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

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

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Example: Video Segmentation

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

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

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

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

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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

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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

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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

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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.

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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.

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

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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

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

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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

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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

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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

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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)

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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

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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!

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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

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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!

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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.

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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

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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

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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

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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

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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)

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

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

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Part II SOCP for MRF

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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}

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

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

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

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

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

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

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

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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))

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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)

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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

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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

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

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

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

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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

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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

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

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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

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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 )

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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

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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 )

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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

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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

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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

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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.

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

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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

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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

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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

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

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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.

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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.

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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

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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

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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

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

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

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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

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

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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

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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

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

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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.

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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.

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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

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

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

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

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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

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

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

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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

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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

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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.

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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.

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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

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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

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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

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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

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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

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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

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

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

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

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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

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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

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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!!

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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

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

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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

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

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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

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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

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

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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

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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

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

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

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

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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

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Pictorial Structures ROC Curves for 450 +ve and ve images

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Pictorial Structures ROC Curves for 450 +ve and ve images

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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

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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

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