, “Difference Equations, Isoperimetric Inequality and the Transience of Certain Random Walks,” Trans. Am. Math. Soc., vol. 284, pp. 787-794, 1984. J. Roth, “An Application of Algebraic Topology to Numerical Analysis: On the Existence of a Solution to the Network Problem,” Proc. Nat"l Academy of Science of Am., vol. 41, pp. 518-521, 1955. C. Mattiussi, “The Finite Volume, Finite Element and Finite Difference Methods as Numerical Methods for Physical Field Problems,” Advances in Imaging and Electron Physics, pp. 1-146, Apr. 2000. F.R.K. Chung, Spectral Graph Theory, no. 92, Providence, R.I.: Am. Math. Soc., 1997.

  35. N. Biggs, Algebraic Graph Theory, no. 67, Cambridge Univ. Press, 1974. G. Golub and C. Van Loan, Matrix Computations, third ed. The Johns Hopkins Univ. Press, 1996. W. Hackbusch, Iterative Solution of Large Sparse Systems of Equations. Springer-Verlag, 1994. J.J. Dongarra, I.S. Duff, D.C. Sorenson and H.A. van der Vorst, Solving Linear Systems on Vector and Shared Memory Computers. Philadelphia: SIAM, 1991. K. Gremban, “Combinatorial Preconditioners for Sparse, Symmetric Diagonally Dominant Linear Systems,” PhD dissertation, Carnegie Mellon Univ., Pittsburgh, Penn. Oct. 1996. J. Bolz, I. Farmer, E. Grinspun and P. Schröder, “Sparse Matrix Solvers on the GPU: Conjugate Gradients and Multigrid,” ACM Trans. Graphics, vol. 22, no. 3, pp. 917-924, July 2003. J. Krüger and R. Westermann, “Linear Algebra Operators for GPU Implementation of Numerical Algorithms,” ACM Trans. Graphics, vol. 22, no. 3, pp. 908-916, July 2003. Multigrid, U. Trottenberg, C.W. Oosterlee, and A. Schuller, eds. San Diego, Calif.: Academic Press, 2000. J.E. Dendy, “Black Box Multigrid,” J. Computational Physics, vol. 48, pp. 366-386, 1982. L. Grady and E.L. Schwartz, “Faster Graph-Theoretic Image Processing via Small-World and Quadtree Topologies,” Proc. 2004 IEEE CS Conf. Computer Vision and Pattern Recognition, pp. 360-365, June-July 2004.

  36. L. Grady and E.L. Schwartz, “The Graph Analysis Toolbox: Image Processing on Arbitrary Graphs,” Technical Report TR-03-021, Boston Univ., Aug. 2003. G. Kirchhoff, “Über die AuflösungderGleichungen, auf Welche man beiderUntersuchungderLinearenVerteilungGalvanischerStrömeGeführtWird,” Poggendorf"sAnnalenderPhysikChemie, vol. 72, pp. 497-508, 1847. M. Fiedler, Special Matrices and Their Applications in Numerical Mathematics.MartinusNijhoff Publishers, 1986. L. Grady, T. Schiwietz, S. Aharon and R. Westermann, “Random Walks for Interactive Organ Segmentation in Two and Three Dimensions: Implementation and Validation,” Proc. Conf. Medical Image Computing and Computer-Assisted Intervention 2005 II, pp. 773-780, Oct. 2005.

  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

More Related