Table Of Content

HANDBOOK OF MATHEMATICAL MODELS IN COMPUTER VISION HANDBOOK OF MATHEMATICAL MODELS IN COMPUTER VISION Edited by Nikos Paragios Ecole Nationale des Fonts et Chaussees Yunmei Chen University of Florida Olivier Faugeras INRIA Springer Library of Congress Cataloging-in-Publication Data A CLP. Catalogue record for this book is available from the Library of Congress. Handbook of Mathematical Models in Computer Vision, Edited by Nikos Paragios, Yunmei Chen and Olivier Faugeras p.cm. ISBN-10: (HB) 0-387-26371-3 ISBN-13: (HB) 978-0387-26371-7 ISBN-10: (eBook) 0-387-28831-7 ISBN-13: (eBook) 978-0387-28831-4 Printed on acid-free paper. Copyright © 2006 by Springer Science+Business Media, Inc. All rights reserved. This work may not be translated or copied m whole or in part without the written permission of the publisher [Springer Science+Business Media, Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. 9 8 7 6 5 4 3 2 1 SPIN 11055662 (HC) / 11552987 (eBook) springeronline.com Contents Preface xix List of Contributors xxiii I Image Reconstruction 1 1 Diffusion Filters and Wavelets: What Can They Learn from Each Other? 3 J. Weickert, G. Steidl, P. Mrazek, M. Welk, and T. Brox 1.1 Introduction 3 1.2 Basic Methods 4 1.2.1 Wavelet Shrinkage 4 1.2.2 Nonlinear Diffusion Filtering 5 1.3 Relations for Space-Discrete Diffusion 6 1.3.1 Equivalence for Two-Pixel Signals 6 1.3.2 A Wavelet-Inspired Scheme for TV Diffusion of Signals 7 1.3.3 Generalisations to Images 8 1.4 Relations for Fully Discrete Diffusion 9 1.4.1 Diffusion-Inspired Shrinkage Functions 9 1.4.2 Wavelet Shrinkage with Improved Rotation Invariance . 10 1.4.3 Diffusion-Inspired Wavelet Shrinkage of Colour Images 13 1.5 Wavelets with Higher Vanishing Moments 13 1.6 Summary 16 2 Total Variation Image Restoration: Overview and Recent Develop ments 17 T. Chan, S. Esedoglu, F. Park and A. Yip 2.1 Introduction 17 2.2 Properties and Extensions 19 2.2.1 BV Space and Basic Properties 19 2.2.2 Multi-channel TV 20 vi Contents 2.2.3 Scale 20 2.3 Caveats 21 2.4 Variants 22 2.4.1 Iterated Refinement 22 2.4.2 L^ Fitting 23 2.4.3 Anisotropic TV 24 2.4.4 //^'PRegularization and Inf Convolution 25 2.5 Further Applications to Image Reconstruction 26 2.5.1 Deconvolution 26 2.5.2 Inpainting 27 2.5.3 Texture and Multiscale Decompositions 28 2.6 Numerical Methods 29 2.6.1 Artificial Time Marching and Fixed Point Iteration . . 29 2.6.2 DuaHty-based Methods 30 3 PDE-Based Image and Surface Inpainting 33 M. Bertalmio, V. Caselles, G. Haro, and G. Sapiro 3.1 Introduction 33 3.2 Inpaintingby Propagation of Information 36 3.2.1 Image Inpainting 36 3.2.2 Navier-Stokes Inpainting 40 3.3 Variational Models for Filling-In 42 3.3.1 Elastica-based Reconstruction of Level Lines 43 3.3.2 Joint Interpolation of Vector Fields and Gray Levels . . 45 3.3.3 A Variant and Mathematical Results 48 3.3.4 Experimental Results 50 3.4 Surface Reconstruction: The Laplace and the Absolute Minimizing Lipschitz Extension Interpolation 52 3.4.1 Experimental Results 54 3.5 Dealing with texture 55 3.5.1 Texture Synthesis by Non-Parametric Sampling . . .. 56 3.5.2 Inpainting with Image Decomposition 56 3.5.3 Exemplar-based Inpainting 58 3.6 Other Approaches 58 3.6.1 Other PDE-based Models 58 3.6.2 Miscellaneous 59 3.7 Concluding Remarks 60 3.8 Appendix 60 3.9 Acknowledgments 61 II Boundary Extraction, Segmentation and Grouping 63 4 Levelings: Theory and Practice 65 Contents vii F. Meyer 4.1 Introduction 65 4.2 Binary connected operators 66 4.3 Flat grey-tone connected operators 67 4.3.1 Level by level construction 67 4.3.2 A morphological characterization 68 4.4 Extended connected operators 68 4.4.1 Construction of floodings, razings, flattenings and levelings 70 4.4.1.1 Construction of floodings, razings, flattenings and levelings 70 4.5 Levelings for image simplification 71 4.5.1 Varying (a,/?) 72 4.5.2 Varying the marker function h 73 4.5.3 Multiscale filtering 74 4.5.3.1 Construction of a hierarchy based on increasing floodings 74 4.5.3.2 Construction of a hierarchy based on quasi-flat zones 76 4.6 Conclusion 77 5 Graph Cuts in Vision and Graphics: Theories and Applications 79 Y. Boykov and O. Veksler 5.1 Introduction 79 5.2 Graph Cuts Basics 80 5.2.1 The Min-Cut and Max-Flow Problem 81 5.2.2 Algorithms for the Min-Cut and Max-Flow Problem . . 81 5.3 Graph Cuts for Binary Optimization 82 5.3.1 Example: Binary Image Restoration 82 5.3.2 General Case of Binary Energy Minimization 84 5.4 Graph Cuts as Hypersurfaces 84 5.4.1 Basic idea 85 5.4.2 Topological properties of graph cuts 86 5.4.3 Applications of graph cuts as hypersurfaces 87 5.4.4 Theories connecting graph-cuts and hypersurfaces in R^ 90 5.5 Generalizing Graph Cuts for Multi-Label Problems 92 5.5.1 Exact Multi-Label Optimization 92 5.5.2 Approximate Optimization 94 5.5.2.1 Local Minimum with Respect to Expansion and Swap Moves 95 6 Minimal Paths and Fast Marching Methods for Image Analysis 97 L. Cohen 6.1 Introduction 97 viii Contents 6.2 Minimal Paths 98 6.2.1 Geometrical optics 98 6.2.2 Global Minimum for active contours 99 6.2.3 Problem formulation 99 6.2.4 Fast Marching Resolution 100 6.2.5 2D Up-Wind Scheme 102 6.2.6 Minimal Paths in 3D 102 6.2.7 Simultaneous Front Propagation 103 6.2.8 Simultaneous estimate of the path length 104 6.3 Minimal paths from a set of endpointsp/c 105 6.4 Multiple minimal paths between regions/?it 107 6.5 Segmentation by Fast Marching 108 6.6 Centered Minimal Paths and virtual endoscopy 110 6.7 Conclusion Ill 7 Integrating Shape and Texture in Deformable Models: from Hybrid Methods to Metamorphs 113 D. Metaxas, X. Huang and T. Chen 7.1 Introduction 113 7.2 Hybrid Segmentation Method 116 7.2.1 Gibbs Models 116 7.2.2 Deformable models in the Hybrid Framework 118 7.2.3 Integration ofDeformable Models and Gibbs Models . 119 7.3 Metamorphs: Deformable Shape and Texture Models 120 7.3.1 The Metamorphs Model representations 120 7.3.1.1 The Model's Shape Representation 120 7.3.1.2 The Model's Deformations 121 7.3.1.3 The Model's Texture 122 7.3.2 The Metamorph Dynamics 123 7.3.2.1 The Shape Data Terms 123 7.3.2.2 The Intensity Data Terms 125 7.3.3 Model Evolution 126 7.3.4 The Model Fitting Algorithm and Experimental Results 127 7.4 Conclusions 128 8 Variational Segmentation with Shape Priors 131 M. Bergtholdt, D. Cremers and C. Schnorr 8.1 Introduction 131 8.2 Shape Representation 133 8.2.1 Parametric Contour Representations, Geometric Distances, and Invariance 133 8.2.2 Matching Functionals and Psychophysical Distance Measures 134 8.3 Learning Shape Statistics 136 Contents ix 8.3.1 Shape Distances in Kernel Feature Space 136 8.3.2 Structure-Preserving Embedding and Clustering . . .. 137 8.4 Variational Segmentation and Shape Priors 139 8.4.1 Variational Approach 139 8.4.2 Kernel-based Invariant Shape Priors 141 8.4.3 Shape Priors based on the Matching Distance 141 8.4.4 Experimental Results 142 8.5 Conclusion and Further Work 142 9 Curve Propagation, Level Set Methods and Grouping 145 N. Paragios 9.1 Introduction 145 9.2 On the Propagation of Curves 146 9.2.1 Level Set Method 147 9.2.2 Optimisation and Level Set Methods 149 9.3 Data-driven Segmentation 151 9.3.1 Boundary-based Segmentation 151 9.3.2 Region-based Segmentation 152 9.4 Prior Knowledge 154 9.4.1 Average Models 154 9.4.2 Prior Knowledge through Linear Shape Spaces . . .. 157 9.5 Discussion 159 10 On a Stochastic Model of Geometric Snalces 161 A. Yezzi, D. Nain, G. Unal, O. Zeitouni and A. Tannenbaum 10.1 Introduction 161 10.2 Overview of Geodesic Snake Models 163 10.3 Birth and Death Zero Range Particle Systems 163 10.4 Poisson System Simulation 164 10.5 Choosing a Random Event 166 10.5.1 Using a List of Event Tokens 166 10.5.2 Virtual Token List Method 167 10.6 Similarity Invariant Flows 168 10.6.1 Heat Equation and Similarity Flows 169 10.6.2 Gradient Flow 170 10.7 Stochastic Snakes 171 10.7.1 Polygon representation and construction 171 10.8 Experimental Results 173 10.9 Conclusions and Future Research 174 III Shape Modeling & Registration 175 11 Invariant Processing and Occlusion Resistant Recognition of Pla nar Shapes 177 X Contents A. Bruckstein 11.1 Introduction 177 11.2 Invariant Point Locations and Displacements 178 11.3 Invariant Boundary Signatures for Recognition under Partial Occlusions 182 11.4 Invariant Processing of Planar Shapes 184 11.5 Concluding Remarks 188 12 Planar Shape Analysis and Its Applications in Image-Based Infer ences 189 A. Srivastava, S. Joshi, D. Kaziska and D. Wilson 12.1 Introduction 189 12.2 A Framework for Planar Shape Analysis 191 12.3 Clustering of Shapes 194 12.4 Interpolation of Shapes in Echocardiographic Image-Sequences 196 12.5 Study of Human Silhouettes in Infrared Images 200 12.5.1 TPCA Shape Model 200 12.5.2 Bayesian Shape Estimation 202 12.6 Summary & Discussion 202 13 Diffeomorphic Point Matching 205 H. Guo, A. Rangarajan and S. Joshi 13.1 Introduction 205 13.2 Diffeomorphic Landmark Matching 206 13.3 Diffeomorphic Point Shape Matching 214 13.4 Discussion 219 14 Uncertainty-Driven, Point-Based Image Registration 221 C. Stewart 14.1 Introduction 221 14.2 Objective Function, ICP and Normal Distances 223 14.3 Parameter Estimates and Covariance Matrices 226 14.4 Stable Sampling of ICP Constraints 228 14.5 Dual-Bootstrap ICP 230 14.6 Discussion and Conclusion 234 IV Motion Analysis, Optical Flow & Tracking 237 15 Optical Flow Estimation 239 D. Fleet and Y. Weiss 15.1 Introduction 239 15.2 Basic Gradient-Based Estimation 240 15.3 Iterative Optical Flow Estimation 243