ECCV2008: MAP Estimation Algorithms in Computer Vision - Part 1
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  • 1. MAP Estimation Algorithms in M. Pawan Kumar, University of Oxford Pushmeet Kohli, Microsoft Research Computer Vision - Part I
  • 2. Aim of the Tutorial
    • Description of some successful algorithms
    • Computational issues
    • Enough details to implement
    • Some proofs will be skipped :-(
    • But references to them will be given :-)
  • 3. A Vision Application Binary Image Segmentation How ? Cost function Models our knowledge about natural images Optimize cost function to obtain the segmentation
  • 4. A Vision Application Object - white, Background - green/grey Graph G = (V,E) Each vertex corresponds to a pixel Edges define a 4-neighbourhood grid graph Assign a label to each vertex from L = {obj,bkg} Binary Image Segmentation
  • 5. A Vision Application Graph G = (V,E) Cost of a labelling f : V  L Per Vertex Cost Cost of label ‘obj’ low Cost of label ‘bkg’ high Object - white, Background - green/grey Binary Image Segmentation
  • 6. A Vision Application Graph G = (V,E) Cost of a labelling f : V  L Cost of label ‘obj’ high Cost of label ‘bkg’ low Per Vertex Cost UNARY COST Object - white, Background - green/grey Binary Image Segmentation
  • 7. A Vision Application Graph G = (V,E) Cost of a labelling f : V  L Per Edge Cost Cost of same label low Cost of different labels high Object - white, Background - green/grey Binary Image Segmentation
  • 8. A Vision Application Graph G = (V,E) Cost of a labelling f : V  L Cost of same label high Cost of different labels low Per Edge Cost PAIRWISE COST Object - white, Background - green/grey Binary Image Segmentation
  • 9. A Vision Application Graph G = (V,E) Problem: Find the labelling with minimum cost f* Object - white, Background - green/grey Binary Image Segmentation
  • 10. A Vision Application Graph G = (V,E) Problem: Find the labelling with minimum cost f* Binary Image Segmentation
  • 11. Another Vision Application Object Detection using Parts-based Models How ? Once again, by defining a good cost function
  • 12. Another Vision Application H T L1 Each vertex corresponds to a part - ‘Head’, ‘Torso’, ‘Legs’ 1 Edges define a TREE Assign a label to each vertex from L = {positions} Graph G = (V,E) L2 L3 L4 Object Detection using Parts-based Models
  • 13. Another Vision Application 2 Each vertex corresponds to a part - ‘Head’, ‘Torso’, ‘Legs’ Assign a label to each vertex from L = {positions} Graph G = (V,E) Edges define a TREE H T L1 L2 L3 L4 Object Detection using Parts-based Models
  • 14. Another Vision Application 3 Each vertex corresponds to a part - ‘Head’, ‘Torso’, ‘Legs’ Assign a label to each vertex from L = {positions} Graph G = (V,E) Edges define a TREE H T L1 L2 L3 L4 Object Detection using Parts-based Models
  • 15. Another Vision Application Cost of a labelling f : V  L Unary cost : How well does part match image patch? Pairwise cost : Encourages valid configurations Find best labelling f* Graph G = (V,E) 3 H T L1 L2 L3 L4 Object Detection using Parts-based Models
  • 16. Another Vision Application Cost of a labelling f : V  L Unary cost : How well does part match image patch? Pairwise cost : Encourages valid configurations Find best labelling f* Graph G = (V,E) 3 H T L1 L2 L3 L4 Object Detection using Parts-based Models
  • 17. Yet Another Vision Application Stereo Correspondence Disparity Map How ? Minimizing a cost function
  • 18. Yet Another Vision Application Stereo Correspondence Graph G = (V,E) Vertex corresponds to a pixel Edges define grid graph L = {disparities}
  • 19. Yet Another Vision Application Stereo Correspondence Cost of labelling f : Unary cost + Pairwise Cost Find minimum cost f*
  • 20. The General Problem b a e d c f Graph G = ( V, E ) Discrete label set L = {1,2,…,h} Assign a label to each vertex f: V  L 1 1 2 2 2 3 Cost of a labelling Q(f) Unary Cost Pairwise Cost Find f* = arg min Q(f)
  • 21. Outline
    • Problem Formulation
      • Energy Function
      • MAP Estimation
      • Computing min-marginals
    • Reparameterization
    • Belief Propagation
    • Tree-reweighted Message Passing
  • 22. Energy Function V a V b V c V d Label l 0 Label l 1 D a D b D c D d Random Variables V = {V a , V b , ….} Labels L = {l 0 , l 1 , ….} Data D Labelling f: {a, b, …. }  {0,1, …}
  • 23. Energy Function V a V b V c V d D a D b D c D d Q(f) = ∑ a  a;f(a) Unary Potential 2 5 4 2 6 3 3 7 Label l 0 Label l 1 Easy to minimize Neighbourhood
  • 24. Energy Function V a V b V c V d D a D b D c D d E : (a,b)  E iff V a and V b are neighbours E = { (a,b) , (b,c) , (c,d) } 2 5 4 2 6 3 3 7 Label l 0 Label l 1
  • 25. Energy Function V a V b V c V d D a D b D c D d +∑ (a,b)  ab;f(a)f(b) Pairwise Potential 0 1 1 0 0 2 1 1 4 1 0 3 2 5 4 2 6 3 3 7 Label l 0 Label l 1 Q(f) = ∑ a  a;f(a)
  • 26. Energy Function V a V b V c V d D a D b D c D d 0 1 1 0 0 2 1 1 4 1 0 3 Parameter 2 5 4 2 6 3 3 7 Label l 0 Label l 1 +∑ (a,b)  ab;f(a)f(b) Q(f;  ) = ∑ a  a;f(a)
  • 27. Outline
    • Problem Formulation
      • Energy Function
      • MAP Estimation
      • Computing min-marginals
    • Reparameterization
    • Belief Propagation
    • Tree-reweighted Message Passing
  • 28. MAP Estimation V a V b V c V d 2 5 4 2 6 3 3 7 0 1 1 0 0 2 1 1 4 1 0 3 Q(f;  ) = ∑ a  a;f(a) + ∑ (a,b)  ab;f(a)f(b) Label l 0 Label l 1
  • 29. MAP Estimation V a V b V c V d 2 5 4 2 6 3 3 7 0 1 1 0 0 2 1 1 4 1 0 3 Q(f;  ) = ∑ a  a;f(a) + ∑ (a,b)  ab;f(a)f(b) 2 + 1 + 2 + 1 + 3 + 1 + 3 = 13 Label l 0 Label l 1
  • 30. MAP Estimation V a V b V c V d 2 5 4 2 6 3 3 7 0 1 1 0 0 2 1 1 4 1 0 3 Q(f;  ) = ∑ a  a;f(a) + ∑ (a,b)  ab;f(a)f(b) Label l 0 Label l 1
  • 31. MAP Estimation V a V b V c V d 2 5 4 2 6 3 3 7 0 1 1 0 0 2 1 1 4 1 0 3 Q(f;  ) = ∑ a  a;f(a) + ∑ (a,b)  ab;f(a)f(b) 5 + 1 + 4 + 0 + 6 + 4 + 7 = 27 Label l 0 Label l 1
  • 32. MAP Estimation V a V b V c V d 2 5 4 2 6 3 3 7 0 1 1 0 0 2 1 1 4 1 0 3 Q(f;  ) = ∑ a  a;f(a) + ∑ (a,b)  ab;f(a)f(b) f* = arg min Q(f;  ) q* = min Q(f;  ) = Q(f*;  ) Label l 0 Label l 1
  • 33. MAP Estimation 16 possible labellings f* = {1, 0, 0, 1} q* = 13 20 1 1 1 0 27 0 1 1 0 19 1 0 1 0 22 0 0 1 0 20 1 1 0 0 27 0 1 0 0 15 1 0 0 0 18 0 0 0 0 Q(f;  ) f(d) f(c) f(b) f(a) 16 1 1 1 1 23 0 1 1 1 15 1 0 1 1 18 0 0 1 1 18 1 1 0 1 25 0 1 0 1 13 1 0 0 1 16 0 0 0 1 Q(f;  ) f(d) f(c) f(b) f(a)
  • 34. Computational Complexity Segmentation 2 |V| |V| = number of pixels ≈ 320 * 480 = 153600
  • 35. Computational Complexity |L| = number of pixels ≈ 153600 Detection |L| |V|
  • 36. Computational Complexity |V| = number of pixels ≈ 153600 Stereo |L| |V| Can we do better than brute-force? MAP Estimation is NP-hard !!
  • 37. Computational Complexity |V| = number of pixels ≈ 153600 Stereo |L| |V| Exact algorithms do exist for special cases Good approximate algorithms for general case But first … two important definitions
  • 38. Outline
    • Problem Formulation
      • Energy Function
      • MAP Estimation
      • Computing min-marginals
    • Reparameterization
    • Belief Propagation
    • Tree-reweighted Message Passing
  • 39. Min-Marginals V a V b V c V d 2 5 4 2 6 3 3 7 0 1 1 0 0 2 1 1 4 1 0 3 f* = arg min Q(f;  ) such that f(a) = i Min-marginal q a;i Label l 0 Label l 1 Not a marginal (no summation)
  • 40. Min-Marginals 16 possible labellings q a;0 = 15 20 1 1 1 0 27 0 1 1 0 19 1 0 1 0 22 0 0 1 0 20 1 1 0 0 27 0 1 0 0 15 1 0 0 0 18 0 0 0 0 Q(f;  ) f(d) f(c) f(b) f(a) 16 1 1 1 1 23 0 1 1 1 15 1 0 1 1 18 0 0 1 1 18 1 1 0 1 25 0 1 0 1 13 1 0 0 1 16 0 0 0 1 Q(f;  ) f(d) f(c) f(b) f(a)
  • 41. Min-Marginals 16 possible labellings q a;1 = 13 16 1 1 1 1 23 0 1 1 1 15 1 0 1 1 18 0 0 1 1 18 1 1 0 1 25 0 1 0 1 13 1 0 0 1 16 0 0 0 1 Q(f;  ) f(d) f(c) f(b) f(a) 20 1 1 1 0 27 0 1 1 0 19 1 0 1 0 22 0 0 1 0 20 1 1 0 0 27 0 1 0 0 15 1 0 0 0 18 0 0 0 0 Q(f;  ) f(d) f(c) f(b) f(a)
  • 42. Min-Marginals and MAP
    • Minimum min-marginal of any variable =
    • energy of MAP labelling
    min f Q(f;  ) such that f(a) = i q a;i min i min i ( ) V a has to take one label min f Q(f;  )
  • 43. Summary MAP Estimation f* = arg min Q(f;  ) Q(f;  ) = ∑ a  a;f(a) + ∑ (a,b)  ab;f(a)f(b) Min-marginals q a;i = min Q(f;  ) s.t. f(a) = i Energy Function
  • 44. Outline
    • Problem Formulation
    • Reparameterization
    • Belief Propagation
    • Tree-reweighted Message Passing
  • 45. Reparameterization V a V b 2 5 4 2 0 1 1 0 2 + 2 + - 2 - 2 Add a constant to all  a;i Subtract that constant from all  b;k 6 1 1 5 0 1 10 1 0 7 0 0 Q(f;  ) f(b) f(a)
  • 46. Reparameterization Add a constant to all  a;i Subtract that constant from all  b;k Q(f;  ’) = Q(f;  ) V a V b 2 5 4 2 0 0 2 + 2 + - 2 - 2 1 1 6 + 2 - 2 1 1 5 + 2 - 2 0 1 10 + 2 - 2 1 0 7 + 2 - 2 0 0 Q(f;  ) f(b) f(a)
  • 47. Reparameterization V a V b 2 5 4 2 0 1 1 0 - 3 + 3 Add a constant to one  b;k Subtract that constant from  ab;ik for all ‘i’ - 3 6 1 1 5 0 1 10 1 0 7 0 0 Q(f;  ) f(b) f(a)
  • 48. Reparameterization V a V b 2 5 4 2 0 1 1 0 - 3 + 3 - 3 Q(f;  ’) = Q(f;  ) Add a constant to one  b;k Subtract that constant from  ab;ik for all ‘i’ 6 - 3 + 3 1 1 5 0 1 10 - 3 + 3 1 0 7 0 0 Q(f;  ) f(b) f(a)
  • 49. Reparameterization - 2 - 2 - 2 + 2 + 1 + 1 + 1 - 1 - 4 + 4 - 4 - 4  ’ a;i =  a;i  ’ b;k =  b;k  ’ ab;ik =  ab;ik + M ab;k - M ab;k + M ba;i - M ba;i Q(f;  ’) = Q(f;  ) V a V b 2 5 4 2 3 1 0 1 2 V a V b 2 5 4 2 3 1 1 0 1 V a V b 2 5 4 2 3 1 2 1 0
  • 50. Reparameterization Q(f;  ’) = Q(f;  ), for all f  ’ is a reparameterization of  , iff  ’    ’ b;k =  b;k Kolmogorov, PAMI, 2006  ’ a;i =  a;i  ’ ab;ik =  ab;ik + M ab;k - M ab;k + M ba;i - M ba;i Equivalently V a V b 2 5 4 2 0 0 2 + 2 + - 2 - 2 1 1
  • 51. Recap MAP Estimation f* = arg min Q(f;  ) Q(f;  ) = ∑ a  a;f(a) + ∑ (a,b)  ab;f(a)f(b) Min-marginals q a;i = min Q(f;  ) s.t. f(a) = i Q(f;  ’) = Q(f;  ), for all f  ’   Reparameterization
  • 52. Outline
    • Problem Formulation
    • Reparameterization
    • Belief Propagation
      • Exact MAP for Chains and Trees
      • Approximate MAP for general graphs
      • Computational Issues and Theoretical Properties
    • Tree-reweighted Message Passing
  • 53. Belief Propagation
    • Belief Propagation gives exact MAP for chains
    • Remember, some MAP problems are easy
    • Exact MAP for trees
    • Clever Reparameterization
  • 54. Two Variables V a V b 2 5 2 1 0 V a V b 2 5 4 0 1 Choose the right constant  ’ b;k = q b;k Add a constant to one  b;k Subtract that constant from  ab;ik for all ‘i’
  • 55. Two Variables V a V b 2 5 2 1 0 V a V b 2 5 4 0 1 Choose the right constant  ’ b;k = q b;k  a;0 +  ab;00 = 5 + 0  a;1 +  ab;10 = 2 + 1 min M ab;0 =
  • 56. Two Variables V a V b 2 5 5 -2 -3 V a V b 2 5 4 0 1 Choose the right constant  ’ b;k = q b;k
  • 57. Two Variables V a V b 2 5 5 -2 -3 V a V b 2 5 4 0 1 Choose the right constant  ’ b;k = q b;k f(a) = 1  ’ b;0 = q b;0 Potentials along the red path add up to 0
  • 58. Two Variables V a V b 2 5 5 -2 -3 V a V b 2 5 4 0 1 Choose the right constant  ’ b;k = q b;k  a;0 +  ab;01 = 5 + 1  a;1 +  ab;11 = 2 + 0 min M ab;1 =
  • 59. Two Variables V a V b 2 5 5 -2 -3 V a V b 2 5 6 -2 -1 Choose the right constant  ’ b;k = q b;k f(a) = 1  ’ b;0 = q b;0 f(a) = 1  ’ b;1 = q b;1 Minimum of min-marginals = MAP estimate
  • 60. Two Variables V a V b 2 5 5 -2 -3 V a V b 2 5 6 -2 -1 Choose the right constant  ’ b;k = q b;k f(a) = 1  ’ b;0 = q b;0 f(a) = 1  ’ b;1 = q b;1 f*(b) = 0 f*(a) = 1
  • 61. Two Variables V a V b 2 5 5 -2 -3 V a V b 2 5 6 -2 -1 Choose the right constant  ’ b;k = q b;k f(a) = 1  ’ b;0 = q b;0 f(a) = 1  ’ b;1 = q b;1 We get all the min-marginals of V b
  • 62. Recap We only need to know two sets of equations General form of Reparameterization Reparameterization of (a,b) in Belief Propagation M ab;k = min i {  a;i +  ab;ik } M ba;i = 0  ’ a;i =  a;i  ’ ab;ik =  ab;ik + M ab;k - M ab;k + M ba;i - M ba;i  ’ b;k =  b;k
  • 63. Three Variables V a V b 2 5 2 1 0 V c 4 6 0 1 0 1 3 2 3 Reparameterize the edge (a,b) as before l 0 l 1
  • 64. Three Variables V a V b 2 5 5 -3 V c 6 6 0 1 -2 3 Reparameterize the edge (a,b) as before f(a) = 1 f(a) = 1 -2 -1 2 3 l 0 l 1
  • 65. Three Variables V a V b 2 5 5 -3 V c 6 6 0 1 -2 3 Reparameterize the edge (a,b) as before f(a) = 1 f(a) = 1 Potentials along the red path add up to 0 -2 -1 2 3 l 0 l 1
  • 66. Three Variables V a V b 2 5 5 -3 V c 6 6 0 1 -2 3 Reparameterize the edge (b,c) as before f(a) = 1 f(a) = 1 Potentials along the red path add up to 0 -2 -1 2 3 l 0 l 1
  • 67. Three Variables V a V b 2 5 5 -3 V c 6 12 -6 -5 -2 9 Reparameterize the edge (b,c) as before f(a) = 1 f(a) = 1 Potentials along the red path add up to 0 f(b) = 1 f(b) = 0 -2 -1 -4 -3 l 0 l 1
  • 68. Three Variables V a V b 2 5 5 -3 V c 6 12 -6 -5 -2 9 Reparameterize the edge (b,c) as before f(a) = 1 f(a) = 1 Potentials along the red path add up to 0 f(b) = 1 f(b) = 0 q c;0 q c;1 -2 -1 -4 -3 l 0 l 1
  • 69. Three Variables V a V b 2 5 5 -3 V c 6 12 -6 -5 -2 9 f(a) = 1 f(a) = 1 f(b) = 1 f(b) = 0 q c;0 q c;1 f*(c) = 0 f*(b) = 0 f*(a) = 1 Generalizes to any length chain -2 -1 -4 -3 l 0 l 1
  • 70. Three Variables V a V b 2 5 5 -3 V c 6 12 -6 -5 -2 9 f(a) = 1 f(a) = 1 f(b) = 1 f(b) = 0 q c;0 q c;1 f*(c) = 0 f*(b) = 0 f*(a) = 1 Only Dynamic Programming -2 -1 -4 -3 l 0 l 1
  • 71. Why Dynamic Programming? 3 variables  2 variables + book-keeping n variables  (n-1) variables + book-keeping Start from left, go to right Reparameterize current edge (a,b) M ab;k = min i {  a;i +  ab;ik } Repeat  ’ ab;ik =  ab;ik + M ab;k - M ab;k  ’ b;k =  b;k
  • 72. Why Dynamic Programming? Start from left, go to right Reparameterize current edge (a,b) M ab;k = min i {  a;i +  ab;ik } Repeat Messages Message Passing Why stop at dynamic programming?  ’ ab;ik =  ab;ik + M ab;k - M ab;k  ’ b;k =  b;k
  • 73. Three Variables V a V b 2 5 5 -3 V c 6 12 -6 -5 -2 9 Reparameterize the edge (c,b) as before -2 -1 -4 -3 l 0 l 1
  • 74. Three Variables V a V b 2 5 9 -3 V c 11 12 -11 -9 -2 9 Reparameterize the edge (c,b) as before -2 -1 -9 -7  ’ b;i = q b;i l 0 l 1
  • 75. Three Variables V a V b 2 5 9 -3 V c 11 12 -11 -9 -2 9 Reparameterize the edge (b,a) as before -2 -1 -9 -7 l 0 l 1
  • 76. Three Variables V a V b 9 11 9 -9 V c 11 12 -11 -9 -9 9 Reparameterize the edge (b,a) as before -9 -7 -9 -7  ’ a;i = q a;i l 0 l 1
  • 77. Three Variables V a V b 9 11 9 -9 V c 11 12 -11 -9 -9 9 Forward Pass   Backward Pass -9 -7 -9 -7 All min-marginals are computed l 0 l 1
  • 78. Belief Propagation on Chains Start from left, go to right Reparameterize current edge (a,b) M ab;k = min i {  a;i +  ab;ik } Repeat till the end of the chain Start from right, go to left Repeat till the end of the chain  ’ ab;ik =  ab;ik + M ab;k - M ab;k  ’ b;k =  b;k
  • 79. Belief Propagation on Chains
    • A way of computing reparam constants
    • Generalizes to chains of any length
    • Forward Pass - Start to End
      • MAP estimate
      • Min-marginals of final variable
    • Backward Pass - End to start
      • All other min-marginals
    Won’t need this .. But good to know
  • 80. Computational Complexity
    • Each constant takes O(|L|)
    • Number of constants - O(|E||L|)
    O(|E||L| 2 )
    • Memory required ?
    O(|E||L|)
  • 81. Belief Propagation on Trees V b V a Forward Pass: Leaf  Root All min-marginals are computed Backward Pass: Root  Leaf V c V d V e V g V h
  • 82. Outline
    • Problem Formulation
    • Reparameterization
    • Belief Propagation
      • Exact MAP for Chains and Trees
      • Approximate MAP for general graphs
      • Computational Issues and Theoretical Properties
    • Tree-reweighted Message Passing
  • 83. Belief Propagation on Cycles V a V b V d V c Where do we start? Arbitrarily  a;0  a;1  b;0  b;1  d;0  d;1  c;0  c;1 Reparameterize (a,b)
  • 84. Belief Propagation on Cycles V a V b V d V c  a;0  a;1  ’ b;0  ’ b;1  d;0  d;1  c;0  c;1 Potentials along the red path add up to 0
  • 85. Belief Propagation on Cycles V a V b V d V c  a;0  a;1  ’ b;0  ’ b;1  d;0  d;1  ’ c;0  ’ c;1 Potentials along the red path add up to 0
  • 86. Belief Propagation on Cycles V a V b V d V c  a;0  a;1  ’ b;0  ’ b;1  ’ d;0  ’ d;1  ’ c;0  ’ c;1 Potentials along the red path add up to 0
  • 87. Belief Propagation on Cycles V a V b V d V c  ’ a;0  ’ a;1  ’ b;0  ’ b;1  ’ d;0  ’ d;1  ’ c;0  ’ c;1 Potentials along the red path add up to 0
  • 88. Belief Propagation on Cycles V a V b V d V c  ’ a;0  ’ a;1  ’ b;0  ’ b;1  ’ d;0  ’ d;1  ’ c;0  ’ c;1 Potentials along the red path add up to 0 -  a;0 -  a;1  ’ a;0 -  a;0 = q a;0  ’ a;1 -  a;1 = q a;1
  • 89. Belief Propagation on Cycles V a V b V d V c  ’ a;0  ’ a;1  ’ b;0  ’ b;1  ’ d;0  ’ d;1  ’ c;0  ’ c;1 Pick minimum min-marginal. Follow red path. -  a;0 -  a;1  ’ a;0 -  a;0 = q a;0  ’ a;1 -  a;1 = q a;1
  • 90. Belief Propagation on Cycles V a V b V d V c  a;0  a;1  ’ b;0  ’ b;1  d;0  d;1  c;0  c;1 Potentials along the red path add up to 0
  • 91. Belief Propagation on Cycles V a V b V d V c  a;0  a;1  ’ b;0  ’ b;1  d;0  d;1  ’ c;0  ’ c;1 Potentials along the red path add up to 0
  • 92. Belief Propagation on Cycles V a V b V d V c  a;0  a;1  ’ b;0  ’ b;1  ’ d;0  ’ d;1  ’ c;0  ’ c;1 Potentials along the red path add up to 0
  • 93. Belief Propagation on Cycles V a V b V d V c  ’ a;0  ’ a;1  ’ b;0  ’ b;1  ’ d;0  ’ d;1  ’ c;0  ’ c;1 Potentials along the red path add up to 0 -  a;0 -  a;1  ’ a;1 -  a;1 = q a;1  ’ a;0 -  a;0 ≤ q a;0 ≤
  • 94. Belief Propagation on Cycles V a V b V d V c  ’ a;0  ’ a;1  ’ b;0  ’ b;1  ’ d;0  ’ d;1  ’ c;0  ’ c;1 Problem Solved -  a;0 -  a;1  ’ a;1 -  a;1 = q a;1  ’ a;0 -  a;0 ≤ q a;0 ≤
  • 95. Belief Propagation on Cycles V a V b V d V c  ’ a;0  ’ a;1  ’ b;0  ’ b;1  ’ d;0  ’ d;1  ’ c;0  ’ c;1 Problem Not Solved -  a;0 -  a;1  ’ a;1 -  a;1 = q a;1  ’ a;0 -  a;0 ≤ q a;0 ≥
  • 96. Belief Propagation on Cycles V a V b V d V c  ’ a;0  ’ a;1  ’ b;0  ’ b;1  ’ d;0  ’ d;1  ’ c;0  ’ c;1 -  a;0 -  a;1 Reparameterize (a,b) again
  • 97. Belief Propagation on Cycles V a V b V d V c  ’ a;0  ’ a;1  ’’ b;0  ’’ b;1  ’ d;0  ’ d;1  ’ c;0  ’ c;1 Reparameterize (a,b) again But doesn’t this overcount some potentials?
  • 98. Belief Propagation on Cycles V a V b V d V c  ’ a;0  ’ a;1  ’’ b;0  ’’ b;1  ’ d;0  ’ d;1  ’ c;0  ’ c;1 Reparameterize (a,b) again Yes. But we will do it anyway
  • 99. Belief Propagation on Cycles V a V b V d V c  ’ a;0  ’ a;1  ’’ b;0  ’’ b;1  ’ d;0  ’ d;1  ’ c;0  ’ c;1 Keep reparameterizing edges in some order Hope for convergence and a good solution
  • 100. Belief Propagation
    • Generalizes to any arbitrary random field
    • Complexity per iteration ?
    O(|E||L| 2 )
    • Memory required ?
    O(|E||L|)
  • 101. Outline
    • Problem Formulation
    • Reparameterization
    • Belief Propagation
      • Exact MAP for Chains and Trees
      • Approximate MAP for general graphs
      • Computational Issues and Theoretical Properties
    • Tree-reweighted Message Passing
  • 102. Computational Issues of BP Complexity per iteration O(|E||L| 2 ) Special Pairwise Potentials  ab;ik = w ab d(|i-k|) O(|E||L|) Felzenszwalb & Huttenlocher, 2004 i - k d Potts i - k d Truncated Linear i - k d Truncated Quadratic
  • 103. Computational Issues of BP Memory requirements O(|E||L|) Half of original BP Kolmogorov, 2006 Some approximations exist But memory still remains an issue Yu, Lin, Super and Tan, 2007 Lasserre, Kannan and Winn, 2007
  • 104. Computational Issues of BP Order of reparameterization Randomly Residual Belief Propagation In some fixed order The one that results in maximum change Elidan et al. , 2006
  • 105. Theoretical Properties of BP Exact for Trees Pearl, 1988 What about any general random field? Run BP. Assume it converges.
  • 106. Theoretical Properties of BP Exact for Trees Pearl, 1988 What about any general random field? Choose variables in a tree. Change their labels. Value of energy does not decrease
  • 107. Theoretical Properties of BP Exact for Trees Pearl, 1988 What about any general random field? Choose variables in a cycle. Change their labels. Value of energy does not decrease
  • 108. Theoretical Properties of BP Exact for Trees Pearl, 1988 What about any general random field? For cycles, if BP converges then exact MAP Weiss and Freeman, 2001
  • 109. Results Object Detection Felzenszwalb and Huttenlocher, 2004 Labels - Poses of parts Unary Potentials: Fraction of foreground pixels Pairwise Potentials: Favour Valid Configurations H T A1 A2 L1 L2
  • 110. Results Object Detection Felzenszwalb and Huttenlocher, 2004
  • 111. Results Binary Segmentation Szeliski et al. , 2008 Labels - {foreground, background} Unary Potentials: -log(likelihood) using learnt fg/bg models Pairwise Potentials: 0, if same labels 1 -  exp(|D a - D b |), if different labels
  • 112. Results Binary Segmentation Labels - {foreground, background} Unary Potentials: -log(likelihood) using learnt fg/bg models Szeliski et al. , 2008 Pairwise Potentials: 0, if same labels 1 -  exp(|D a - D b |), if different labels Belief Propagation
  • 113. Results Binary Segmentation Labels - {foreground, background} Unary Potentials: -log(likelihood) using learnt fg/bg models Szeliski et al. , 2008 Global optimum Pairwise Potentials: 0, if same labels 1 -  exp(|D a - D b |), if different labels
  • 114. Results Szeliski et al. , 2008 Labels - {disparities} Unary Potentials: Similarity of pixel colours Pairwise Potentials: 0, if same labels 1 -  exp(|D a - D b |), if different labels Stereo Correspondence
  • 115. Results Szeliski et al. , 2008 Labels - {disparities} Unary Potentials: Similarity of pixel colours Pairwise Potentials: 0, if same labels 1 -  exp(|D a - D b |), if different labels Belief Propagation Stereo Correspondence
  • 116. Results Szeliski et al. , 2008 Labels - {disparities} Unary Potentials: Similarity of pixel colours Global optimum Pairwise Potentials: 0, if same labels 1 -  exp(|D a - D b |), if different labels Stereo Correspondence
  • 117. Summary of BP Exact for chains Exact for trees Approximate MAP for general cases Not even convergence guaranteed So can we do something better?
  • 118. Outline
    • Problem Formulation
    • Reparameterization
    • Belief Propagation
    • Tree-reweighted Message Passing
      • Integer Programming Formulation
      • Linear Programming Relaxation and its Dual
      • Convergent Solution for Dual
      • Computational Issues and Theoretical Properties
  • 119. TRW Message Passing
    • Convex (not Combinatorial) Optimization
    • A different look at the same problem
    • A similar solution
    • Combinatorial (not Convex) Optimization
    We will look at the most general MAP estimation Not trees No assumption on potentials
  • 120. Things to Remember
    • Forward-pass computes min-marginals of root
    • BP is exact for trees
    • Every iteration provides a reparameterization
    • Basics of Mathematical Optimization
  • 121. Mathematical Optimization min g 0 (x) subject to g i (x) ≤ 0 i=1, … , N
    • Objective function
    • Constraints
    • Feasible region = {x | g i (x) ≤ 0}
    x* = arg Optimal Solution g 0 (x*) Optimal Value
  • 122. Integer Programming min g 0 (x) subject to g i (x) ≤ 0 i=1, … , N
    • Objective function
    • Constraints
    • Feasible region = {x | g i (x) ≤ 0}
    x* = arg Optimal Solution g 0 (x*) Optimal Value x k  Z
  • 123. Feasible Region Generally NP-hard to optimize
  • 124. Linear Programming min g 0 (x) subject to g i (x) ≤ 0 i=1, … , N
    • Objective function
    • Constraints
    • Feasible region = {x | g i (x) ≤ 0}
    x* = arg Optimal Solution g 0 (x*) Optimal Value
  • 125. Linear Programming min g 0 (x) subject to g i (x) ≤ 0 i=1, … , N
    • Linear objective function
    • Linear constraints
    • Feasible region = {x | g i (x) ≤ 0}
    x* = arg Optimal Solution g 0 (x*) Optimal Value
  • 126. Linear Programming min c T x subject to Ax ≤ b i=1, … , N
    • Linear objective function
    • Linear constraints
    • Feasible region = {x | Ax ≤ b}
    x* = arg Optimal Solution c T x* Optimal Value Polynomial-time Solution
  • 127. Feasible Region Polynomial-time Solution
  • 128. Feasible Region Optimal solution lies on a vertex (obj func linear)
  • 129. Outline
    • Problem Formulation
    • Reparameterization
    • Belief Propagation
    • Tree-reweighted Message Passing
      • Integer Programming Formulation
      • Linear Programming Relaxation and its Dual
      • Convergent Solution for Dual
      • Computational Issues and Theoretical Properties
  • 130. Integer Programming Formulation V a V b Label l 0 Label l 1 2 5 4 2 0 1 1 0 2 Unary Potentials  a;0 = 5  a;1 = 2  b;0 = 2  b;1 = 4 Labelling f(a) = 1 f(b) = 0 y a;0 = 0 y a;1 = 1 y b;0 = 1 y b;1 = 0 Any f(.) has equivalent boolean variables y a;i
  • 131. Integer Programming Formulation V a V b 2 5 4 2 0 1 1 0 2 Unary Potentials  a;0 = 5  a;1 = 2  b;0 = 2  b;1 = 4 Labelling f(a) = 1 f(b) = 0 y a;0 = 0 y a;1 = 1 y b;0 = 1 y b;1 = 0 Find the optimal variables y a;i Label l 0 Label l 1
  • 132. Integer Programming Formulation V a V b 2 5 4 2 0 1 1 0 2 Unary Potentials  a;0 = 5  a;1 = 2  b;0 = 2  b;1 = 4 Sum of Unary Potentials ∑ a ∑ i  a;i y a;i y a;i  {0,1}, for all V a , l i ∑ i y a;i = 1, for all V a Label l 0 Label l 1
  • 133. Integer Programming Formulation V a V b 2 5 4 2 0 1 1 0 2 Pairwise Potentials  ab;00 = 0  ab;10 = 1  ab;01 = 1  ab;11 = 0 Sum of Pairwise Potentials ∑ (a,b) ∑ ik  ab;ik y a;i y b;k y a;i  {0,1} ∑ i y a;i = 1 Label l 0 Label l 1
  • 134. Integer Programming Formulation V a V b 2 5 4 2 0 1 1 0 2 Pairwise Potentials  ab;00 = 0  ab;10 = 1  ab;01 = 1  ab;11 = 0 Sum of Pairwise Potentials ∑ (a,b) ∑ ik  ab;ik y ab;ik y a;i  {0,1} ∑ i y a;i = 1 y ab;ik = y a;i y b;k Label l 0 Label l 1
  • 135. Integer Programming Formulation min ∑ a ∑ i  a;i y a;i + ∑ (a,b) ∑ ik  ab;ik y ab;ik y a;i  {0,1} ∑ i y a;i = 1 y ab;ik = y a;i y b;k
  • 136. Integer Programming Formulation min  T y y a;i  {0,1} ∑ i y a;i = 1 y ab;ik = y a;i y b;k  = [ …  a;i …. ; …  ab;ik ….] y = [ … y a;i …. ; … y ab;ik ….]
  • 137. One variable, two labels y a;0  {0,1} y a;1  {0,1} y a;0 + y a;1 = 1 y = [ y a;0 y a;1 ]  = [  a;0  a;1 ] y a;0 y a;1
  • 138. Two variables, two labels
    • = [  a;0  a;1  b;0  b;1
    •  ab;00  ab;01  ab;10  ab;11 ]
    y = [ y a;0 y a;1 y b;0 y b;1 y ab;00 y ab;01 y ab;10 y ab;11 ] y a;0  {0,1} y a;1  {0,1} y a;0 + y a;1 = 1 y b;0  {0,1} y b;1  {0,1} y b;0 + y b;1 = 1 y ab;00 = y a;0 y b;0 y ab;01 = y a;0 y b;1 y ab;10 = y a;1 y b;0 y ab;11 = y a;1 y b;1
  • 139. In General Marginal Polytope
  • 140. In General
    •  R (|V||L| + |E||L| 2 )
    y  {0,1} (|V||L| + |E||L| 2 ) Number of constraints |V||L| + |V| + |E||L| 2 y a;i  {0,1} ∑ i y a;i = 1 y ab;ik = y a;i y b;k
  • 141. Integer Programming Formulation min  T y y a;i  {0,1} ∑ i y a;i = 1 y ab;ik = y a;i y b;k  = [ …  a;i …. ; …  ab;ik ….] y = [ … y a;i …. ; … y ab;ik ….]
  • 142. Integer Programming Formulation min  T y y a;i  {0,1} ∑ i y a;i = 1 y ab;ik = y a;i y b;k Solve to obtain MAP labelling y*
  • 143. Integer Programming Formulation min  T y y a;i  {0,1} ∑ i y a;i = 1 y ab;ik = y a;i y b;k But we can’t solve it in general
  • 144. Outline
    • Problem Formulation
    • Reparameterization
    • Belief Propagation
    • Tree-reweighted Message Passing
      • Integer Programming Formulation
      • Linear Programming Relaxation and its Dual
      • Convergent Solution for Dual
      • Computational Issues and Theoretical Properties
  • 145. Linear Programming Relaxation min  T y y a;i  {0,1} ∑ i y a;i = 1 y ab;ik = y a;i y b;k Two reasons why we can’t solve this
  • 146. Linear Programming Relaxation min  T y y a;i  [0,1] ∑ i y a;i = 1 y ab;ik = y a;i y b;k One reason why we can’t solve this
  • 147. Linear Programming Relaxation min  T y y a;i  [0,1] ∑ i y a;i = 1 ∑ k y ab;ik = ∑ k y a;i y b;k One reason why we can’t solve this
  • 148. Linear Programming Relaxation min  T y y a;i  [0,1] ∑ i y a;i = 1 One reason why we can’t solve this = 1 ∑ k y ab;ik = y a;i ∑ k y b;k
  • 149. Linear Programming Relaxation min  T y y a;i  [0,1] ∑ i y a;i = 1 ∑ k y ab;ik = y a;i One reason why we can’t solve this
  • 150. Linear Programming Relaxation min  T y y a;i  [0,1] ∑ i y a;i = 1 ∑ k y ab;ik = y a;i No reason why we can’t solve this * * memory requirements, time complexity
  • 151. One variable, two labels y a;0  {0,1} y a;1  {0,1} y a;0 + y a;1 = 1 y = [ y a;0 y a;1 ]  = [  a;0  a;1 ] y a;0 y a;1
  • 152. One variable, two labels y a;0  [0,1] y a;1  [0,1] y a;0 + y a;1 = 1 y = [ y a;0 y a;1 ]  = [  a;0  a;1 ] y a;0 y a;1
  • 153. Two variables, two labels
    • = [  a;0  a;1  b;0  b;1
    •  ab;00  ab;01  ab;10  ab;11 ]
    y = [ y a;0 y a;1 y b;0 y b;1 y ab;00 y ab;01 y ab;10 y ab;11 ] y a;0  {0,1} y a;1  {0,1} y a;0 + y a;1 = 1 y b;0  {0,1} y b;1  {0,1} y b;0 + y b;1 = 1 y ab;00 = y a;0 y b;0 y ab;01 = y a;0 y b;1 y ab;10 = y a;1 y b;0 y ab;11 = y a;1 y b;1
  • 154. Two variables, two labels
    • = [  a;0  a;1  b;0  b;1
    •  ab;00  ab;01  ab;10  ab;11 ]
    y = [ y a;0 y a;1 y b;0 y b;1 y ab;00 y ab;01 y ab;10 y ab;11 ] y a;0  [0,1] y a;1  [0,1] y a;0 + y a;1 = 1 y b;0  [0,1] y b;1  [0,1] y b;0 + y b;1 = 1 y ab;00 = y a;0 y b;0 y ab;01 = y a;0 y b;1 y ab;10 = y a;1 y b;0 y ab;11 = y a;1 y b;1
  • 155. Two variables, two labels
    • = [  a;0  a;1  b;0  b;1
    •  ab;00  ab;01  ab;10  ab;11 ]
    y = [ y a;0 y a;1 y b;0 y b;1 y ab;00 y ab;01 y ab;10 y ab;11 ] y a;0  [0,1] y a;1  [0,1] y a;0 + y a;1 = 1 y b;0  [0,1] y b;1  [0,1] y b;0 + y b;1 = 1 y ab;00 + y ab;01 = y a;0 y ab;10 = y a;1 y b;0 y ab;11 = y a;1 y b;1
  • 156. Two variables, two labels
    • = [  a;0  a;1  b;0  b;1
    •  ab;00  ab;01  ab;10  ab;11 ]
    y = [ y a;0 y a;1 y b;0 y b;1 y ab;00 y ab;01 y ab;10 y ab;11 ] y a;0  [0,1] y a;1  [0,1] y a;0 + y a;1 = 1 y b;0  [0,1] y b;1  [0,1] y b;0 + y b;1 = 1 y ab;00 + y ab;01 = y a;0 y ab;10 + y ab;11 = y a;1
  • 157. In General Marginal Polytope Local Polytope
  • 158. In General
    •  R (|V||L| + |E||L| 2 )
    y  [0,1] (|V||L| + |E||L| 2 ) Number of constraints |V||L| + |V| + |E||L|
  • 159. Linear Programming Relaxation min  T y y a;i  [0,1] ∑ i y a;i = 1 ∑ k y ab;ik = y a;i No reason why we can’t solve this
  • 160. Linear Programming Relaxation Extensively studied Optimization Schlesinger, 1976 Koster, van Hoesel and Kolen, 1998 Theory Chekuri et al, 2001 Archer et al, 2004 Machine Learning Wainwright et al., 2001
  • 161. Linear Programming Relaxation Many interesting Properties
    • Global optimal MAP for trees
    Wainwright et al., 2001 But we are interested in NP-hard cases
    • Preserves solution for reparameterization
  • 162. Linear Programming Relaxation
    • Large class of problems
      • Metric Labelling
      • Semi-metric Labelling
    Many interesting Properties - Integrality Gap Manokaran et al., 2008
    • Most likely, provides best possible integrality gap
  • 163. Linear Programming Relaxation
    • A computationally useful dual
    Many interesting Properties - Dual Optimal value of dual = Optimal value of primal Easier-to-solve
  • 164. Dual of the LP Relaxation Wainwright et al., 2001 V a V b V c V d V e V f V g V h V i  min  T y y a;i  [0,1] ∑ i y a;i = 1 ∑ k y ab;ik = y a;i
  • 165. Dual of the LP Relaxation Wainwright et al., 2001 V a V b V c V d V e V f V g V h V i  V a V b V c V d V e V f V g V h V i V a V b V c V d V e V f V g V h V i  1  2  3  4  5  6  1  2  3  4  5  6   i  i =   i ≥ 0
  • 166. Dual of the LP Relaxation Wainwright et al., 2001  1  2  3  4  5  6 q*(  1 )   i  i =  q*(  2 ) q*(  3 ) q*(  4 ) q*(  5 ) q*(  6 )   i q*(  i ) Dual of LP  V a V b V c V d V e V f V g V h V i V a V b V c V d V e V f V g V h V i V a V b V c V d V e V f V g V h V i  i ≥ 0 max
  • 167. Dual of the LP Relaxation Wainwright et al., 2001  1  2  3  4  5  6 q*(  1 )   i  i   q*(  2 ) q*(  3 ) q*(  4 ) q*(  5 ) q*(  6 ) Dual of LP  V a V b V c V d V e V f V g V h V i V a V b V c V d V e V f V g V h V i V a V b V c V d V e V f V g V h V i  i ≥ 0   i q*(  i ) max
  • 168. Dual of the LP Relaxation Wainwright et al., 2001   i  i   max   i q*(  i ) I can easily compute q*(  i ) I can easily maintain reparam constraint So can I easily solve the dual?
  • 169. Outline
    • Problem Formulation
    • Reparameterization
    • Belief Propagation
    • Tree-reweighted Message Passing
      • Integer Programming Formulation
      • Linear Programming Relaxation and its Dual
      • Convergent Solution for Dual
      • Computational Issues and Theoretical Properties
  • 170. TRW Message Passing Kolmogorov, 2006 V a V b V c V d V e V f V g V h V i V a V b V c V d V e V f V g V h V i  1  2  3  1  2  3  4  5  6  4  5  6   i  i     i q*(  i ) Pick a variable V a
  • 171. TRW Message Passing Kolmogorov, 2006   i  i     i q*(  i ) V c V b V a  1 c;0  1 c;1  1 b;0  1 b;1  1 a;0  1 a;1 V a V d V g  4 a;0  4 a;1  4 d;0  4 d;1  4 g;0  4 g;1
  • 172. TRW Message Passing Kolmogorov, 2006  1  1 +  4  4 +  rest    1 q*(  1 ) +  4 q*(  4 ) + K V c V b V a V a V d V g Reparameterize to obtain min-marginals of V a  1 c;0  1 c;1  1 b;0  1 b;1  1 a;0  1 a;1  4 a;0  4 a;1  4 d;0  4 d;1  4 g;0  4 g;1
  • 173. TRW Message Passing Kolmogorov, 2006  1  ’ 1 +  4  ’ 4 +  rest V c V b V a  ’ 1 c;0  ’ 1 c;1  ’ 1 b;0  ’ 1 b;1  ’ 1 a;0  ’ 1 a;1 V a V d V g  ’ 4 a;0  ’ 4 a;1  ’ 4 d;0  ’ 4 d;1  ’ 4 g;0  ’ 4 g;1 One pass of Belief Propagation  1 q*(  ’ 1 ) +  4 q*(  ’ 4 ) + K
  • 174. TRW Message Passing Kolmogorov, 2006  1  ’ 1 +  4  ’ 4 +  rest   V c V b V a V a V d V g Remain the same  1 q*(  ’ 1 ) +  4 q*(  ’ 4 ) + K  ’ 1 c;0  ’ 1 c;1  ’ 1 b;0  ’ 1 b;1  ’ 1 a;0  ’ 1 a;1  ’ 4 a;0  ’ 4 a;1  ’ 4 d;0  ’ 4 d;1  ’ 4 g;0  ’ 4 g;1
  • 175. TRW Message Passing Kolmogorov, 2006  1  ’ 1 +  4  ’ 4 +  rest    1 min{  ’ 1 a;0 ,  ’ 1 a;1 } +  4 min{  ’ 4 a;0 ,  ’ 4 a;1 } + K V c V b V a V a V d V g  ’ 1 c;0  ’ 1 c;1  ’ 1 b;0  ’ 1 b;1  ’ 1 a;0  ’ 1 a;1  ’ 4 a;0  ’ 4 a;1  ’ 4 d;0  ’ 4 d;1  ’ 4 g;0  ’ 4 g;1
  • 176. TRW Message Passing Kolmogorov, 2006  1  ’ 1 +  4  ’ 4 +  rest   V c V b V a V a V d V g Compute weighted average of min-marginals of V a  ’ 1 c;0  ’ 1 c;1  ’ 1 b;0  ’ 1 b;1  ’ 1 a;0  ’ 1 a;1  ’ 4 a;0  ’ 4 a;1  ’ 4 d;0  ’ 4 d;1  ’ 4 g;0  ’ 4 g;1  1 min{  ’ 1 a;0 ,  ’ 1 a;1 } +  4 min{  ’ 4 a;0 ,  ’ 4 a;1 } + K
  • 177. TRW Message Passing Kolmogorov, 2006  1  ’ 1 +  4  ’ 4 +  rest   V c V b V a V a V d V g  ’’ a;0 =  1  ’ 1 a;0 +  4  ’ 4 a;0  1 +  4  ’’ a;1 =  1  ’ 1 a;1 +  4  ’ 4 a;1  1 +  4  ’ 1 c;0  ’ 1 c;1  ’ 1 b;0  ’ 1 b;1  ’ 1 a;0  ’ 1 a;1  ’ 4 a;0  ’ 4 a;1  ’ 4 d;0  ’ 4 d;1  ’ 4 g;0  ’ 4 g;1  1 min{  ’ 1 a;0 ,  ’ 1 a;1 } +  4 min{  ’ 4 a;0 ,  ’ 4 a;1 } + K
  • 178. TRW Message Passing Kolmogorov, 2006  1  ’’ 1 +  4  ’’ 4 +  rest V c V b V a V a V d V g  ’ 1 c;0  ’ 1 c;1  ’ 1 b;0  ’ 1 b;1  ’’ a;0  ’’ a;1  ’’ a;0  ’’ a;1  ’ 4 d;0  ’ 4 d;1  ’ 4 g;0  ’ 4 g;1  1 min{  ’ 1 a;0 ,  ’ 1 a;1 } +  4 min{  ’ 4 a;0 ,  ’ 4 a;1 } + K  ’’ a;0 =  1  ’ 1 a;0 +  4  ’ 4 a;0  1 +  4  ’’ a;1 =  1  ’ 1 a;1 +  4  ’ 4 a;1  1 +  4
  • 179. TRW Message Passing Kolmogorov, 2006  1  ’’ 1 +  4  ’’ 4 +  rest   V c V b V a V a V d V g  ’ 1 c;0  ’ 1 c;1  ’ 1 b;0  ’ 1 b;1  ’’ a;0  ’’ a;1  ’’ a;0  ’’ a;1  ’ 4 d;0  ’ 4 d;1  ’ 4 g;0  ’ 4 g;1  1 min{  ’ 1 a;0 ,  ’ 1 a;1 } +  4 min{  ’ 4 a;0 ,  ’ 4 a;1 } + K  ’’ a;0 =  1  ’ 1 a;0 +  4  ’ 4 a;0  1 +  4  ’’ a;1 =  1  ’ 1 a;1 +  4  ’ 4 a;1  1 +  4
  • 180. TRW Message Passing Kolmogorov, 2006  1  ’’ 1 +  4  ’’ 4 +  rest   V c V b V a V a V d V g  1 min{  ’’ a;0 ,  ’’ a;1 } +  4 min{  ’’ a;0 ,  ’’ a;1 } + K  ’ 1 c;0  ’ 1 c;1  ’ 1 b;0  ’ 1 b;1  ’’ a;0  ’’ a;1  ’’ a;0  ’’ a;1  ’ 4 d;0  ’ 4 d;1  ’ 4 g;0  ’ 4 g;1  ’’ a;0 =  1  ’ 1 a;0 +  4  ’ 4 a;0  1 +  4  ’’ a;1 =  1  ’ 1 a;1 +  4  ’ 4 a;1  1 +  4
  • 181. TRW Message Passing Kolmogorov, 2006  1  ’’ 1 +  4  ’’ 4 +  rest   V c V b V a V a V d V g (  1 +  4 ) min{  ’’ a;0 ,  ’’ a;1 } + K  ’ 1 c;0  ’ 1 c;1  ’ 1 b;0  ’ 1 b;1  ’’ a;0  ’’ a;1  ’’ a;0  ’’ a;1  ’ 4 d;0  ’ 4 d;1  ’ 4 g;0  ’ 4 g;1  ’’ a;0 =  1  ’ 1 a;0 +  4  ’ 4 a;0  1 +  4  ’’ a;1 =  1  ’ 1 a;1 +  4  ’ 4 a;1  1 +  4
  • 182. TRW Message Passing Kolmogorov, 2006  1  ’’ 1 +  4  ’’ 4 +  rest   V c V b V a V a V d V g (  1 +  4 ) min{  ’’ a;0 ,  ’’ a;1 } + K  ’ 1 c;0  ’ 1 c;1  ’ 1 b;0  ’ 1 b;1  ’’ a;0  ’’ a;1  ’’ a;0  ’’ a;1  ’ 4 d;0  ’ 4 d;1  ’ 4 g;0  ’ 4 g;1 min {p 1 +p 2 , q 1 +q 2 } min {p 1 , q 1 } + min {p 2 , q 2 } ≥
  • 183. TRW Message Passing Kolmogorov, 2006  1  ’’ 1 +  4  ’’ 4 +  rest   V c V b V a V a V d V g Objective function increases or remains constant  ’ 1 c;0  ’ 1 c;1  ’ 1 b;0  ’ 1 b;1  ’’ a;0  ’’ a;1  ’’ a;0  ’’ a;1  ’ 4 d;0  ’ 4 d;1  ’ 4 g;0  ’ 4 g;1 (  1 +  4 ) min{  ’’ a;0 ,  ’’ a;1 } + K
  • 184. TRW Message Passing Initialize  i . Take care of reparam constraint Choose random variable V a Compute min-marginals of V a for all trees Node-average the min-marginals REPEAT Kolmogorov, 2006 Can also do edge-averaging
  • 185. Example 1 V a V b 0 1 1 0 2 5 4 2 l 0 l 1 V b V c 0 2 3 1 4 2 6 3 V c V a 1 4 1 0 6 3 6 4  2 =1  3 =1  1 =1 5 6 7 Pick variable V a . Reparameterize.
  • 186. Example 1 V a V b -3 -2 -1 -2 5 7 4 2 V b V c 0 2 3 1 4 2 6 3 V c V a -3 1 -3 -3 6 3 10 7  2 =1  3 =1  1 =1 5 6 7 Average the min-marginals of V a l 0 l 1
  • 187. Example 1 V a V b -3 -2 -1 -2 7.5 7 4 2 V b V c 0 2 3 1 4 2 6 3 V c V a -3 1 -3 -3 6 3 7.5 7  2 =1  3 =1  1 =1 7 6 7 Pick variable V b . Reparameterize. l 0 l 1
  • 188. Example 1 V a V b -7.5 -7 -5.5 -7 7.5 7 8.5 7 V b V c -5 -3 -1 -3 9 6 6 3 V c V a -3 1 -3 -3 6 3 7.5 7  2 =1  3 =1  1 =1 7 6 7 Average the min-marginals of V b l 0 l 1
  • 189. Example 1 V a V b -7.5 -7 -5.5 -7 7.5 7 8.75 6.5 V b V c -5 -3 -1 -3 8.75 6.5 6 3 V c V a -3 1 -3 -3 6 3 7.5 7  2 =1  3 =1  1 =1 6.5 6.5 7 Value of dual does not increase l 0 l 1
  • 190. Example 1 V a V b -7.5 -7 -5.5 -7 7.5 7 8.75 6.5 V b V c -5 -3 -1 -3 8.75 6.5 6 3 V c V a -3 1 -3 -3 6 3 7.5 7  2 =1  3 =1  1 =1 6.5 6.5 7 Maybe it will increase for V c NO l 0 l 1
  • 191. Example 1 V a V b -7.5 -7 -5.5 -7 7.5 7 8.75 6.5 V b V c -5 -3 -1 -3 8.75 6.5 6 3 V c V a -3 1 -3 -3 6 3 7.5 7  2 =1  3 =1  1 =1 Strong Tree Agreement Exact MAP Estimate f 1 (a) = 0 f 1 (b) = 0 f 2 (b) = 0 f 2 (c) = 0 f 3 (c) = 0 f 3 (a) = 0 l 0 l 1
  • 192. Example 2 V a V b 0 1 1 0 2 5 2 2 V b V c 1 0 0 1 0 0 0 0 V c V a 0 1 1 0 0 3 4 8  2 =1  3 =1  1 =1 4 0 4 Pick variable V a . Reparameterize. l 0 l 1
  • 193. Example 2 V a V b -2 -1 -1 -2 4 7 2 2 V b V c 1 0 0 1 0 0 0 0 V c V a 0 0 1 -1 0 3 4 9  2 =1  3 =1  1 =1 4 0 4 Average the min-marginals of V a l 0 l 1
  • 194. Example 2 V a V b -2 -1 -1 -2 4 8 2 2 V b V c 1 0 0 1 0 0 0 0 V c V a 0 0 1 -1 0 3 4 8  2 =1  3 =1  1 =1 4 0 4 Value of dual does not increase l 0 l 1
  • 195. Example 2 V a V b -2 -1 -1 -2 4 8 2 2 V b V c 1 0 0 1 0 0 0 0 V c V a 0 0 1 -1 0 3 4 8  2 =1  3 =1  1 =1 4 0 4 Maybe it will decrease for V b or V c NO l 0 l 1
  • 196. Example 2 V a V b -2 -1 -1 -2 4 8 2 2 V b V c 1 0 0 1 0 0 0 0 V c V a 0 0 1 -1 0 3 4 8  2 =1  3 =1  1 =1 f 1 (a) = 1 f 1 (b) = 1 f 2 (b) = 1 f 2 (c) = 0 f 3 (c) = 1 f 3 (a) = 1 f 2 (b) = 0 f 2 (c) = 1 Weak Tree Agreement Not Exact MAP Estimate l 0 l 1
  • 197. Example 2 V a V b -2 -1 -1 -2 4 8 2 2 V b V c 1 0 0 1 0 0 0 0 V c V a 0 0 1 -1 0 3 4 8  2 =1  3 =1  1 =1 Weak Tree Agreement Convergence point of TRW l 0 l 1 f 1 (a) = 1 f 1 (b) = 1 f 2 (b) = 1 f 2 (c) = 0 f 3 (c) = 1 f 3 (a) = 1 f 2 (b) = 0 f 2 (c) = 1
  • 198. Obtaining the Labelling Only solves the dual. Primal solutions? V a V b V c V d V e V f V g V h V i  ’ =   i  i   Fix the label Of V a
  • 199. Obtaining the Labelling Only solves the dual. Primal solutions? V a V b V c V d V e V f V g V h V i  ’ =   i  i   Fix the label Of V b Continue in some fixed order Meltzer et al., 2006
  • 200. Outline
    • Problem Formulation
    • Reparameterization
    • Belief Propagation
    • Tree-reweighted Message Passing
      • Integer Programming Formulation
      • Linear Programming Relaxation and its Dual
      • Convergent Solution for Dual
      • Computational Issues and Theoretical Properties
  • 201. Computational Issues of TRW
    • Speed-ups for some pairwise potentials
    Basic Component is Belief Propagation Felzenszwalb & Huttenlocher, 2004
    • Memory requirements cut down by half
    Kolmogorov, 2006
    • Further speed-ups using monotonic chains
    Kolmogorov, 2006
  • 202. Theoretical Properties of TRW
    • Always converges, unlike BP
    Kolmogorov, 2006
    • Strong tree agreement implies exact MAP
    Wainwright et al., 2001
    • Optimal MAP for two-label submodular problems
    Kolmogorov and Wainwright, 2005  ab;00 +  ab;11 ≤  ab;01 +  ab;10
  • 203. Results Binary Segmentation Szeliski et al. , 2008 Labels - {foreground, background} Unary Potentials: -log(likelihood) using learnt fg/bg models Pairwise Potentials: 0, if same labels 1 -  exp(|D a - D b |), if different labels
  • 204. Results Binary Segmentation Labels - {foreground, background} Unary Potentials: -log(likelihood) using learnt fg/bg models Szeliski et al. , 2008 Pairwise Potentials: 0, if same labels 1 -  exp(|D a - D b |), if different labels TRW
  • 205. Results Binary Segmentation Labels - {foreground, background} Unary Potentials: -log(likelihood) using learnt fg/bg models Szeliski et al. , 2008 Belief Propagation Pairwise Potentials: 0, if same labels 1 -  exp(|D a - D b |), if different labels
  • 206. Results Stereo Correspondence Szeliski et al. , 2008 Labels - {disparities} Unary Potentials: Similarity of pixel colours Pairwise Potentials: 0, if same labels 1 -  exp(|D a - D b |), if different labels
  • 207. Results Szeliski et al. , 2008 Labels - {disparities} Unary Potentials: Similarity of pixel colours Pairwise Potentials: 0, if same labels 1 -  exp(|D a - D b |), if different labels TRW Stereo Correspondence
  • 208. Results Szeliski et al. , 2008 Labels - {disparities} Unary Potentials: Similarity of pixel colours Belief Propagation Pairwise Potentials: 0, if same labels 1 -  exp(|D a - D b |), if different labels Stereo Correspondence
  • 209. Results Non-submodular problems Kolmogorov, 2006 BP TRW-S 30x30 grid K 50 BP TRW-S BP outperforms TRW-S
  • 210. Summary
    • Trees can be solved exactly - BP
    • No guarantee of convergence otherwise - BP
    • Strong Tree Agreement - TRW-S
    • Submodular energies solved exactly - TRW-S
    • TRW-S solves an LP relaxation of MAP estimation
    • Loopier graphs give worse results
    • Rother and Kolmogorov, 2006
  • 211. Related New(er) Work
    • Solving the Dual
    Globerson and Jaakkola, 2007 Komodakis, Paragios and Tziritas 2007 Weiss et al., 2006 Schlesinger and Giginyak, 2007
    • Solving the Primal
    Ravikumar, Agarwal and Wainwright, 2008
  • 212. Related New(er) Work
    • More complex relaxations
    Sontag and Jaakkola, 2007 Komodakis and Paragios, 2008 Kumar, Kolmogorov and Torr, 2007 Werner, 2008 Sontag et al., 2008 Kumar and Torr, 2008
  • 213. Questions on Part I ? Code + Standard Data http://vision.middlebury.edu/MRF