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Solving Markov Random Fields using Second Order Cone Programming Relaxations M. Pawan Kumar Philip Torr Andrew Zisserman

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Aim Accurate MAP estimation of pairwise Markov random fields 2 5 4 2 6 3 3 7 0 1 1 0 0 2 3 1 1 41 0 V1V1 V2V2 V3V3 V4V4 Label 0 Label 1 Labelling m = {1, 0, 0, 1} Random Variables V = {V 1,..,V 4 } Label Set L = {0,1}

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Aim Accurate MAP estimation of pairwise Markov random fields 2 5 4 2 6 3 3 7 0 1 1 0 0 2 3 1 1 41 0 V1V1 V2V2 V3V3 V4V4 Label 0 Label 1 Cost(m) = 2

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Aim Accurate MAP estimation of pairwise Markov random fields 2 5 4 2 6 3 3 7 0 1 1 0 0 2 3 1 1 41 0 V1V1 V2V2 V3V3 V4V4 Label 0 Label 1 Cost(m) = 2 + 1

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Aim Accurate MAP estimation of pairwise Markov random fields 2 5 4 2 6 3 3 7 0 1 1 0 0 2 3 1 1 41 0 V1V1 V2V2 V3V3 V4V4 Label 0 Label 1 Cost(m) = 2 + 1 + 2

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Aim Accurate MAP estimation of pairwise Markov random fields 2 5 4 2 6 3 3 7 0 1 1 0 0 2 3 1 1 41 0 V1V1 V2V2 V3V3 V4V4 Label 0 Label 1 Cost(m) = 2 + 1 + 2 + 1

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Aim Accurate MAP estimation of pairwise Markov random fields 2 5 4 2 6 3 3 7 0 1 1 0 0 2 3 1 1 41 0 V1V1 V2V2 V3V3 V4V4 Label 0 Label 1 Cost(m) = 2 + 1 + 2 + 1 + 3

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Aim Accurate MAP estimation of pairwise Markov random fields 2 5 4 2 6 3 3 7 0 1 1 0 0 2 3 1 1 41 0 V1V1 V2V2 V3V3 V4V4 Label 0 Label 1 Cost(m) = 2 + 1 + 2 + 1 + 3 + 1

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Aim Accurate MAP estimation of pairwise Markov random fields 2 5 4 2 6 3 3 7 0 1 1 0 0 2 3 1 1 41 0 V1V1 V2V2 V3V3 V4V4 Label 0 Label 1 Cost(m) = 2 + 1 + 2 + 1 + 3 + 1 + 3

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Aim Accurate MAP estimation of pairwise Markov random fields 2 5 4 2 6 3 3 7 0 1 1 0 0 2 3 1 1 41 0 V1V1 V2V2 V3V3 V4V4 Label 0 Label 1 Cost(m) = 2 + 1 + 2 + 1 + 3 + 1 + 3 = 13 Minimum Cost Labelling = MAP estimate Pr(m) exp(-Cost(m))

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Aim Accurate MAP estimation of pairwise Markov random fields 2 5 4 2 6 3 3 7 0 1 1 0 0 2 3 1 1 41 0 V1V1 V2V2 V3V3 V4V4 Label 0 Label 1 Objectives Applicable for all neighbourhood relationships Applicable for all forms of pairwise costs Guaranteed to converge

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Motivation Subgraph Matching - Torr - 2003, Schellewald et al - 2005 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 - 2003, Schellewald et al - 2005 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 - 2003, Schellewald et al - 2005

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Motivation V2V2 V3V3 V1V1 MRF A B C D A B C D A B C D Subgraph Matching - Torr - 2003, Schellewald et al - 2005

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Motivation Matching Pictorial Structures - Felzenszwalb et al - 2001 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 - 2001 1 2 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 - 2001 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 2 5 4 2 0 1 3 0 V1V1 V2V2 Label 0 Label 1 Unary Cost Unary Cost Vector u = [ 5 Cost of V 1 = 0 2 Cost of V 1 = 1 ; 2 4 ] Labelling m = {1, 0}

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Integer Programming Formulation 2 5 4 2 0 1 3 0 V1V1 V2V2 Label 0 Label 1 Unary Cost Unary Cost Vector u = [ 5 2 ; 2 4 ] T Labelling m = {1, 0} Label vector x = [ -1 V 1 0 1 V 1 = 1 ; 1 -1 ] T Recall that the aim is to find the optimal x

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Integer Programming Formulation 2 5 4 2 0 1 3 0 V1V1 V2V2 Label 0 Label 1 Unary Cost Unary Cost Vector u = [ 5 2 ; 2 4 ] T Labelling m = {1, 0} 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 2 5 4 2 0 1 3 0 V1V1 V2V2 Label 0 Label 1 Pairwise Cost Labelling m = {1, 0} 0 Cost of V 1 = 0 and V 1 = 0 0 00 0 Cost of V 1 = 0 and V 2 = 0 3 Cost of V 1 = 0 and V 2 = 1 10 00 00 10 30 Pairwise Cost Matrix P

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Integer Programming Formulation 2 5 4 2 0 1 3 0 V1V1 V2V2 Label 0 Label 1 Pairwise Cost Labelling m = {1, 0} Pairwise Cost Matrix P 00 00 0 3 10 00 00 10 30 Sum of Pairwise Costs 1 4 ij P ij (1 + x i )(1+x j )

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Integer Programming Formulation 2 5 4 2 0 1 3 0 V1V1 V2V2 Label 0 Label 1 Pairwise Cost Labelling m = {1, 0} Pairwise Cost Matrix P 00 00 0 3 10 00 00 10 30 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 ) + 1 4 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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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 ) + 1 4 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 ) + 1 4 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 ) + 1 4 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

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

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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

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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

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Bounded algorithms proposed by Chekuri et al, SODA 2001 -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 ) + 1 4 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 ) + 1 4 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 1... 1x1x1 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 1... 1x1x1 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 ) + 1 4 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 ) + 1 4 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

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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

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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

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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 Accurate Not efficient because of Semidefinite Programming

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Previous Work - Overview LPSDP Examples TRW-S, -expansion Max-k-Cut 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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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 ) + 1 4 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 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

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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 Same as the SDP formulation

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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 2. 0 10 00 1 - x 2 2 x 1 2 1 -1 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 2. 0. 0 00 01 X 12 -x 1 x 2 1 - x 2 2 x 2 2 1 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 3. 0. 0 11 11 X 12 -x 1 x 2 1 - x 2 2 (x 1 + x 2 ) 2 2 + 2X 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 4. 0. 0 1 1 X 12 -x 1 x 2 1 - x 2 2 (x 1 - x 2 ) 2 2 - 2X 12 SOC of the form || v || 2 st C 4 0

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SOCP Relaxation 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, 2001 10 00 00 01 11 11 1 1 Constraints hold for the above semidefinite matrices

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SOCP Relaxation Muramatsu and Suzuki, 2001 10 00 00 01 11 11 1 1 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 ) + 1 4 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 ) + 1 4 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 ) 2 2 + 2X ij (x i - x j ) 2 2 - 2X ij Specified only when P ij 0

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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 = [0.5 0.5 0.3 0.3 0.5] Q = [0 0 -0.2 -0.2 0] 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 - 2003, Schellewald et al - 2005 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 1 - 0.25 * #vertices in G 2 5% noise to the position of vertices NP-hard problem

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Subgraph Matching Method Time (sec)Accuracy (%) LP0.856.64 SDP-A35.093.11 SOCP-A3.092.01 SOCP-B4.594.79 SOCP-C4.896.18

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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 - 2001 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 LBP GBP SOCP

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Pictorial Structures LBP GBP SOCP

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Pictorial Structures LBP GBP SOCP

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Pictorial Structures LBP GBP SOCP LBP and GBP do not converge

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Pictorial Structures ROC Curves for 450 +ve and 2400 -ve images

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Pictorial Structures ROC Curves for 450 +ve and 2400 -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 ?? Message passing algorithm ?? –Similar to TRW-S or -expansion –To handle image sized MRF

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