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MAP Estimation Algorithms in

MAP Estimation Algorithms in. Computer Vision - Part II. M. Pawan Kumar, University of Oxford Pushmeet Kohli, Microsoft Research. Example: Image Segmentation. E: {0,1} n → R 0 → fg 1 → bg. E(x) = ∑ c i x i + ∑ c ij x i (1-x j ). i. i,j. n = number of pixels. Image (D).

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MAP Estimation Algorithms in

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  1. MAP Estimation Algorithms in Computer Vision - Part II M. Pawan Kumar, University of Oxford Pushmeet Kohli, Microsoft Research

  2. Example: Image Segmentation E: {0,1}n→R 0 → fg 1 → bg E(x) = ∑ ci xi + ∑cij xi(1-xj) i i,j n = number of pixels Image (D)

  3. Example: Image Segmentation E: {0,1}n→R 0 → fg 1 → bg E(x) = ∑ ci xi + ∑cij xi(1-xj) i i,j n = number of pixels Unary Cost (ci) Dark (negative) Bright (positive)

  4. Example: Image Segmentation E: {0,1}n→R 0 → fg 1 → bg E(x) = ∑ ci xi + ∑cij xi(1-xj) i i,j n = number of pixels Discontinuity Cost (cij)

  5. Example: Image Segmentation E: {0,1}n→R 0 → fg 1 → bg E(x) = ∑ ci xi + ∑cij xi(1-xj) i i,j n = number of pixels x*= arg min E(x) x How to minimize E(x)? Global Minimum (x*)

  6. Outline of the Tutorial The st-mincut problem Connection between st-mincut and energy minimization? What problems can we solve using st-mincut? st-mincut based Move algorithms Recent Advances and Open Problems

  7. Outline of the Tutorial The st-mincut problem Connection between st-mincut and energy minimization? What problems can we solve using st-mincut? st-mincut based Move algorithms Recent Advances and Open Problems

  8. The st-Mincut Problem • Graph (V, E, C) • Vertices V = {v1, v2 ... vn} • Edges E = {(v1, v2) ....} • Costs C = {c(1, 2) ....} Source 2 9 1 v1 v2 2 5 4 Sink

  9. The st-Mincut Problem What is a st-cut? Source 2 9 1 v1 v2 2 5 4 Sink

  10. The st-Mincut Problem What is a st-cut? An st-cut (S,T) divides the nodes between source and sink. Source 2 9 What is the cost of a st-cut? Sum of cost of all edges going from S to T 1 v1 v2 2 5 4 Sink 5 + 2 + 9 = 16

  11. The st-Mincut Problem What is a st-cut? An st-cut (S,T) divides the nodes between source and sink. Source 2 9 What is the cost of a st-cut? Sum of cost of all edges going from S to T 1 v1 v2 2 5 4 What is the st-mincut? Sink st-cut with the minimum cost 2 + 1 + 4 = 7

  12. How to compute the st-mincut? Solve the dual maximum flow problem Compute the maximum flow between Source and Sink Source Constraints Edges: Flow < Capacity Nodes: Flow in = Flow out 2 9 1 v1 v2 2 5 4 Min-cut\Max-flow Theorem Sink In every network, the maximum flow equals the cost of the st-mincut

  13. Maxflow Algorithms Flow = 0 Augmenting Path Based Algorithms Source Find path from source to sink with positive capacity Push maximum possible flow through this path Repeat until no path can be found 2 9 1 v1 v2 2 5 4 Sink Algorithms assume non-negative capacity

  14. Maxflow Algorithms Flow = 0 Augmenting Path Based Algorithms Source Find path from source to sink with positive capacity Push maximum possible flow through this path Repeat until no path can be found 2 9 1 v1 v2 2 5 4 Sink Algorithms assume non-negative capacity

  15. Maxflow Algorithms Flow = 0 + 2 Augmenting Path Based Algorithms Source Find path from source to sink with positive capacity Push maximum possible flow through this path Repeat until no path can be found 2-2 9 1 v1 v2 2 4 5-2 Sink Algorithms assume non-negative capacity

  16. Maxflow Algorithms Flow = 2 Augmenting Path Based Algorithms Source Find path from source to sink with positive capacity Push maximum possible flow through this path Repeat until no path can be found 0 9 1 v1 v2 2 3 4 Sink Algorithms assume non-negative capacity

  17. Maxflow Algorithms Flow = 2 Augmenting Path Based Algorithms Source Find path from source to sink with positive capacity Push maximum possible flow through this path Repeat until no path can be found 0 9 1 v1 v2 2 3 4 Sink Algorithms assume non-negative capacity

  18. Maxflow Algorithms Flow = 2 Augmenting Path Based Algorithms Source Find path from source to sink with positive capacity Push maximum possible flow through this path Repeat until no path can be found 0 9 1 v1 v2 2 3 4 Sink Algorithms assume non-negative capacity

  19. Maxflow Algorithms Flow = 2 + 4 Augmenting Path Based Algorithms Source Find path from source to sink with positive capacity Push maximum possible flow through this path Repeat until no path can be found 0 5 1 v1 v2 2 3 0 Sink Algorithms assume non-negative capacity

  20. Maxflow Algorithms Flow = 6 Augmenting Path Based Algorithms Source Find path from source to sink with positive capacity Push maximum possible flow through this path Repeat until no path can be found 0 5 1 v1 v2 2 3 0 Sink Algorithms assume non-negative capacity

  21. Maxflow Algorithms Flow = 6 Augmenting Path Based Algorithms Source Find path from source to sink with positive capacity Push maximum possible flow through this path Repeat until no path can be found 0 5 1 v1 v2 2 3 0 Sink Algorithms assume non-negative capacity

  22. Maxflow Algorithms Flow = 6 + 1 Augmenting Path Based Algorithms Source Find path from source to sink with positive capacity Push maximum possible flow through this path Repeat until no path can be found 0 4 1-1 v1 v2 2+1 2 0 Sink Algorithms assume non-negative capacity

  23. Maxflow Algorithms Flow = 7 Augmenting Path Based Algorithms Source Find path from source to sink with positive capacity Push maximum possible flow through this path Repeat until no path can be found 0 4 0 v1 v2 3 2 0 Sink Algorithms assume non-negative capacity

  24. Maxflow Algorithms Flow = 7 Augmenting Path Based Algorithms Source Find path from source to sink with positive capacity Push maximum possible flow through this path Repeat until no path can be found 0 4 0 v1 v2 3 2 0 Sink Algorithms assume non-negative capacity

  25. History of Maxflow Algorithms n: #nodes m: #edges U: maximum edge weight Augmenting Path and Push-Relabel Algorithms assume non-negative edge weights [Slide credit: Andrew Goldberg]

  26. History of Maxflow Algorithms n: #nodes m: #edges U: maximum edge weight Augmenting Path and Push-Relabel Algorithms assume non-negative edge weights [Slide credit: Andrew Goldberg]

  27. Augmenting Path based Algorithms Ford Fulkerson: Choose any augmenting path Source 1000 1000 1 a1 a2 0 1000 1000 Sink

  28. Augmenting Path based Algorithms Ford Fulkerson: Choose any augmenting path Source 1000 1000 1 a1 a2 Bad Augmenting Paths 0 1000 1000 Sink

  29. Augmenting Path based Algorithms Ford Fulkerson: Choose any augmenting path Source 1000 1000 1 a1 a2 Bad Augmenting Path 0 1000 1000 Sink

  30. Augmenting Path based Algorithms Ford Fulkerson: Choose any augmenting path Source 999 1000 0 a1 a2 1 1000 999 Sink

  31. Augmenting Path based Algorithms n: #nodes m: #edges Ford Fulkerson: Choose any augmenting path Source 999 1000 0 a1 a2 1 1000 999 Sink We will have to perform 2000 augmentations! Worst case complexity: O (m x Total_Flow) (Pseudo-polynomial bound: depends on flow)

  32. Augmenting Path based Algorithms n: #nodes m: #edges Dinic: Choose shortest augmenting path Source 1000 1000 1 a1 a2 0 1000 1000 Sink Worst case Complexity: O (m n2)

  33. Maxflow in Computer Vision • Specialized algorithms for vision problems • Grid graphs • Low connectivity (m ~ O(n)) • Dual search tree augmenting path algorithm [Boykov and Kolmogorov PAMI 2004] • Finds approximate shortest augmenting paths efficiently • High worst-case time complexity • Empirically outperforms other algorithms on vision problems

  34. Maxflow in Computer Vision • Specialized algorithms for vision problems • Grid graphs • Low connectivity (m ~ O(n)) • Dual search tree augmenting path algorithm [Boykov and Kolmogorov PAMI 2004] • Finds approximate shortest augmenting paths efficiently • High worst-case time complexity • Empirically outperforms other algorithms on vision problems • Efficient code available on the web http://www.adastral.ucl.ac.uk/~vladkolm/software.html

  35. Outline of the Tutorial The st-mincut problem Connection between st-mincut and energy minimization? What problems can we solve using st-mincut? st-mincut based Move algorithms Recent Advances and Open Problems

  36. st-mincut S T St-mincut and Energy Minimization Minimizing a Qudratic Pseudoboolean function E(x) Functions of boolean variables Pseudoboolean? E: {0,1}n→R E(x) = ∑ ci xi + ∑cij xi(1-xj) cij≥0 i i,j Polynomial time st-mincut algorithms require non-negative edge weights

  37. st-mincut S Solution T So how does this work? Construct a graph such that: Any st-cut corresponds to an assignment of x The cost of the cut is equal to the energy of x : E(x) E(x)

  38. Graph Construction E(a1,a2) Source (0) a1 a2 Sink (1)

  39. Graph Construction E(a1,a2) = 2a1 Source (0) 2 a1 a2 Sink (1)

  40. Graph Construction E(a1,a2) = 2a1 + 5ā1 Source (0) 2 a1 a2 5 Sink (1)

  41. Graph Construction E(a1,a2) = 2a1 + 5ā1+ 9a2 + 4ā2 Source (0) 9 2 a1 a2 5 4 Sink (1)

  42. Graph Construction E(a1,a2) = 2a1 + 5ā1+ 9a2 + 4ā2 + 2a1ā2 Source (0) 9 2 a1 a2 2 5 4 Sink (1)

  43. Graph Construction E(a1,a2) = 2a1 + 5ā1+ 9a2 + 4ā2 + 2a1ā2 +ā1a2 Source (0) 9 2 1 a1 a2 2 5 4 Sink (1)

  44. Graph Construction E(a1,a2) = 2a1 + 5ā1+ 9a2 + 4ā2 + 2a1ā2 +ā1a2 Source (0) 9 2 1 a1 a2 2 5 4 Sink (1)

  45. Graph Construction E(a1,a2) = 2a1+ 5ā1+ 9a2+ 4ā2+ 2a1ā2+ā1a2 Source (0) 9 2 Cost of cut = 11 1 a1 = 1 a2 = 1 a1 a2 2 E(1,1) = 11 5 4 Sink (1)

  46. Graph Construction E(a1,a2) = 2a1+ 5ā1+ 9a2+ 4ā2+ 2a1ā2+ā1a2 Source (0) 9 2 st-mincut cost = 8 1 a1 = 1 a2 = 0 a1 a2 2 E(1,0) = 8 5 4 Sink (1)

  47. Energy Function Reparameterization Two functions E1 and E2 are reparameterizations if E1 (x) = E2 (x) for all x For instance: E1 (a1) = 1+ 2a1 + 3ā1 E2 (a1) = 3 + ā1

  48. Flow and Reparametrization E(a1,a2) = 2a1 + 5ā1+ 9a2 + 4ā2 + 2a1ā2 +ā1a2 Source (0) 9 2 1 a1 a2 2 5 4 Sink (1)

  49. Flow and Reparametrization E(a1,a2) = 2a1 + 5ā1+ 9a2 + 4ā2 + 2a1ā2 +ā1a2 Source (0) 9 2 2a1 + 5ā1 1 a1 a2 = 2(a1+ā1) + 3ā1 2 = 2 + 3ā1 5 4 Sink (1)

  50. Flow and Reparametrization E(a1,a2) = 2+ 3ā1+ 9a2 + 4ā2 + 2a1ā2 +ā1a2 Source (0) 9 0 2a1 + 5ā1 1 a1 a2 = 2(a1+ā1) + 3ā1 2 = 2 + 3ā1 3 4 Sink (1)

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