Nikos Paragios, Yunmei Chen, Olivier D. Faugeras-Handbook of Mathematical Models in Computer Vision- - HANDBOOK OF MATHEMATICAL MODELS IN COMPUTER

Nikos Paragios, Yunmei Chen, Olivier D. Faugeras-Handbook of Mathematical Models in Computer Vision-

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Unformatted text preview: 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 springeronline.com SPIN 11055662 (HC) / 11552987 (eBook) Contents Preface List of Contributors I Image Reconstruction 1 2 Diffusion Filters and Wavelets: What Can They Learn from Each Other? J. Weickert, G. Steidl, P. Mrazek, M. Welk, and T. Brox 1.1 Introduction 1.2 Basic Methods 1.2.1 Wavelet Shrinkage 1.2.2 Nonlinear Diffusion Filtering 1.3 Relations for Space-Discrete Diffusion 1.3.1 Equivalence for Two-Pixel Signals 1.3.2 A Wavelet-Inspired Scheme for TV Diffusion of Signals 1.3.3 Generalisations to Images 1.4 Relations for Fully Discrete Diffusion 1.4.1 Diffusion-Inspired Shrinkage Functions 1.4.2 Wavelet Shrinkage with Improved Rotation Invariance . 1.4.3 Diffusion-Inspired Wavelet Shrinkage of Colour Images 1.5 Wavelets with Higher Vanishing Moments 1.6 Summary Total Variation Image Restoration: Overview and Recent Developments T. Chan, S. Esedoglu, F. Park and A. Yip 2.1 Introduction 2.2 Properties and Extensions 2.2.1 BV Space and Basic Properties 2.2.2 Multi-channel TV xix xxiii 1 3 3 4 4 5 6 6 7 8 9 9 10 13 13 16 17 17 19 19 20 vi Contents 2.2.3 Scale Caveats Variants 2.4.1 Iterated Refinement 2.4.2 L^ Fitting 2.4.3 Anisotropic TV 2.4.4 //^'PRegularization and Inf Convolution Further Applications to Image Reconstruction 2.5.1 Deconvolution 2.5.2 Inpainting 2.5.3 Texture and Multiscale Decompositions Numerical Methods 2.6.1 Artificial Time Marching and Fixed Point Iteration . . 2.6.2 DuaHty-based Methods 20 21 22 22 23 24 25 26 26 27 28 29 29 30 3 PDE-Based Image and Surface Inpainting M. Bertalmio, V. Caselles, G. Haro, and G. Sapiro 3.1 Introduction 3.2 Inpaintingby Propagation of Information 3.2.1 Image Inpainting 3.2.2 Navier-Stokes Inpainting 3.3 Variational Models for Filling-In 3.3.1 Elastica-based Reconstruction of Level Lines 3.3.2 Joint Interpolation of Vector Fields and Gray Levels . . 3.3.3 A Variant and Mathematical Results 3.3.4 Experimental Results 3.4 Surface Reconstruction: The Laplace and the Absolute Minimizing Lipschitz Extension Interpolation 3.4.1 Experimental Results 3.5 Dealing with texture 3.5.1 Texture Synthesis by Non-Parametric Sampling . . . . 3.5.2 Inpainting with Image Decomposition 3.5.3 Exemplar-based Inpainting 3.6 Other Approaches 3.6.1 Other PDE-based Models 3.6.2 Miscellaneous 3.7 Concluding Remarks 3.8 Appendix 3.9 Acknowledgments 33 2.3 2.4 2.5 2.6 33 36 36 40 42 43 45 48 50 52 54 55 56 56 58 58 58 59 60 60 61 II Boundary Extraction, Segmentation and Grouping 63 4 Levelings: Theory and Practice 65 Contents F. Meyer 4.1 Introduction 4.2 Binary connected operators 4.3 Flat grey-tone connected operators 4.3.1 Level by level construction 4.3.2 A morphological characterization 4.4 Extended connected operators 4.4.1 Construction of floodings, razings, flattenings and levelings 4.4.1.1 Construction of floodings, razings, flattenings and levelings 4.5 Levelings for image simplification 4.5.1 Varying (a,/?) 4.5.2 Varying the marker function h 4.5.3 Multiscale filtering 4.5.3.1 Construction of a hierarchy based on increasing floodings 4.5.3.2 Construction of a hierarchy based on quasi-flat zones 4.6 Conclusion 5 6 Graph Cuts in Vision and Graphics: Theories and Applications Y. Boykov and O. Veksler 5.1 Introduction 5.2 Graph Cuts Basics 5.2.1 The Min-Cut and Max-Flow Problem 5.2.2 Algorithms for the Min-Cut and Max-Flow Problem . . 5.3 Graph Cuts for Binary Optimization 5.3.1 Example: Binary Image Restoration 5.3.2 General Case of Binary Energy Minimization 5.4 Graph Cuts as Hypersurfaces 5.4.1 Basic idea 5.4.2 Topological properties of graph cuts 5.4.3 Applications of graph cuts as hypersurfaces 5.4.4 Theories connecting graph-cuts and hypersurfaces in R^ 5.5 Generalizing Graph Cuts for Multi-Label Problems 5.5.1 Exact Multi-Label Optimization 5.5.2 Approximate Optimization 5.5.2.1 Local Minimum with Respect to Expansion and Swap Moves Minimal Paths and Fast Marching Methods for Image Analysis L. Cohen 6.1 Introduction vii 65 66 67 67 68 68 70 70 71 72 73 74 74 76 77 79 79 80 81 81 82 82 84 84 85 86 87 90 92 92 94 95 97 97 viii Contents 6.2 6.3 6.4 6.5 6.6 6.7 7 8 Minimal Paths 6.2.1 Geometrical optics 6.2.2 Global Minimum for active contours 6.2.3 Problem formulation 6.2.4 Fast Marching Resolution 6.2.5 2D Up-Wind Scheme 6.2.6 Minimal Paths in 3D 6.2.7 Simultaneous Front Propagation 6.2.8 Simultaneous estimate of the path length Minimal paths from a set of endpointsp/c Multiple minimal paths between regions/?it Segmentation by Fast Marching Centered Minimal Paths and virtual endoscopy Conclusion Integrating Shape and Texture in Deformable Models: from Hybrid Methods to Metamorphs D. Metaxas, X. Huang and T. Chen 7.1 Introduction 7.2 Hybrid Segmentation Method 7.2.1 Gibbs Models 7.2.2 Deformable models in the Hybrid Framework 7.2.3 Integration ofDeformable Models and Gibbs Models . 7.3 Metamorphs: Deformable Shape and Texture Models 7.3.1 The Metamorphs Model representations 7.3.1.1 The Model's Shape Representation 7.3.1.2 The Model's Deformations 7.3.1.3 The Model's Texture 7.3.2 The Metamorph Dynamics 7.3.2.1 The Shape Data Terms 7.3.2.2 The Intensity Data Terms 7.3.3 Model Evolution 7.3.4 The Model Fitting Algorithm and Experimental Results 7.4 Conclusions Variational Segmentation with Shape Priors M. Bergtholdt, D. Cremers and C. Schnorr 8.1 Introduction 8.2 Shape Representation 8.2.1 Parametric Contour Representations, Geometric Distances, and Invariance 8.2.2 Matching Functionals and Psychophysical Distance Measures 8.3 Learning Shape Statistics 98 98 99 99 100 102 102 103 104 105 107 108 110 Ill 113 113 116 116 118 119 120 120 120 121 122 123 123 125 126 127 128 131 131 133 133 134 136 Contents 8.4 8.5 9 ix 8.3.1 Shape Distances in Kernel Feature Space 8.3.2 Structure-Preserving Embedding and Clustering . . . . Variational Segmentation and Shape Priors 8.4.1 Variational Approach 8.4.2 Kernel-based Invariant Shape Priors 8.4.3 Shape Priors based on the Matching Distance 8.4.4 Experimental Results Conclusion and Further Work Curve Propagation, Level Set Methods and Grouping N. Paragios 9.1 Introduction 9.2 On the Propagation of Curves 9.2.1 Level Set Method 9.2.2 Optimisation and Level Set Methods 9.3 Data-driven Segmentation 9.3.1 Boundary-based Segmentation 9.3.2 Region-based Segmentation 9.4 Prior Knowledge 9.4.1 Average Models 9.4.2 Prior Knowledge through Linear Shape Spaces 9.5 Discussion 136 137 139 139 141 141 142 142 145 .... 145 146 147 149 151 151 152 154 154 157 159 10 On a Stochastic Model of Geometric Snalces A. Yezzi, D. Nain, G. Unal, O. Zeitouni and A. Tannenbaum 10.1 Introduction 10.2 Overview of Geodesic Snake Models 10.3 Birth and Death Zero Range Particle Systems 10.4 Poisson System Simulation 10.5 Choosing a Random Event 10.5.1 Using a List of Event Tokens 10.5.2 Virtual Token List Method 10.6 Similarity Invariant Flows 10.6.1 Heat Equation and Similarity Flows 10.6.2 Gradient Flow 10.7 Stochastic Snakes 10.7.1 Polygon representation and construction 10.8 Experimental Results 10.9 Conclusions and Future Research 161 III Shape Modeling & Registration 175 11 Invariant Processing and Occlusion Resistant Recognition of Planar Shapes 177 161 163 163 164 166 166 167 168 169 170 171 171 173 174 X Contents A. Bruckstein 11.1 Introduction 11.2 Invariant Point Locations and Displacements 11.3 Invariant Boundary Signatures for Recognition under Partial Occlusions 11.4 Invariant Processing of Planar Shapes 11.5 Concluding Remarks 12 Planar Shape Analysis and Its Applications in Image-Based Inferences A. Srivastava, S. Joshi, D. Kaziska and D. Wilson 12.1 Introduction 12.2 A Framework for Planar Shape Analysis 12.3 Clustering of Shapes 12.4 Interpolation of Shapes in Echocardiographic Image-Sequences 12.5 Study of Human Silhouettes in Infrared Images 12.5.1 TPCA Shape Model 12.5.2 Bayesian Shape Estimation 12.6 Summary & Discussion 13 Diffeomorphic Point Matching H. Guo, A. Rangarajan and S. Joshi 13.1 Introduction 13.2 Diffeomorphic Landmark Matching 13.3 Diffeomorphic Point Shape Matching 13.4 Discussion 177 178 182 184 188 189 189 191 194 196 200 200 202 202 205 205 206 214 219 14 Uncertainty-Driven, Point-Based Image Registration C. Stewart 14.1 Introduction 14.2 Objective Function, ICP and Normal Distances 14.3 Parameter Estimates and Covariance Matrices 14.4 Stable Sampling of ICP Constraints 14.5 Dual-Bootstrap ICP 14.6 Discussion and Conclusion 221 223 226 228 230 234 IV Motion Analysis, Optical Flow & Tracking 237 15 Optical Flow Estimation D. Fleet and Y. Weiss 15.1 Introduction 15.2 Basic Gradient-Based Estimation 15.3 Iterative Optical Flow Estimation 221 239 239 240 243 Contents 15.4 15.5 15.6 15.7 15.8 15.9 15.10 xi Robust Motion Estimation Motion Models Global Smoothing Conservation Assumptions Probabilistic Formulations Layered Motion Conclusions 16 From Bayes to PDEs in Image Warping M. Nielsen and B. Markussen 16.1 Motivation and problem statement 16.2 Admissible warps 16.3 Bayesian formulation of warp estimation 16.4 Likelihood: Matching criteria 16.5 Prior: Smoothness criteria 16.6 Warp time and computing time 16.7 From fluid registration to diffeomorphic minimizers 16.8 Discussion and open problems 17 Image Alignment and Stitching R. Szeliski 17.1 Introduction 17.2 Motion models 17.3 Direct and feature-based alignment 17.3.1 Direct methods 17.3.2 Feature-based registration 17.3.3 Direct vs. feature-based 17.4 Global registration 17.4.1 Bundle adjustment 17.4.2 Parallax removal 17.4.3 Recognizing panoramas 17.5 Choosing a compositing surface 17.6 Seam selection and pixel blending 17.7 Extensions and open issues 18 Visual Tracking: A Short Research Roadmap A. Blake 18.1 Introduction 18.2 Simple appearance models 18.2.1 Simple patches 18.2.2 Blobs 18.2.3 Background maintenance 18.3 Active contours 18.3.1 Snakes 18.3.2 Parametric structures 246 247 249 250 252 253 256 259 259 260 262 264 266 269 270 271 273 273 274 277 277 279 282 283 283 285 285 286 287 291 293 293 294 294 295 295 296 296 297 xii Contents 18.4 18.5 18.3.3 Affine contours 18.3.4 Nonrigidity 18.3.5 Robust curve distances Spatio-temporal filtering 18.4.1 Dynamical models 18.4.2 Kalman filter for point features 18.4.3 Kalman filter for contours 18.4.4 Particle filter Further topics 298 300 300 301 301 302 303 303 306 19 Shape Gradient for Image and Video Segmentation S. Jehan-Besson, A. Herbulot, M. Barlaud, G. Aubert 19.1 Introduction 19.2 Problem Statement 19.3 From shape derivation tools towards region-based active contours models 19.3.1 Shape derivation tools 19.3.1.1 Introduction of transformations 19.3.1.2 Relations between the derivatives 19.3.2 Derivation of boundary-based terms 19.3.3 Derivation of region-based terms 19.3.3.1 Region-independent descriptors 19.3.3.2 Region-dependent descriptors 19.4 Segmentation using Statistical Region-dependent descriptors . 19.4.1 Examples of Descriptors based on parametric statistics . 19.4.1.1 Region-dependent descriptors using the mean 19.4.1.2 Region-dependent descriptors based on the variance 19.4.2 Descriptors based on non parametric statistics 19.4.2.1 Region-dependent descriptors based on non parametric pdfs of image features 19.4.2.2 Minimization of the distance between pdfs for tracking 19.5 Discussion 309 20 Model-Based Human Motion Capture I. Kakadiaris and C. Barron 20.1 Introduction 20.2 Methods 20.2.1 Human body model acquisition 20.2.2 Model-based tracking 20.3 Results 20.4 Discussion 325 309 310 312 313 313 313 314 315 315 315 317 319 319 319 320 320 321 322 325 327 328 331 334 338 Contents 21 Modeling Dynamic Scenes: An Overview of Dynamic Textures G. Doretto and S. Soatto 21.1 Introduction 21.1.1 Related work 21.2 Representation of dynamic textures 21.3 Learning dynamic textures 21.3.1 Closed-form solution 21.4 Model validation 21.5 Recognition 21.5.1 Distances between dynamic texture models 21.5.2 Performance of the nearest neighbor classifier 21.6 Segmentation 21.7 Discussion xiii 341 341 343 344 344 346 347 349 349 350 351 355 V 3D from Images, Projective Geometry & Stereo Reconstruction 357 22 Differential Geometry from the Frenet Point of View: Boundary Detection, Stereo, Texture and Color S. Zucker 22.1 Introduction 22.2 Introduction to Frenet-Serret 22.3 Co-Circularity in M^ x 5^ 22.3.1 Multiple Orientations and Product Spaces 22.4 Stereo: Inferring Frenet 3-Frames from 2-Frames 22.5 Covariant Derivatives, Oriented Textures, and Color 22.5.1 Hue Flows 22.6 Discussion 23 Shape From Shading E. Prados and O. Faugeras 23.1 Introduction 23.2 Mathematical formulation of the SFS problem 23.2.1 "Orthographic SFS" with a far light source 23.2.2 "Perspective SFS" with a far light source 23.2.3 "Perspective SFS" with a point light source at the optical center 23.2.4 A generic Hamiltonian 23.3 Mathematical study ofthe SFS problem 23.3.1 Related work 23.3.2 Nonuniqueness and characterization of a solution . . . 23.4 Numerical solutions by "Propagation and PDEs methods" . . . 23.4.1 Related work 359 359 361 363 364 365 367 371 372 375 375 377 377 378 378 379 379 379 380 382 382 xiv Contents 23.5 23.6 23.4.2 An example of provably convergent numerical method: Prados and Faugeras' method Examples of numerical results 23.5.1 Document restoration using SFS 23.5.2 Face reconstruction from SFS 23.5.3 Potential applications to medical images Conclusion 24 3D from Image Sequences: Calibration, Motion and Shape Recovery M. Pollefeys 24.1 Introduction 24.1.1 Notations and background 24.2 Relating images 24.2.1 Epipolar geometry computation 24.3 Structure and motion recovery 24.3.1 Initial structure and motion 24.3.2 Updating the structure and motion 24.3.3 Refining structure and motion 24.3.4 Upgrading from projective to metric 24.4 Dense surface estimation 24.4.1 Rectification and stereo matching 24.4.2 Multi-view linking 24.5 3D surface reconstruction 24.6 Conclusion 25 Multi-view Reconstruction of Static and Dynamic Scenes M. Agrawal, A. Mittal and L. Davis 25.1 Introduction 25.2 Reconstruction of Static Scenes 25.2.1 Visual Hull 25.2.2 Voxel Coloring 25.2.3 Space Carving 25.2.4 ProbabiHstic Approaches 25.2.5 ProbabiHstic Space Carving 25.2.6 Roxels: Responsibility Weighted Voxels 25.2.7 ProbabiHstic Surface Reconstruction 25.2.8 ProbabiHstic Image-Based Stereo 25.3 Reconstruction of Dynamic Scenes 25.3.1 Visual Hull Algorithms 25.3.2 Approximate 3D Localization of Targets for Surveillance 25.4 Sensor Planning 25.5 Conclusion 383 385 385 387 387 388 389 389 390 392 392 393 394 395 396 396 398 398 399 400 402 405 405 406 407 407 409 411 411 412 412 415 416 416 416 419 421 Contents 26 Graph Cut Algorithms for Binocular Stereo with Occlusions V. Kolmogorov and R. Zabih 26.1 Traditional stereo methods 26.1.1 Energy minimization via graph cuts 26.2 Stereo with occlusions 26.2.1 Notation 26.3 Voxel labeling algorithm 26.4 Pixel labeling algorithm 26.5 Minimizing the energy 26.6 Experimental results 26.6.1 Implementational details 26.6.2 Algorithm performance 26.7 Conclusions 27 Modelling Non-Rigid Dynamic Scenes from Multi-View Image Sequences J.-P. Pons, R. Keriven and O. Faugeras 27.1 Introduction 27.2 Previous Work 27.2.1 Multi-view complete stereovision 27.2.2 Scene flow estimation 27.2.3 Shape-motion integration 27.3 The Prediction Error as a New Metric for Stereovision and Scene Flow Estimation 27.3.1 Stereovision 27.3.2 Scene flow 27.3.3 Some similarity measures 27.4 Experimental Results 27.4.1 Stereovision 27.4.2 Stereovision + scene flow 27.5 Conclusion and Future Work VI Applications: Medical Image Analysis 28 Interactive Graph-Based Segmentation Methods in Cardiovascular Imaging L. Grady, Y. Sun and J. Williams 28.1 Introduction 28.2 Characteristic Behaviors of the Algorithms 28.3 Applications on CT Cardiovascular data 28.3.1 Segmenting Individual Heart Chambers using Graph Cuts 28.3.2 Multi-Resolution Banded Graph Cuts 28.3.3 Empirical Results xv 423 423 425 426 428 429 430 431 432 432 433 434 439 439 440 440 442 443 443 445 446 447 448 449 450 451 453 455 455 456 459 460 460 461 xvi Contents 28.4 28.3.4 Random Walks for Simultaneous Chamber Segmentation 28.3.5 The Random Walker Algorithm 28.3.6 Numerical solution 28.3.7 Empirical Results 28.3.8 Isoperimetric algorithm 28.3.9 Bone-Vessel Separation Conclusions 29 3D Active Shape and Appearance Models in Cardiac Image Analysis B. Lelieveldt, A. Frangi, S. Mitchell, H. van Assen, S. Ordas, J. Reiber and M. Sonka 29.1 Introduction 29.1.1 Background 29.1.2 Issues inherent to 3D extension 29.2 Methods 29.2.1 3D Point Distribution Models 29.2.2 3D Active Shape Models 29.2.3 3D and 4D Active Appearance Models 29.2.3.1 2D + time Active Appearance Models . . . . 29.2.3.2 3D Active Appearance Models: Modeling Volume Appearance 29.2.3.3 3D Active Appearance Models: Matching . . 29.2.3.4 Multi-view Active Appearance Models . . . 29.2.3.5 3D + time Active Appearance Models . . . . 29.3 Discussion and Conclusion 30 Characterization of Diffusion Anisotropy in DWI Y. Chen 30.1 Introduction 30.2 Estimation of PDF 30.3 Estimation of ADC profiles 30.4 Conclusion 31 Segmentation of Diffusion Tensor Images Z. Wang and B. Vemuri 31.1 Introduction 31.2 K-means for DTI segmentation 31.3 Boundary-based active contours for DTI segmentation 31.4 Region-based active contour for DTI segmentation 31.4.1 An information theoretic diffusion tensor "distance" . . 31.4.2 The DTI Segmentation Model 31.4.3 The Piecewise Constant Model for DTI Segmentation . 31.4.4 The Piecewise Smooth DTI Segmentation Model . . . 462 463 464 465 466 467 469 471 471 472 474 475 475 476 479 479 480 481 482 484 484 487 487 489 493 499 503 503 505 505 507 507 509 510 511 Contents 31.5 xvii 31.4.5 Experimental Results Conclusion 32 Variational Approaches to the Estimation, Regularization and Segmentation of Diffusion Tensor Images R. Deriche, D. Tschumperle, C. Lenglet and M. Rousson 32.1 Introduction 32.2 Estimation of Diffusion Tensor Images 32.2.1 Data acquisition 32.2.2 Linear estimation 32.2.3 Variational estimation 32.3 Regularizationof Diffusion Tensor Images 32.3.1 On some non-spectral methods and their limitations . . 32.3.2 A fast isospectral method 32.4 Segmentation ofDiffusion Tensor Images 32.4.1 Level-set and region-based surface evolution 32.4.2 Multivariate Gaussian distributions as a Hnear space . . 32.4.3 Information-theoretic statistics between distributions . 32.4.4 A Riemannian approach to DTI segmentation 32.5 Conclusion 33 An Introduction to Statistical Methods of Medical Image Registration L. Zollei, J. Fisher and W. Wells 33.1 Introduction 33.2 The Similari...
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