1. Graph Cut 韋弘 2010/2/22

2. Outline Background Graph cut Ford–Fulkerson algorithm Application Extended reading

3. Graph G = ex ： G= G= Undirected Directed

4. Weighted graph a number (weight) is assigned to each edge weights might represent : costs, lengths or capacities, etc. depending on the problem

5. Flow network a directed graph where each edge has a positive capacity(weight) two special vertices are designated the source s and the sink t ex:

6. Flow network a flow in G is a real-valued function f : VXV→R that satisfies the following three properties:

7. Max-flow a feasible flow through a single-source, single-sink that is maximum

8. Outline Background Graph cut Ford–Fulkerson algorithm Application Extended reading

9. a subset of edges such that source &sink become separated G(C)= the cost of a cut : Minimum cut : a cut whose cost is the least over all cuts Cut

10. The cost of a cut -1+12-1+14-(13+16+9+20-4)=-30

11. The max-flow min-cut theorem If f is a flow in a flow network G = (V;E) with source s and sink t then the value of the maximum flow is equal to the capacity of a minimum cut. Refer to T.H. Cormen, C.E. Leiserson and R.L. Rivest,.Introduction to Algorithms., McGraw-Hill, 1990. for the prove.

12. Outline Background Graph cut Ford–Fulkerson algorithm Application Extended reading

13. Ford–Fulkerson algorithm s v4v4 v2v2 v3v3 v1v1 t 16 13 104 9 7 4 20 12 14

14. Ford–Fulkerson algorithm

15. s v4v4 v2v2 v3v3 v1v1 t 16 13 104 9 7 4 20 12 14

16. Ford–Fulkerson algorithm s v4v4 v2v2 v3v3 v1v1 t 16 13 104 9 7 14 4 20 12

17. Ford–Fulkerson algorithm s v4v4 v2v2 v3v3 v1v1 t 4/16 13 104 4/9 7 4/14 4/4 20 4/12

18. Ford–Fulkerson algorithm s v4v4 v2v2 v3v3 v1v1 t 12 13 10 4 5 8 7 20 10

19. Ford–Fulkerson algorithm 7/10 s v4v4 v2v2 v3v3 v1v1 t 7/12 13 7/10 4 5 7/7 7/20 8

20. Ford–Fulkerson algorithm s v4v4 v2v2 v3v3 v1v1 t 5 13 3 4 5 8 3

21. Ford–Fulkerson algorithm s v4v4 v2v2 v3v3 v1v1 t 5 4/13 34/4 5 3 4/13 4/8

22. Ford–Fulkerson algorithm s v4v4 v2v2 v3v3 v1v1 t 5 9 3 5 4 9 3

23. s v4v4 v2v2 v3v3 v1v1 t 4/5 9 3 5 4/4 4/9 3

24. Ford–Fulkerson algorithm s v4v4 v2v2 v3v3 v1v1 t 1 9 3 5 5 34/13 7/10 15/16 4/9 11/14 15/20

25. Ford–Fulkerson algorithm s v4v4 v2v2 v3v3 v1v1 t 4/13 7/10 15/16 4/9 11/14 15/20 8/13 0/9 19/20

26. Ford–Fulkerson algorithm s v4v4 v2v2 v3v3 v1v1 t 15/16 8/13 3/104 9 7/7 4/4 19/20 12/12 11/14 a convenient tool: http://www.lix.polytechnique.fr/~durr/MaxFlow/

27. Outline Background Graph cut Ford–Fulkerson algorithm Application Extended reading

28. Combinatorial Optimization Determine a combination (pattern, set of labels) such that the energy of this combination is minimum Example: 4-bit binary label problem  Find a label-set which yields the minimal energy Each individual bit can be set as 0 or 1  Each label corresponds to an energy cost Each neighboring bit pair is better to have the same label (smoothness) ???? 0 123 10 9992 100 101 10079 114 98 0 1 0 1 0 1 0 1 Energy(0000) Energy(0001) = = 99+92+100+101 = 392 = 99+92+100+98+10 = 399

29. 100 114 98 79 101 100 99 92 14 Graph-Cut Formulate the previous problem into a graph-cut problem Find the cut with minimum total cost(energy) Solving the graph-cut: Ford-Fulkerson Method ???? 0 123 10 0 1 1 13 3 9 122 4 7 1 99+79+100+98+1+10+3 =390  Max Flow (Energy of the cut 1100) Total Flow Pushed = 1100

30. Exhaustive Search List all the combinations and corresponding energy Example: 1100 has the minimal energy of 390 Label setEnergyLabel setEnergy 00003921000403 00013991001410 00104261010437 00114131011424 01003991100390 01014141101397 01104131110404 01114001111391 ???? 0 123 10 9992 100 101 10079 114 98 0 1 0 1 0 1 0 1

31. Outline Background Graph cut Ford–Fulkerson algorithm Application Extended reading

32. Graph Cut algorithms used in image reconstruction Greig, D., Porteous, B.Seheult, A., Exact Maximum A Posteriori Estimation for Binary Images, J. Royal Statistical Soc., Series B, vol. 51, no. 2, pp. 271- 279, 1989 Boykov, Y., Veksler, O.Zabih, R., Fast Approximate Energy Minimization via Graph Cuts, Proc. IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 23, no. 11, pp. 1222-123, 2001 Boykov, Y., Veksler, O.Zabih, R., Markov Random Fields with Efficient Approximations, Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 648-655, 1998 Ishikawa, H., Geiger, D., Segmentation by Grouping Junctions, Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 125-131, 1998

33. Graph Cut algorithms used in stereo vision Birchfield, S., Tomasi, C., Multiway Cut for Stereo and Motion with Slanted Surfaces, Proc. Int’l Conf. Computer Vision, pp. 489-495, 1999 Ishikawa, H., Geiger, D.Zabih, R., Occlusions, Discontinuities, and Epipolar Lines in Stereo, Proc. Int’l Conf. Computer Vision, pp. 1033-1040, 2003 Kim, J., Kolmogorov, V., Visual Correspondence Using Energy Minimization and Mutual Information, Proc. Int’l Conf. Computer Vision, pp. 508-515, 2001 Kolmogorov, V., Zabih, R., Visual Correspondence with Occlusions Using Graph Cuts, PhD thesis, Stanford Univ., Dec. 2002 Lin, M.H., Surfaces with Occlusions from Layered Stereo, Int’l J. Computer Vision, vol. 1, no. 2, pp. 1-15, 1999

34. Graph Cut algorithms used in image segmentation Boykov, Y., Kolmogorov, V., Computing Geodesics and Minimal Surfaces via Graph Cuts,Proc. European Conf. Computer Vision, pp. 232-248, 1998

35. Graph Cut algorithms used in multi-camera scene reconstruction Roy, S., Cox, I., A Maximum-Flow Formulation of the n-Camera Stereo Correspondence Problem, Proc. Int’l Conf. Computer Vision, pp. 26-33, 2003 Kolmogorov, V., Zabih, R., Multi-Camera Scene Reconstruction via Graph Cuts, Proc. European Conf. Computer Vision, vol. 3, pp. 82-96, 2002