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Convex Mathematical Programs for Relational Matching of Object Views

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Convex Mathematical Programs for Relational Matching of Object Views Convex Mathematical Programs for Relational Matching of Object Views Inauguraldissertation zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften der UniversitÄat Mannheim vorgelegt von Dipl.-Phys. Christian Schellewald aus Hohenlimburg Mannheim, 2004 Dekan: Professor Dr. Matthias Krause, UniversitÄat Mannheim Referent: Professor Dr. Christoph SchnÄorr, UniversitÄat Mannheim Korreferent: Professor Dr. Joachim Denzler, UniversitÄat Jena Tag der mundlicÄ hen Prufung:Ä 14. Juli 2005 Contents 1 Introduction 1 1.1 Motivation . 1 1.2 Overview . 2 1.2.1 Graph Matching . 2 1.2.2 Exact and Inexact Graph Matching . 3 1.2.3 What this thesis is about . 3 1.2.4 What this thesis is not about . 4 1.3 Convex Optimization . 4 1.3.1 Semide¯nite Programming . 5 1.4 Contribution . 5 1.4.1 QAP Convex Relaxation used for Graph Matching . 5 1.4.2 Subgraph Matching by Semide¯nite Programming . 6 1.5 Outline . 6 1.6 Related Work . 7 1.6.1 Basic Graph Matching Approaches . 7 1.6.2 More General Graph Matching Approaches . 11 2 Preliminaries 15 2.1 Basic Graph Theory . 15 2.2 Graph Matching . 16 2.2.1 Bipartite Matching . 16 2.2.2 Representation of Matchings . 16 2.2.3 Classi¯cation of Graph Matching Problems . 18 2.3 Complexity . 19 2.3.1 Complexity Classes . 20 2.3.2 Complexity of Discussed Graph Matching Approaches . 21 2.4 Terminology and Mathematical Preliminaries . 21 2.4.1 Notation . 21 2.4.2 Operator De¯nitions . 22 2.5 Basic Convexity Concepts . 23 2.5.1 Convex functions and sets . 24 3 Combinatorial Relaxation and Convex Optimization 27 3.1 Relaxations and Lower Bounds . 27 3.1.1 Interpretation of the Relaxation as Approximation . 28 3.1.2 Lagrangian Relaxation . 29 iii iv Contents 3.1.3 Convex Relaxations . 30 3.2 Optimality Conditions . 30 3.2.1 Lagrangian Duality . 31 3.2.2 Karush-Kuhn-Tucker Conditions . 31 3.3 Convex Optimization Problems . 32 3.3.1 General Convex optimization problems . 32 3.3.2 Convex optimization problems in this thesis . 33 3.3.3 Linear Programming . 33 3.3.4 Quadratic Programming . 34 3.3.5 Semide¯nite Programming . 35 3.4 Solving Convex Optimization Problems . 36 3.4.1 Interior Point Methods . 36 3.4.2 Solver for Convex Optimization Problems . 36 3.5 Computing Integer Solutions for Assignments . 37 4 Weighted Graph Matching 41 4.1 Problem Statement . 41 4.1.1 The Quadratic Assignment Problem . 42 4.1.2 Graph Matching as QAP . 42 4.2 Relaxations and Lower Bounds . 45 4.2.1 Permutation Matrices and Relaxations . 45 4.2.2 Orthogonal Relaxation . 46 4.2.3 Projected Eigenvalue Bound . 47 4.2.4 Convex Relaxation . 48 4.2.5 Combinatorial Solutions . 50 4.2.6 The 2opt Post-Processing Heuristics . 51 4.3 Other Approaches . 51 4.3.1 The Approach by Umeyama . 51 4.3.2 Graduated Assignment . 52 4.3.3 SDP Relaxation . 53 4.4 Convex Relaxation: An Illustrative Numerical Example . 54 4.4.1 A Small Graph Matching Problem . 55 4.4.2 Relaxations and Bounds . 55 4.4.3 Visualization . 57 4.5 Implementation Details . 58 4.6 Experiments . 59 4.6.1 QAPLIB Benchmark Experiments . 60 4.6.2 Random Ground-Truth Experiments . 61 4.6.3 Real World Example . 64 4.7 Discussion . 66 4.7.1 Invariants . 66 4.7.2 Limitations of the QAP Graph Matching . 67 4.8 Towards Subgraph Matching . 68 4.8.1 A Simple Extension Approach . 68 4.8.2 A Promising Subgraph Matching Approach . 69 4.8.3 Comparison of the two Approaches . 71 4.8.4 Relaxation Attempt . 72 Contents v 4.8.5 Projection Approach . 74 5 Subgraph Matching 75 5.1 Problem Statement . 75 5.2 Notation and Bipartite Matching . 76 5.2.1 Matching in Bipartite Graphs . 76 5.2.2 Linear Problem Formulation of the Bipartite Matching . 77 5.3 Combinatorial Subgraph Matching Approach . 77 5.3.1 Quadratic Integer Program . 78 5.3.2 Regularization Parameter . 81 5.3.3 Hidden Parameters . 82 5.3.4 Classi¯cation of the Approach . 82 5.4 Semide¯nite Program Formulation . 82 5.4.1 Objective Function . 83 5.4.2 Equality Constraints . 84 5.4.3 Inequality Constraints . 87 5.4.4 Post-Processing to obtain an Integer Solution . 90 5.5 Illustrative Example . 91 5.5.1 The Subgraph Matching Example . 91 5.5.2 Linear Bipartite Matching Approach . 92 5.5.3 SDP Subgraph Matching Approach . 92 5.5.4 Problem Nature . 94 5.6 Subgraph Matching Experiments . 95 5.6.1 Creation of the Problem Instances . 95 5.6.2 Problem Nature . 97 5.6.3 Influence of the Regularization Parameter . 98 5.6.4 Statistical Performance Investigation . 103 5.7 Random Subgraph Matching Experiments . 108 5.7.1 Creation of the Problem Instances . 109 5.7.2 Problem Nature . 110 5.7.3 Performance Investigation . 110 5.8 Real World Examples . 115 5.8.1 Creation of the Real World Examples . 115 5.8.2 Results . 116 5.9 Discussion . 124 5.9.1 Computational E®ort . 125 5.9.2 Reducing the Computational E®ort . 127 5.9.3 Structural Perturbations . 134 5.9.4 Bimodal Experiments . 136 5.9.5 A New Bound for Subgraph Non-Isomorphism . 138 6 Conclusion 141 6.1 Summary . 141 6.1.1 Weighted Graph Matching . 142 6.1.2 Subgraph Matching . 144 6.2 Future Work . 147 vi Contents A Quadratic Assignment Supplements 151 A.1 The Dual of the relaxed homogeneous QAP . 151 A.1.1 QAP Dual as linear program . 152 A.1.2 Non unique solutions for the EVB . 155 A.2 QAP-Bounds . 157 B Subgraph Matching Supplements 159 B.1 Example Data . 159 B.1.1 Illustrative Example Data . 159 B.1.2 Graph Data for the Parameter Dependency Example . 160 B.2 Earth Movers Distance . 161 Bibliography 164 Abstract vii Abstract Automatic recognition of objects in images is a di±cult and challenging task in computer vision which has been tackled in many di®erent ways. Based on the powerful and widely used concept to represent objects and scenes as relational structures, the problem of graph matching, i.e. to ¯nd correspondences between two graphs is a part of the object recognition problem. Belonging to the ¯eld of combinatorial optimization graph matching is considered to be one of the most complex problems in computer vision: It is known to be NP-complete in the general case. In this thesis, two novel approaches to the graph matching problem are proposed and investigated. They are based on recent progress in the mathematical liter- ature on convex programming. Starting out from describing the desired match- ings by suitable objective functions in terms of binary variables, relaxations of combinatorial constraints and an adequate adaption of the objective function lead to continuous convex optimization problems which can be solved without parameter tuning and in polynomial time. A subsequent post-processing step results in feasible, sub-optimal combinatorial solutions to the original decision problem. In the ¯rst part of this thesis, the connection between speci¯c graph-matching problems and the quadratic assignment problem is explored. In this case, the convex relaxation leads to a convex quadratic program , which is combined with a linear program for post-processing. Conditions under which the quadratic assignment representation is adequate from the computer vision point of view are investigated, along with attempts to relax these conditions by modifying the approach accordingly. The second part of this work focuses directly on the matching of subgraphs { representing a model { to a considerably larger scene graph. A bipartite matching is extended with a quadratic regularization term to take into ac- count relations within each set of vertices. Based on this convex relaxation, post-processing and the application to computer vision are investigated and discussed. Numerical experiments reveal both the power and the limitations of the ap- proach. For problems of sizes which occur in applications the approach is quite reasonable and often the combinatorial optimal solution is found. For larger instances the intrinsic combinatorial nature of the problem comes out and leads to sub-optimal solutions which, however, are still good. viii Zusammenfassung Zusammenfassung Die automatische Erkennung von Objekten in Bildern ist eine der grÄo¼ten Herausforderungen in der Bildverarbeitung. Werden die Objekte und Szenar- ien durch Graphen reprÄasentiert, ist ein Teilproblem der Objekterkennung die gewunscÄ hte Zuordnung der Knoten zweier Graphen zu ¯nden. Dabei soll mÄoglichst die Graphstruktur und evtl. zusÄatzlich vorhandene Information berucÄ ksichtigt werden. Die Bestimmung der besten Korrespondenzen der Graphknoten, auch Graph Matching genannt, ist furÄ allgemeine Graphen ein NP-Hartes kombina- torisches Problem und gehÄort damit zu den schwierigsten aller Probleme in der Bildverarbeitung. In dieser Arbeit fuhrenÄ wir zwei neue AnsÄatze ein, um Graph Matching und Subgraph Matching Probleme approximativ zu lÄosen. Dazu nutzen wir neuere Methoden und Erkentnisse der konvexen Optimierung. Die Graph Matching Probleme kÄonnen als 0=1-Integer Optimierungsproblem formuliert werden und lassen sich mit Hilfe einer geeigneten Relaxierung in ein kontinuierliches und konvexes Problem transformieren. Ein Vorteil konvexer Optimierungsprobleme liegt in der Tatsache, dass sie ohne zusÄatzliche Parameteroptimierung e±zient mit Standardverfahren gelÄost werden kÄonnen. Ein Nachverarbeitungsschritt sorgt dafur,Ä dass mit Hilfe der Approximierten LÄosung eine furÄ das Original- problem passende und gute 0=1-Integer LÄosung gefunden wird. In dem ersten Teil.
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