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Probabilistic Inference Lecture 2

Probabilistic Inference Lecture 2. M. Pawan Kumar pawan.kumar@ecp.fr. Slides available online http:// cvc.centrale-ponts.fr /personnel/ pawan /. Pose Estimation. Courtesy Pedro Felzenszwalb. Pose Estimation. Courtesy Pedro Felzenszwalb. Pose Estimation. Variables are body parts.

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Probabilistic Inference Lecture 2

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  1. Probabilistic InferenceLecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online http://cvc.centrale-ponts.fr/personnel/pawan/

  2. Pose Estimation Courtesy Pedro Felzenszwalb

  3. Pose Estimation Courtesy Pedro Felzenszwalb

  4. Pose Estimation Variables are body parts Labels are positions

  5. Pose Estimation Variables are body parts Labels are positions Unary potentials θa;i proportional to fraction of foreground pixels

  6. Pose Estimation Head Joint location according to ‘head’ part d Joint location according to ‘torso’ part Torso Pairwise potentials θab;ik proportional to d2

  7. Pose Estimation Head Head d > d Torso Torso Pairwise potentials θab;ik proportional to d2

  8. Outline • Problem Formulation • Energy Function • Energy Minimization • Computing min-marginals • Reparameterization • Energy Minimization for Trees • Loopy Belief Propagation

  9. Energy Function Label l1 Label l0 Vb Vc Vd Va Random Variables V= {Va, Vb, ….} Labels L= {l0, l1, ….} Labelling f: {a, b, …. }  {0,1, …}

  10. Energy Function 6 3 2 4 Label l1 Label l0 5 3 7 2 Vb Vc Vd Va Easy to minimize Q(f) = ∑a a;f(a) Neighbourhood Unary Potential

  11. Energy Function 6 3 2 4 Label l1 Label l0 5 3 7 2 Vb Vc Vd Va E : (a,b)  E iff Va and Vb are neighbours E = { (a,b) , (b,c) , (c,d) }

  12. Energy Function 0 6 1 3 2 0 4 Label l1 1 2 3 4 1 1 Label l0 1 0 5 0 3 7 2 Vb Vc Vd Va Pairwise Potential Q(f) = ∑a a;f(a) +∑(a,b) ab;f(a)f(b)

  13. Energy Function 0 6 1 3 2 0 4 Label l1 1 2 3 4 1 1 Label l0 1 0 5 0 3 7 2 Vb Vc Vd Va Q(f; ) = ∑a a;f(a) +∑(a,b) ab;f(a)f(b) Parameter

  14. Outline • Problem Formulation • Energy Function • Energy Minimization • Computing min-marginals • Reparameterization • Energy Minimization for Trees • Loopy Belief Propagation

  15. Energy Minimization 0 6 1 3 2 0 4 Label l1 1 2 3 4 1 1 Label l0 1 0 5 0 3 7 2 Vb Vc Vd Va Q(f; ) = ∑a a;f(a) + ∑(a,b) ab;f(a)f(b)

  16. Energy Minimization 0 6 1 3 2 0 4 Label l1 1 2 3 4 1 1 Label l0 1 0 5 0 3 7 2 Vb Vc Vd Va Q(f; ) = ∑a a;f(a) + ∑(a,b) ab;f(a)f(b) 2 + 1 + 2 + 1 + 3 + 1 + 3 = 13

  17. Energy Minimization 0 6 1 3 2 0 4 Label l1 1 2 3 4 1 1 Label l0 1 0 5 0 3 7 2 Vb Vc Vd Va Q(f; ) = ∑a a;f(a) + ∑(a,b) ab;f(a)f(b)

  18. Energy Minimization 0 6 1 3 2 0 4 Label l1 1 2 3 4 1 1 Label l0 1 0 5 0 3 7 2 Vb Vc Vd Va Q(f; ) = ∑a a;f(a) + ∑(a,b) ab;f(a)f(b) 5 + 1 + 4 + 0 + 6 + 4 + 7 = 27

  19. Energy Minimization 0 6 1 3 2 0 4 Label l1 1 2 3 4 1 1 Label l0 1 0 5 0 3 7 2 Vb Vc Vd Va q* = min Q(f; ) = Q(f*; ) Q(f; ) = ∑a a;f(a) + ∑(a,b) ab;f(a)f(b) f* = arg min Q(f; )

  20. Energy Minimization f* = {1, 0, 0, 1} 16 possible labellings q* = 13

  21. Outline • Problem Formulation • Energy Function • Energy Minimization • Computing min-marginals • Reparameterization • Energy Minimization for Trees • Loopy Belief Propagation

  22. Min-Marginals 0 6 1 3 2 0 4 Label l1 1 2 3 4 1 1 Label l0 1 0 5 0 3 7 2 Vb Vc Vd Va such that f(a) = i f* = arg min Q(f; ) Min-marginal qa;i

  23. Min-Marginals qa;0 = 15 16 possible labellings

  24. Min-Marginals qa;1 = 13 16 possible labellings

  25. Min-Marginals and MAP • Minimum min-marginal of any variable = • energy of MAP labelling qa;i mini ) mini ( such that f(a) = i minfQ(f; ) Va has to take one label minfQ(f; )

  26. Summary Energy Function Q(f; ) = ∑a a;f(a) + ∑(a,b) ab;f(a)f(b) Energy Minimization f* = arg min Q(f; ) Min-marginals s.t. f(a) = i qa;i= min Q(f; )

  27. Outline • Problem Formulation • Reparameterization • Energy Minimization for Trees • Loopy Belief Propagation

  28. Reparameterization 2 + - 2 2 0 4 1 1 2 + - 2 5 0 2 Vb Va Add a constant to all a;i Subtract that constant from all b;k

  29. Reparameterization 2 + - 2 2 0 4 1 1 2 + - 2 5 0 2 Vb Va Add a constant to all a;i Subtract that constant from all b;k Q(f; ’) = Q(f; )

  30. Reparameterization + 3 - 3 2 0 4 - 3 1 1 5 0 2 Vb Va Add a constant to one b;k Subtract that constant from ab;ik for all ‘i’

  31. Reparameterization + 3 - 3 2 0 4 - 3 1 1 5 0 2 Vb Va Add a constant to one b;k Subtract that constant from ab;ik for all ‘i’ Q(f; ’) = Q(f; )

  32. 3 1 3 1 3 0 1 1 1 2 2 4 2 4 2 4 0 1 2 1 5 0 5 5 2 2 2 Vb Vb Vb Va Va Va Reparameterization - 4 + 4 + 1 - 4 - 2 - 1 + 1 - 2 - 4 + 1 - 2 + 2 + Mab;k + Mba;i ’b;k = b;k ’a;i = a;i Q(f; ’) = Q(f; ) ’ab;ik = ab;ik - Mab;k - Mba;i

  33. Equivalently 2 + - 2 2 0 4 + Mba;i ’a;i = a;i 1 1 + Mab;k 2 + - 2 5 0 2 Vb Va ’ab;ik = ab;ik - Mab;k - Mba;i Reparameterization ’ is a reparameterization of , iff ’   Q(f; ’) = Q(f; ), for all f Kolmogorov, PAMI, 2006 ’b;k = b;k

  34. Recap Energy Minimization f* = arg min Q(f; ) Q(f; ) = ∑a a;f(a) + ∑(a,b) ab;f(a)f(b) Min-marginals s.t. f(a) = i qa;i= min Q(f; ) Reparameterization ’   Q(f; ’) = Q(f; ), for all f

  35. Outline • Problem Formulation • Reparameterization • Energy Minimization for Trees • Loopy Belief Propagation Pearl, 1988

  36. Belief Propagation • Some problems are easy • Belief Propagation is exact for chains • Exact MAP for trees • Clever Reparameterization

  37. Two Variables 2 2 0 4 1 1 5 0 5 2 Vb Vb Va Va Add a constant to one b;k Subtract that constant from ab;ik for all ‘i’ ’b;k = qb;k Choose the right constant

  38. Two Variables 2 2 0 4 1 1 5 0 5 2 Vb Vb Va Va = 5 + 0 a;0 + ab;00 Mab;0 = min = 2 + 1 a;1 + ab;10 ’b;k = qb;k Choose the right constant

  39. Two Variables 2 2 0 4 1 -2 5 -3 5 5 Vb Vb Va Va ’b;k = qb;k Choose the right constant

  40. Two Variables f(a) = 1 2 2 0 4 1 -2 5 -3 5 5 Vb Vb Va Va ’b;0 = qb;0 Potentials along the red path add up to 0 ’b;k = qb;k Choose the right constant

  41. Two Variables 2 2 0 4 1 -2 5 -3 5 5 Vb Vb Va Va = 5 + 1 a;0 + ab;01 Mab;1 = min = 2 + 0 a;1 + ab;11 ’b;k = qb;k Choose the right constant

  42. Two Variables f(a) = 1 f(a) = 1 2 2 -2 6 -1 -2 5 -3 5 5 Vb Vb Va Va ’b;0 = qb;0 ’b;1 = qb;1 Minimum of min-marginals = MAP estimate ’b;k = qb;k Choose the right constant

  43. Two Variables f(a) = 1 f(a) = 1 2 2 -2 6 -1 -2 5 -3 5 5 Vb Vb Va Va ’b;0 = qb;0 ’b;1 = qb;1 f*(b) = 0 f*(a) = 1 ’b;k = qb;k Choose the right constant

  44. Two Variables f(a) = 1 f(a) = 1 2 2 -2 6 -1 -2 5 -3 5 5 Vb Vb Va Va ’b;0 = qb;0 ’b;1 = qb;1 We get all the min-marginals of Vb ’b;k = qb;k Choose the right constant

  45. + Mab;k ’b;k= b;k + Mba;i ’a;i = a;i ’ab;ik = ab;ik - Mab;k - Mba;i Recap We only need to know two sets of equations General form of Reparameterization Reparameterization of (a,b) in Belief Propagation Mab;k = mini { a;i + ab;ik } Mba;i = 0

  46. Three Variables 0 2 4 0 6 l1 3 1 1 2 l0 5 0 1 3 2 Vb Vc Va Reparameterize the edge (a,b) as before

  47. Three Variables f(a) = 1 -2 2 6 0 6 l1 3 -2 -1 2 l0 5 -3 1 3 5 Vb Vc Va f(a) = 1 Reparameterize the edge (a,b) as before

  48. Three Variables f(a) = 1 -2 2 6 0 6 l1 3 -2 -1 2 l0 5 -3 1 3 5 Vb Vc Va f(a) = 1 Reparameterize the edge (a,b) as before Potentials along the red path add up to 0

  49. Three Variables f(a) = 1 -2 2 6 0 6 l1 3 -2 -1 2 l0 5 -3 1 3 5 Vb Vc Va f(a) = 1 Reparameterize the edge (b,c) as before Potentials along the red path add up to 0

  50. Three Variables f(a) = 1 f(b) = 1 -2 2 6 -6 12 l1 -3 -2 -1 -4 l0 5 -3 -5 9 5 Vb Vc Va f(a) = 1 f(b) = 0 Reparameterize the edge (b,c) as before Potentials along the red path add up to 0

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