Presentation is loading. Please wait.

Presentation is loading. Please wait.

Solving Markov Random Fields using Dynamic Graph Cuts & Second Order Cone Programming Relaxations M. Pawan Kumar, Pushmeet Kohli Philip Torr.

Similar presentations


Presentation on theme: "Solving Markov Random Fields using Dynamic Graph Cuts & Second Order Cone Programming Relaxations M. Pawan Kumar, Pushmeet Kohli Philip Torr."— Presentation transcript:

1 Solving Markov Random Fields using Dynamic Graph Cuts & Second Order Cone Programming Relaxations M. Pawan Kumar, Pushmeet Kohli Philip Torr

2 Talk Outline Dynamic Graph Cuts –Fast reestimation of cut –Useful for video –Object specific segmentation Estimation of non submodular MRFs –Relaxations beyond linear!!

3 Example: Video Segmentation

4 Model Based Segmentation ImageSegmentationPose Estimate [Images courtesy: M. Black, L. Sigal]

5 Min-Marginals Image MAP SolutionBelief - Foreground Low smoothness High smoothness Moderate smoothness Colour Scale

6 Uses of Min marginals Estimate of true marginals (uncertainty) Parameter Learning. Get best n solutions easily.

7 Dynamic Graph Cuts PBPB SBSB cheaper operation computationally expensive operation Simpler problem P B* differences between A and B similar PAPA SASA solve

8 First segmentation problem MAP solution GaGa Our Algorithm GbGb second segmentation problem Maximum flow residual graph ( G r ) G` difference between G a and G b updated residual graph

9 The Max-flow Problem - Edge capacity and flow balance constraints Computing the st-mincut from Max-flow algorithms Notation - Residual capacity (edge capacity – current flow) - Augmenting path Simple Augmenting Path based Algorithms - Repeatedly find augmenting paths and push flow. - Saturated edges constitute the st-mincut. [Ford-Fulkerson Theorem] Sink (1) Source (0) a1a1 a2a

10 9 + α 4 + α Adding a constant to both the t-edges of a node does not change the edges constituting the st-mincut. Key Observation Sink (1) Source (0) a1a1 a2a E (a 1,a 2 ) = 2a 1 + 5ā 1 + 9a 2 + 4ā 2 + 2a 1 ā 2 + ā 1 a 2 E*(a 1,a 2 ) = E(a 1,a 2 ) + α(a 2 +ā 2 ) = E(a 1,a 2 ) + α [a 2 +ā 2 =1] Reparametrization

11 9 + α 4 All reparametrizations of the graph are sums of these two types. Other type of reparametrization Sink (1) Source (0) a1a1 a2a α 2 + α 1 - α Reparametrization, second type Both maintain the solution and add a constant α to the energy.

12 Reparametrization Nice result (easy to prove) All other reparametrizations can be viewed in terms of these two basic operations. Proof in Hammer, and also in one of Vlads recent papers.

13 s G t original graph 0/9 0/7 0/5 0/20/4 0/1 xixi xjxj flow/residual capacity Graph Re-parameterization

14 s G t original graph 0/9 0/7 0/5 0/20/4 0/1 xixi xjxj flow/residual capacity Graph Re-parameterization t residual graph xixi xjxj 0/12 5/2 3/2 1/0 2/04/0 st-mincut Compute Maxflow GrGr Edges cut

15 Update t-edge Capacities s GrGr t residual graph xixi xjxj 0/12 5/2 3/2 1/0 2/04/0

16 Update t-edge Capacities s GrGr t residual graph xixi xjxj 0/12 5/2 3/2 1/0 2/04/0 capacity changes from 7 to 4

17 Update t-edge Capacities s G` t updated residual graph xixi xjxj 0/12 5/-1 3/2 1/0 2/04/0 capacity changes from 7 to 4 edge capacity constraint violated! (flow > capacity) = 5 – 4 = 1 excess flow (e) = flow – new capacity

18 add e to both t-edges connected to node i Update t-edge Capacities s G` t updated residual graph xixi xjxj 0/12 3/2 1/0 2/04/0 capacity changes from 7 to 4 edge capacity constraint violated! (flow > capacity) = 5 – 4 = 1 excess flow (e) = flow – new capacity 5/-1

19 Update t-edge Capacities s G` t updated residual graph xixi xjxj 0/12 3/2 1/0 4/0 capacity changes from 7 to 4 excess flow (e) = flow – new capacity add e to both t-edges connected to node i = 5 – 4 = 1 5/0 2/1 edge capacity constraint violated! (flow > capacity)

20 Update n-edge Capacities s GrGr t residual graph xixi xjxj 0/12 5/2 3/2 1/0 2/04/0 Capacity changes from 5 to 2

21 Update n-edge Capacities s t Updated residual graph xixi xjxj 0/12 5/2 3/-1 1/0 2/04/0 G` Capacity changes from 5 to 2 - edge capacity constraint violated!

22 Update n-edge Capacities s t Updated residual graph xixi xjxj 0/12 5/2 3/-1 1/0 2/04/0 G` Capacity changes from 5 to 2 - edge capacity constraint violated! Reduce flow to satisfy constraint

23 Update n-edge Capacities s t Updated residual graph xixi xjxj 0/11 5/2 2/0 1/0 2/04/0 excess deficiency G` Capacity changes from 5 to 2 - edge capacity constraint violated! Reduce flow to satisfy constraint - causes flow imbalance!

24 Update n-edge Capacities s t Updated residual graph xixi xjxj 0/11 5/2 2/0 1/0 2/04/0 deficiency excess G` Capacity changes from 5 to 2 - edge capacity constraint violated! Reduce flow to satisfy constraint - causes flow imbalance! Push excess flow to/from the terminals Create capacity by adding α = excess to both t-edges.

25 Update n-edge Capacities Updated residual graph Capacity changes from 5 to 2 - edge capacity constraint violated! Reduce flow to satisfy constraint - causes flow imbalance! Push excess flow to the terminals Create capacity by adding α = excess to both t-edges. G` xixi xjxj 0/11 5/3 2/0 3/04/1 s t

26 Update n-edge Capacities Updated residual graph Capacity changes from 5 to 2 - edge capacity constraint violated! Reduce flow to satisfy constraint - causes flow imbalance! Push excess flow to the terminals Create capacity by adding α = excess to both t-edges. G` xixi xjxj 0/11 5/3 2/0 3/04/1 s t

27 Complexity analysis of MRF Update Operations MRF Energy Operation Graph OperationComplexity modifying a unary term modifying a pair-wise term adding a latent variable delete a latent variable Updating a t-edge capacity Updating a n-edge capacity adding a node set the capacities of all edges of a node zero O(1) O(k)* *requires k edge update operations where k is degree of the node

28 Finding augmenting paths is time consuming. Dual-tree maxflow algorithm [Boykov & Kolmogorov PAMI 2004] -Reuses search trees after each augmentation. -Empirically shown to be substantially faster. Our Idea –Reuse search trees from previous graph cut computation –Saves us search tree creation tree time [O(#edges)] –Search trees have to be modified to make them consistent with new graphs – Constrain the search of augmenting paths New paths must contain at least one updated edge Improving the Algorithm

29 Reusing Search Trees c = measure of change in the energy –Running time Dynamic algorithm (c + re-create search tree ) Improved dynamic algorithm (c) Video Segmentation Example - Duplicate image frames (No time is needed)

30 Dynamic Graph Cut vs Active Cuts Our method flow recycling AC cut recycling Both methods: Tree recycling

31 Experimental Analysis MRF consisting of 2x10 5 latent variables connected in a 4-neighborhood. Running time of the dynamic algorithm

32 Part II SOCP for MRF

33 Aim Accurate MAP estimation of pairwise Markov random fields V1V1 V2V2 V3V3 V4V4 Label -1 Label 1 Labelling m = {1, -1, -1, 1} Random Variables V = {V 1,..,V 4 } Label Set L = {-1,1}

34 Aim Accurate MAP estimation of pairwise Markov random fields V1V1 V2V2 V3V3 V4V4 Label -1 Label 1 Cost(m) = 2

35 Aim Accurate MAP estimation of pairwise Markov random fields V1V1 V2V2 V3V3 V4V4 Label -1 Label 1 Cost(m) = 2 + 1

36 Aim Accurate MAP estimation of pairwise Markov random fields V1V1 V2V2 V3V3 V4V4 Label -1 Label 1 Cost(m) =

37 Aim Accurate MAP estimation of pairwise Markov random fields V1V1 V2V2 V3V3 V4V4 Label -1 Label 1 Cost(m) =

38 Aim Accurate MAP estimation of pairwise Markov random fields V1V1 V2V2 V3V3 V4V4 Label -1 Label 1 Cost(m) =

39 Aim Accurate MAP estimation of pairwise Markov random fields V1V1 V2V2 V3V3 V4V4 Label -1 Label 1 Cost(m) =

40 Aim Accurate MAP estimation of pairwise Markov random fields V1V1 V2V2 V3V3 V4V4 Label -1 Label 1 Cost(m) =

41 Aim Accurate MAP estimation of pairwise Markov random fields V1V1 V2V2 V3V3 V4V4 Label -1 Label 1 Cost(m) = = 13 Minimum Cost Labelling = MAP estimate Pr(m) exp(-Cost(m))

42 Aim Accurate MAP estimation of pairwise Markov random fields V1V1 V2V2 V3V3 V4V4 Label -1 Label 1 Objectives Applicable to all types of neighbourhood relationships Applicable to all forms of pairwise costs Guaranteed to converge (Convex approximation)

43 Motivation Subgraph Matching - Torr , Schellewald et al G1G1 G2G2 Unary costs are uniform V2V2 V3V3 V1V1 MRF A B C D A B C D A B C D

44 Motivation Subgraph Matching - Torr , Schellewald et al G1G1 G2G2 | d(m i,m j ) - d(V i,V j ) | < 1 2 YESNO Potts Model Pairwise Costs

45 Motivation V2V2 V3V3 V1V1 MRF A B C D A B C D A B C D Subgraph Matching - Torr , Schellewald et al

46 Motivation V2V2 V3V3 V1V1 MRF A B C D A B C D A B C D Subgraph Matching - Torr , Schellewald et al

47 Motivation Matching Pictorial Structures - Felzenszwalb et al Part likelihoodSpatial Prior Outline Texture Image P1P1 P3P3 P2P2 (x,y,, ) MRF

48 Motivation Image P1P1 P3P3 P2P2 (x,y,, ) MRF Unary potentials are negative log likelihoods Valid pairwise configuration Potts Model Matching Pictorial Structures - Felzenszwalb et al YESNO

49 Motivation P1P1 P3P3 P2P2 (x,y,, ) Pr(Cow)Image Unary potentials are negative log likelihoods Matching Pictorial Structures - Felzenszwalb et al Valid pairwise configuration Potts Model 1 2 YESNO

50 Outline Integer Programming Formulation Previous Work Our Approach –Second Order Cone Programming (SOCP) –SOCP Relaxation –Robust Truncated Model Applications –Subgraph Matching –Pictorial Structures

51 Integer Programming Formulation V1V1 V2V2 Label -1 Label 1 Unary Cost Unary Cost Vector u = [ 5 2 ; 2 4 ] T Labelling m = {1, -1} Label vector x = [ -1 V 1 =-1 1 V 1 = 1 ; 1 -1 ] T Recall that the aim is to find the optimal x

52 Integer Programming Formulation V1V1 V2V2 Label -1 Label 1 Unary Cost Unary Cost Vector u = [ 5 2 ; 2 4 ] T Labelling m = {1, -1} Label vector x = [ -11; 1 -1 ] T Sum of Unary Costs = 1 2 i u i (1 + x i )

53 Integer Programming Formulation V1V1 V2V2 Label -1 Label 1 Pairwise Cost Labelling m = {1, -1} 0 Cost of V 1 = -1 and V 1 = Cost of V 1 = -1 and V 2 = -1 3 Cost of V 1 = 0-1and V 2 = Pairwise Cost Matrix P

54 Integer Programming Formulation V1V1 V2V2 Label -1 Label 1 Pairwise Cost Labelling m = {1, -1} Pairwise Cost Matrix P Sum of Pairwise Costs 1 4 ij P ij (1 + x i )(1+x j )

55 Integer Programming Formulation V1V1 V2V2 Label 0 Label 1 Pairwise Cost Labelling m = {1, 0} Pairwise Cost Matrix P Sum of Pairwise Costs 1 4 ij P ij (1 + x i +x j + x i x j ) 1 4 ij P ij (1 + x i + x j + X ij )= X = x x T X ij = x i x j

56 Integer Programming Formulation Constraints Each variable should be assigned a unique label x i = 2 - |L| i V a Marginalization constraint X ij = (2 - |L|) x i j V b

57 Integer Programming Formulation Chekuri et al., SODA 2001 x* = argmin 1 2 u i (1 + x i ) P ij (1 + x i + x j + X ij ) x i = 2 - |L| i V a X ij = (2 - |L|) x i j V b x i {-1,1} X = x x T Convex Non-Convex

58 Key Point In modern optimization the issue is not linear vs non linear but convex vs nonconvex We want to find a convex and good relaxation of the integer program.

59 Outline Integer Programming Formulation Previous Work Our Approach –Second Order Cone Programming (SOCP) –SOCP Relaxation –Robust Truncated Model Applications –Subgraph Matching –Pictorial Structures

60 Linear Programming Formulation x* = argmin 1 2 u i (1 + x i ) P ij (1 + x i + x j + X ij ) x i = 2 - |L| i V a X ij = (2 - |L|) x i j V b x i {-1,1} X = x x T Chekuri et al., SODA 2001 Retain Convex Part Relax Non-convex Constraint

61 Linear Programming Formulation x* = argmin 1 2 u i (1 + x i ) P ij (1 + x i + x j + X ij ) x i = 2 - |L| i V a X ij = (2 - |L|) x i j V b x i [-1,1] X = x x T Chekuri et al., SODA 2001 Retain Convex Part Relax Non-convex Constraint

62 Linear Programming Formulation x* = argmin 1 2 u i (1 + x i ) P ij (1 + x i + x j + X ij ) x i = 2 - |L| i V a X ij = (2 - |L|) x i j V b x i [-1,1] Chekuri et al., SODA 2001 Retain Convex Part X becomes a variable to be optimized

63 Feasible Region (IP) x {-1,1}, X = x 2 Linear Programming Formulation Feasible Region for X.

64 Feasible Region (IP) Feasible Region (Relaxation 1) x {-1,1}, X = x 2 x [-1,1], X = x 2 Linear Programming Formulation Feasible Region for X.

65 Feasible Region (IP) Feasible Region (Relaxation 1) Feasible Region (Relaxation 2) x {-1,1}, X = x 2 x [-1,1], X = x 2 x [-1,1] Linear Programming Formulation Feasible Region for X.

66 Linear Programming Formulation Bounded algorithms proposed by Chekuri et al, SODA expansion - Komodakis and Tziritas, ICCV 2005 TRW - Wainwright et al., NIPS 2002 TRW-S - Kolmogorov, AISTATS 2005 Efficient because it uses Linear Programming Not accurate

67 Semidefinite Programming Formulation x* = argmin 1 2 u i (1 + x i ) P ij (1 + x i + x j + X ij ) x i = 2 - |L| i V a X ij = (2 - |L|) x i j V b x i {-1,1} X = x x T Lovasz and Schrijver, SIAM Optimization, 1990 Retain Convex Part Relax Non-convex Constraint

68 x* = argmin 1 2 u i (1 + x i ) P ij (1 + x i + x j + X ij ) x i = 2 - |L| i V a X ij = (2 - |L|) x i j V b x i [-1,1] X = x x T Semidefinite Programming Formulation Retain Convex Part Relax Non-convex Constraint Lovasz and Schrijver, SIAM Optimization, 1990

69 Semidefinite Programming Formulation x1x1 x2x2 xnxn x1x1 x2x2... xnxn 1xTxT x X = Rank = 1 X ii = 1 Positive Semidefinite Convex Non-Convex

70 Semidefinite Programming Formulation x1x1 x2x2 xnxn x1x1 x2x2... xnxn 1xTxT x X = X ii = 1 Positive Semidefinite Convex

71 Schurs Complement AB BTBT C = I0 B T A -1 I A0 0 C - B T A -1 B IA -1 B 0 I 0 A 0 C -B T A -1 B 0

72 Semidefinite Programming Formulation X - xx T 0 1xTxT x X = 10 x I 10 0 X - xx T IxTxT 0 1 Schurs Complement

73 x* = argmin 1 2 u i (1 + x i ) P ij (1 + x i + x j + X ij ) x i = 2 - |L| i V a X ij = (2 - |L|) x i j V b x i [-1,1] X = x x T Semidefinite Programming Formulation Relax Non-convex Constraint Retain Convex Part Lovasz and Schrijver, SIAM Optimization, 1990

74 x* = argmin 1 2 u i (1 + x i ) P ij (1 + x i + x j + X ij ) x i = 2 - |L| i V a X ij = (2 - |L|) x i j V b x i [-1,1] Semidefinite Programming Formulation X ii = 1 X - xx T 0 Retain Convex Part Lovasz and Schrijver, SIAM Optimization, 1990

75 Feasible Region (IP) x {-1,1}, X = x 2 Semidefinite Programming Formulation Feasible Region for X.

76 Feasible Region (IP) Feasible Region (Relaxation 1) x {-1,1}, X = x 2 x [-1,1], X = x 2 Semidefinite Programming Formulation Feasible Region for X.

77 Feasible Region (IP) Feasible Region (Relaxation 1) Feasible Region (Relaxation 2) x {-1,1}, X = x 2 x [-1,1], X = x 2 x [-1,1], X x 2 Semidefinite Programming Formulation Feasible Region for X.

78 Semidefinite Programming Formulation Formulated by Lovasz and Schrijver, 1990 Finds a full X matrix Max-cut - Goemans and Williamson, JACM 1995 Max-k-cut - de Klerk et al, 2000 Torr AI Stats for labeling problem (2003 TR 2002) Accurate, but not efficient as Semidefinite Programming algorithms slow

79 Previous Work - Overview LPSDP Examples TRW-S, -expansion Max-k-Cut Torr 2003 AccuracyLowHigh EfficiencyHighLow Is there a Middle Path ???

80 Outline Integer Programming Formulation Previous Work Our Approach –Second Order Cone Programming (SOCP) –SOCP Relaxation –Robust Truncated Model Applications –Subgraph Matching –Pictorial Structures

81 Second Order Cone Programming Second Order Cone || v || t OR || v || 2 st x 2 + y 2 z 2

82 Minimize f T x Subject to || A i x+ b i || <= c i T x + d i i = 1, …, L Linear Objective Function Affine mapping of Second Order Cone (SOC) Constraints are SOC of n i dimensions Feasible regions are intersections of conic regions Second Order Cone Programming

83 || v || t tIv vTvT t 0 LP SOCP SDP = 10 vTvT I tI0 0 t 2 - v T v Iv 0 1 Schurs Complement

84 Outline Integer Programming Formulation Previous Work Our Approach –Second Order Cone Programming (SOCP) –SOCP Relaxation –Robust Truncated Model Applications –Subgraph Matching –Pictorial Structures

85 First quick definition: Matrix Dot Product AB = ij A ij B ij A 11 A 12 A 21 A 22 B 11 B 12 B 21 B 22 = A 11 B 11 + A 12 B 12 + A 21 B 21 + A 22 B 22

86 SDP Relaxation x* = argmin 1 2 u i (1 + x i ) P ij (1 + x i + x j + X ij ) x i = 2 - |L| i V a X ij = (2 - |L|) x i j V b x i [-1,1] X ii = 1 X - xx T 0 We will derive SOCP relaxation from the SDP relaxation Further Relaxation

87 1-D Example X - xx T 0 X - x 2 0 For two semidefinite matrices, the dot product is non-negative A A 0 x 2 X SOC of the form || v || 2 st, s is a scalar constant.

88 Feasible Region (IP) Feasible Region (Relaxation 1) Feasible Region (Relaxation 2) x {-1,1}, X = x 2 x [-1,1], X = x 2 x [-1,1], X x 2 SOCP Relaxation For 1D: Same as the SDP formulation Feasible Region for X.

89 2-D Example X 11 X 12 X 21 X 22 1X 12 1 = X = x1x1x1x1 x1x2x1x2 x2x1x2x1 x2x2x2x2 xx T = x12x12 x1x2x1x2 x1x2x1x2 = x22x22

90 2-D Example (X - xx T ) 1 - x 1 2 X 12 -x 1 x x 2 2 x x 1 1 C 1. 0 C 1 0

91 2-D Example (X - xx T ) 1 - x 1 2 X 12 -x 1 x 2 C X 12 -x 1 x x 2 2 x LP Relaxation -1 x 2 1 C 2 0

92 2-D Example (X - xx T ) 1 - x 1 2 X 12 -x 1 x 2 C X 12 -x 1 x x 2 2 (x 1 + x 2 ) X 12 SOC of the form || v || 2 st C 3 0

93 2-D Example (X - xx T ) 1 - x 1 2 X 12 -x 1 x 2 C X 12 -x 1 x x 2 2 (x 1 - x 2 ) X 12 SOC of the form || v || 2 st C 4 0

94 General form of SOC constraints Consider a matrix C 1 = UU T 0 (X - xx T ) ||U T x || 2 X. C 1 C 1. 0 Continue for C 2, C 3, …, C n SOC of the form || v || 2 st Kim and Kojima, 2000

95 SOCP Relaxation How many constraints for SOCP = SDP ? Infinite. For all C 0 We specify constraints similar to the 2-D example

96 SOCP Relaxation Muramatsu and Suzuki, Constraints hold for the above semidefinite matrices

97 SOCP Relaxation Muramatsu and Suzuki, a + b + c+ d a 0 b 0 c 0 d 0 Constraints hold for the linear combination

98 SOCP Relaxation Muramatsu and Suzuki, 2001 a+c+dc-d b+c+d a 0 b 0 c 0 d 0 Includes all semidefinite matrices where Diagonal elements Off-diagonal elements

99 SOCP Relaxation - A x* = argmin 1 2 u i (1 + x i ) P ij (1 + x i + x j + X ij ) x i = 2 - |L| i V a X ij = (2 - |L|) x i j V b x i [-1,1] X ii = 1 X - xx T 0

100 SOCP Relaxation - A x* = argmin 1 2 u i (1 + x i ) P ij (1 + x i + x j + X ij ) x i = 2 - |L| i V a X ij = (2 - |L|) x i j V b x i [-1,1] (x i + x j ) X ij (x i - x j ) X ij Specified only when P ij 0 i.e. sparse!!

101 Triangular Inequality At least two of x i, x j and x k have the same sign At least one of X ij, X jk, X ik is equal to one X ij + X jk + X ik -1 X ij - X jk - X ik -1 -X ij - X jk + X ik -1 -X ij + X jk - X ik -1 SOCP-B = SOCP-A + Triangular Inequalities

102 Outline Integer Programming Formulation Previous Work Our Approach –Second Order Cone Programming (SOCP) –SOCP Relaxation –Robust Truncated Model Applications –Subgraph Matching –Pictorial Structures

103 Robust Truncated Model Pairwise cost of incompatible labels is truncated Potts ModelTruncated Linear Model Truncated Quadratic Model Robust to noise Widely used in Computer Vision - Segmentation, Stereo

104 Robust Truncated Model Pairwise Cost Matrix can be made sparse P = [ ] Q = [ ] Reparameterization Sparse Q matrix Fewer constraints

105 Compatibility Constraint Q(m a, m b ) < 0 for variables V a and V b Relaxation Q ij (1 + x i + x j + X ij ) < 0 SOCP-C = SOCP-B + Compatibility Constraints

106 SOCP Relaxation More accurate than LP More efficient than SDP Time complexity - O( |V| 3 |L| 3 ) Same as LP Approximate algorithms exist for LP relaxation We use |V| 10 and |L| 200

107 Outline Integer Programming Formulation Previous Work Our Approach –Second Order Cone Programming (SOCP) –SOCP Relaxation –Robust Truncated Model Applications –Subgraph Matching –Pictorial Structures

108 Subgraph Matching Subgraph Matching - Torr , Schellewald et al G1G1 G2G2 Unary costs are uniform V2V2 V3V3 V1V1 MRF A B C D A B C D A B C D Pairwise costs form a Potts model

109 Subgraph Matching 1000 pairs of graphs G 1 and G 2 #vertices in G 2 - between 20 and 30 #vertices in G * #vertices in G 2 5% noise to the position of vertices NP-hard problem

110 Subgraph Matching Method Time (sec) Accuracy (%) LP LBP GBP SDP-A SOCP-A SOCP-B SOCP-C

111 Outline Integer Programming Formulation Previous Work Our Approach –Second Order Cone Programming (SOCP) –SOCP Relaxation –Robust Truncated Model Applications –Subgraph Matching –Pictorial Structures

112 Pictorial Structures Image P1P1 P3P3 P2P2 (x,y,, ) MRF Matching Pictorial Structures - Felzenszwalb et al Outline Texture

113 Pictorial Structures Image P1P1 P3P3 P2P2 (x,y,, ) MRF Unary costs are negative log likelihoods Pairwise costs form a Potts model | V | = 10| L | = 200

114 Pictorial Structures ROC Curves for 450 +ve and ve images

115 Pictorial Structures ROC Curves for 450 +ve and ve images

116 Conclusions We presented an SOCP relaxation to solve MRF More efficient than SDP More accurate than LP, LBP, GBP #variables can be reduced for Robust Truncated Model Provides excellent results for subgraph matching and pictorial structures

117 Future Work Quality of solution –Additive bounds exist –Multiplicative bounds for special cases ?? –What are good Cs. Message passing algorithm ?? –Similar to TRW-S or -expansion –To handle image sized MRF


Download ppt "Solving Markov Random Fields using Dynamic Graph Cuts & Second Order Cone Programming Relaxations M. Pawan Kumar, Pushmeet Kohli Philip Torr."

Similar presentations


Ads by Google