Handbook of Mathematical Models in Computer Vision

HANDBOOK OF MATHEMATICAL MODELS IN COMPUTER VISION......Page 1
Springerlink......Page 0
Half-title......Page 2
Title Page......Page 3
Copyright Page......Page 4
Contents......Page 6
Introduction......Page 20
Contributions & Contributors......Page 22
List of Contributors......Page 24
Part I:  Image Reconstruction......Page 35
1.1 Introduction......Page 37
1.2.1 Wavelet Shrinkage......Page 38
1.2.2 Nonlinear Diffusion Filtering......Page 39
1.3.1 Equivalence for Two-Pixel Signals......Page 40
1.3.2 A Wavelet-Inspired Scheme for TV Diffusion of Signals......Page 41
1.3.3 Generalisations to Images......Page 42
1.4.1 Diffusion-Inspired Shrinkage Functions......Page 43
1.4.2 Wavelet Shrinkage with Improved Rotation Invariance......Page 44
1.5 Wavelets with Higher Vanishing Moments......Page 47
Acknowledgements......Page 50
2.1 Introduction......Page 51
2.2.1 BV Space and Basic Properties......Page 53
2.2.3 Scale......Page 54
2.3 Caveats......Page 55
2.4.1 Iterated Refinement......Page 56
2.4.2 L¹ Fitting......Page 57
2.4.3 Anisotropic TV......Page 58
2.4.4 H^1,p Regularization and Inf Convolution......Page 59
2.5.1 Deconvolution......Page 60
2.5.2 Inpainting......Page 61
2.5.3 Texture and Multiscale Decompositions......Page 62
2.6.1 Artificial Time Marching and Fixed Point Iteration......Page 63
2.6.2 Duality-based Methods......Page 64
3.1 Introduction......Page 67
3.2.1 Image Inpainting......Page 70
3.2.2 Navier–Stokes Inpainting......Page 74
3.3 Variational Models for Filling-In......Page 76
3.3.1 Elastica-based Reconstruction of Level Lines......Page 77
3.3.2 Joint Interpolation of Vector Fields and Gray Levels......Page 79
3.3.3 A Variant and Mathematical Results......Page 82
3.3.4 Experimental Results......Page 84
3.4 Surface Reconstruction: The Laplace and the Absolute Minimizing Lipschitz Extension Interpolation......Page 86
3.4.1 Experimental Results......Page 88
3.5 Dealing with texture......Page 89
3.5.2 Inpainting with Image Decomposition......Page 90
3.6.1 Other PDE-based Models......Page 92
3.6.2 Miscellaneous......Page 93
3.8 Appendix......Page 94
3.9 Acknowledgments......Page 95
Part II: Boundary Extraction, Segmentation and Grouping......Page 97
4.1 Introduction......Page 99
4.2 Binary connected operators......Page 100
4.3.1 Level by level construction......Page 101
4.4 Extended connected operators......Page 102
4.4.1.1 Construction of floodings, razings, flattenings and levelings......Page 104
4.5 Levelings for image simplification......Page 105
4.5.1 Varying (α, β)......Page 106
4.5.2 Varying the marker function h......Page 107
4.5.3.1 Construction of a hierarchy based on increasing floodings......Page 108
4.5.3.2 Construction of a hierarchy based on quasi-flat zones......Page 110
4.6 Conclusion......Page 111
5.1 Introduction......Page 113
5.2 Graph Cuts Basics......Page 114
5.2.2 Algorithms for the Min-Cut and Max-Flow Problem......Page 115
5.3.1 Example: Binary Image Restoration......Page 116
5.4 Graph Cuts as Hypersurfaces......Page 118
5.4.1 Basic idea......Page 119
5.4.2 Topological properties of graph cuts......Page 120
5.4.3 Applications of graph cuts as hypersurfaces......Page 121
5.4.4 Theories connecting graph-cuts and hypersurfaces in R^n......Page 124
5.5.1 Exact Multi-Label Optimization......Page 126
5.5.2 Approximate Optimization......Page 128
5.5.2.1 Local Minimum with Respect to Expansion and Swap Moves......Page 129
6.1 Introduction......Page 131
6.2.1 Geometrical optics......Page 132
6.2.3 Problem formulation......Page 133
6.2.4 Fast Marching Resolution......Page 134
6.2.6 Minimal Paths in 3D......Page 136
6.2.7 Simultaneous Front Propagation......Page 137
6.2.8 Simultaneous estimate of the path length......Page 138
6.3 Minimal paths from a set of endpoints p k......Page 139
6.4 Multiple minimal paths between regions R k......Page 141
6.5 Segmentation by Fast Marching......Page 142
6.6 Centered Minimal Paths and virtual endoscopy......Page 144
6.7 Conclusion......Page 145
7.1 Introduction......Page 147
7.2.1 Gibbs Models......Page 150
7.2.2 Deformable models in the Hybrid Framework......Page 152
7.2.3 Integration of Deformable Models and Gibbs Models......Page 153
7.3.1.1 The Model\'s Shape Representation......Page 154
7.3.1.2 The Model\'s Deformations......Page 155
7.3.1.3 The Model\'s Texture......Page 156
7.3.2.1 The Shape Data Terms......Page 157
7.3.2.2 The Intensity Data Terms......Page 159
7.3.3 Model Evolution......Page 160
7.3.4 The Model Fitting Algorithm and Experimental Results......Page 161
7.4 Conclusions......Page 162
Acknowledgement......Page 163
8.1 Introduction......Page 165
8.2.1 Parametric Contour Representations, Geometric Distances, and Invariance......Page 167
8.2.2 Matching Functionals and Psychophysical Distance Measures......Page 168
8.3.1 Shape Distances in Kernel Feature Space......Page 170
8.3.2 Structure-Preserving Embedding and Clustering......Page 171
8.4.1 Variational Approach......Page 173
8.4.3 Shape Priors based on the Matching Distance......Page 175
8.4.4 Experimental Results......Page 176
8.5 Conclusion and Further Work......Page 177
9.1 Introduction......Page 179
9.2 On the Propagation of Curves......Page 180
9.2.1 Level Set Method......Page 181
9.2.2 Optimisation and Level Set Methods......Page 183
9.3.1 Boundary-based Segmentation......Page 185
9.3.2 Region-based Segmentation......Page 186
9.4.1 Average Models......Page 188
9.4.2 Prior Knowledge through Linear Shape Spaces......Page 191
9.5 Discussion......Page 193
10.1 Introduction......Page 195
10.3 Birth and Death Zero Range Particle Systems......Page 197
10.4 Poisson System Simulation......Page 198
10.5.1 Using a List of Event Tokens......Page 200
10.5.2 Virtual Token List Method......Page 201
10.6 Similarity Invariant Flows......Page 202
10.6.1 Heat Equation and Similarity Flows......Page 203
10.6.2 Gradient Flow......Page 204
10.7.1 Polygon representation and construction......Page 205
10.8 Experimental Results......Page 207
Acknowledgement......Page 208
Part III: Shape Modeling & Registration......Page 209
11.1 Introduction......Page 211
11.2 Invariant Point Locations and Displacements......Page 212
11.3 Invariant Boundary Signatures for Recognition under Partial Occlusions......Page 216
11.4 Invariant Processing of Planar Shapes......Page 218
11.5 Concluding Remarks......Page 222
12.1 Introduction......Page 223
12.2 A Framework for Planar Shape Analysis......Page 225
12.3 Clustering of Shapes......Page 228
12.4 Interpolation of Shapes in Echocardiographic Image-Sequences......Page 230
12.5.1 TPCA Shape Model......Page 234
12.6 Summary & Discussion......Page 236
Acknowledgement......Page 237
13.1 Introduction......Page 239
13.2 Diffeomorphic Landmark Matching......Page 240
13.3 Diffeomorphic Point Shape Matching......Page 248
Acknowledgements......Page 253
14.1 Introduction......Page 255
14.2 Objective Function, ICP and Normal Distances......Page 257
14.3 Parameter Estimates and Covariance Matrices......Page 260
14.4 Stable Sampling of ICP Constraints......Page 262
14.5 Dual-Bootstrap ICP......Page 264
14.6 Discussion and Conclusion......Page 268
Acknowledgements......Page 269
Part IV: Motion Analysis, Optical Flow & Tracking......Page 271
15.1 Introduction......Page 273
15.2 Basic Gradient-Based Estimation......Page 274
15.3 Iterative Optical Flow Estimation......Page 277
15.4 Robust Motion Estimation......Page 280
15.5 Motion Models......Page 281
15.6 Global Smoothing......Page 283
15.7 Conservation Assumptions......Page 284
15.8 Probabilistic Formulations......Page 286
15.9 Layered Motion......Page 287
15.10 Conclusions......Page 290
16.1 Motivation and problem statement......Page 293
16.2 Admissible warps......Page 294
16.3 Bayesian formulation of warp estimation......Page 296
16.4 Likelihood: Matching criteria......Page 298
16.5 Prior: Smoothness criteria......Page 300
16.6 Warp time and Computing time......Page 303
16.7 From fluid registration to diffeomorphic minimizers......Page 304
16.8 Discussion and open problems......Page 305
17.1 Introduction......Page 307
17.2 Motion models......Page 308
17.3.1 Direct methods......Page 311
17.3.2 Feature-based registration......Page 313
17.3.3 Direct vs. feature-based......Page 316
17.4.1 Bundle adjustment......Page 317
17.4.3 Recognizing panoramas......Page 319
17.5 Choosing a compositing surface......Page 320
17.6 Seam selection and pixel blending......Page 321
17.7 Extensions and open issues......Page 325
18.1 Introduction......Page 327
18.2.1 Simple patches......Page 328
18.2.3 Background maintenance......Page 329
18.3.1 Snakes......Page 330
18.3.2 Parametric structures......Page 331
18.3.3 Affine contours......Page 332
18.3.5 Robust curve distances......Page 334
18.4.1 Dynamical models......Page 335
18.4.2 Kalman filter for point features......Page 336
18.4.4 Particle filter......Page 337
18.5 Further topics......Page 340
19.1 Introduction......Page 343
19.2 Problem Statement......Page 344
19.3 From shape derivation tools towards region-based active contours models......Page 346
19.3.1.2 Relations between the derivatives......Page 347
19.3.2 Derivation of boundary-based terms......Page 348
19.3.3.2 Region-dependent descriptors......Page 349
19.4 Segmentation using Statistical Region-dependent descriptors......Page 351
19.4.1.2 Region-dependent descriptors based on the variance......Page 353
19.4.2.1 Region-dependent descriptors based on non parametric pdfs of image features......Page 354
19.4.2.2 Minimization of the distance between pdfs for tracking......Page 355
19.5 Discussion......Page 356
20.1 Introduction......Page 359
20.2 Methods......Page 361
20.2.1 Human body model acquisition......Page 362
20.2.2 Model-based tracking......Page 365
20.3 Results......Page 368
20.4 Discussion......Page 372
21.1 Introduction......Page 375
21.1.1 Related work......Page 377
21.3 Leaming dynamic textures......Page 378
21.3.1 Closed-form Solution......Page 380
21.4 Model Validation......Page 381
21.5.1 Distances between dynamic texture models......Page 383
21.5.2 Performance of the nearest neighbor classifier......Page 384
21.6 Segmentation......Page 385
Acknowledgements......Page 389
Part V: 3D from Images, Projective Geometry & Stereo Reconstruction......Page 391
22.1 Introduction......Page 393
22.2 Introduction to Frenet–Serret......Page 395
22.3 Co-Circularity in ℝ² × S¹......Page 397
22.3.1 Multiple Orientations and Product Spaces......Page 398
22.4 Stereo: Inferring Frenet 3-Frames from 2-Frames......Page 399
22.5 Covariant Derivatives, Oriented Textures, and Color......Page 401
22.5.1 Hue Flows......Page 405
22.6 Discussion......Page 406
23.1 Introduction......Page 409
23.2.1 \"Orthographie SFS\" with a far light source......Page 411
23.2.3 \"Perspective SFS\" with a point light source at the optical center......Page 412
23.3.1 Related work......Page 413
23.3.2 Nonuniqueness and characterization ofa Solution......Page 414
23.4.1 Related work......Page 416
23.4.2 An example of provably convergent numerical method: Prados and Faugeras\' method......Page 417
23.5.1 Document restoration using SFS......Page 419
23.5.3 Potential applications to medical images......Page 421
23.6 Conclusion......Page 422
24.1 Introduction......Page 423
24.1.1 Notations and background......Page 424
24.2.1 Epipolar geometry computation......Page 426
24.3 Structure and motion recovery......Page 427
24.3.1 Initial structure and motion......Page 428
24.3.2 Updating the structure and motion......Page 429
24.3.4 Upgradingfrom projective to metric......Page 430
24.4.1 Rectification and stereo matching......Page 432
24.4.2 Multi-view linking......Page 433
24.5 3D surface reconstruction......Page 434
24.6 Conclusion......Page 436
Acknowledgement......Page 437
25.1 Introduction......Page 439
25.2 Reconstruction of Static Scenes......Page 440
25.2.2 Voxel Coloring......Page 441
25.2.3 Space Carving......Page 443
25.2.5 Probabilistic Space Carving......Page 445
25.2.7 Probabilistic Surface Reconstruction......Page 446
25.2.8 Probabilistic Image-Based Stereo......Page 449
25.3.2 Approximate 3D Localization of Targets for Surveillance......Page 450
25.4 Sensor Planning......Page 453
25.5 Conclusion......Page 455
26.1 Traditional stereo methods......Page 457
26.1.1 Energy minimization via graph cuts......Page 459
26.2 Stereo with occlusions......Page 460
26.2.1 Notation......Page 462
26.3 Voxel labeling algorithm......Page 463
26.4 Pixel labeling algorithm......Page 464
26.5 Minimizing the energy......Page 465
26.6.1 Implementational details......Page 466
26.6.2 Algorithm Performance......Page 467
26.7 Conclusions......Page 468
27.1 Introduction......Page 473
27.2.1 Multi-view complete stereovision......Page 474
27.2.2 Scene flow estimation......Page 476
27.3 The Prediction Error as a New Metric for Stereovision and Scene Flow Estimation......Page 477
27.3.1 Stereovision......Page 479
27.3.2 Sceneflow......Page 480
27.3.3 Some similarity measures......Page 481
27.4 Experimental Results......Page 482
27.4.1 Stereovision......Page 483
27.4.2 Stereovision + scene flow......Page 484
27.5 Conclusion and Future Work......Page 485
Part VI: Applications: Medical Image Analysis......Page 487
28.1 Introduction......Page 489
28.2 Characteristic Behaviors of the Algorithms......Page 490
28.3 Applications on CT Cardiovascular data......Page 493
28.3.2 Multi-Resolution Banded Graph Cuts......Page 494
28.3.3 Empirical Results......Page 495
28.3.4 Random Walks for Simultaneous Chamber Segmentation......Page 496
28.3.5 The Random Walker Algorithm......Page 497
28.3.6 Numerical Solution......Page 498
28.3.7 Empirical Results......Page 499
28.3.8 Isoperimetric algorithm......Page 500
28.3.9 Bone-Vessel Separation......Page 501
28.4 Conclusions......Page 503
29.1 Introduction......Page 505
29.1.1 Background......Page 506
29.1.2 Issues inherent to 3D extension......Page 508
29.2.1 3D Point Distribution Models......Page 509
29.2.2 3D Active Shape Models......Page 510
29.2.3.1 2D + time Active Appearance Models......Page 513
29.2.3.2 3D Active Appearance Models: Modeling Volume Appearance......Page 514
29.2.3.3 3D Active Appearance Models: Matching......Page 515
29.2.3.4 Multi-view Active Appearance Models......Page 516
29.3 Discussion and Conclusion......Page 518
Acknowledgments......Page 519
30.1 Introduction......Page 521
30.2 Estimation of PDF......Page 523
30.3 Estimation of ADC profiles......Page 527
30.4 Conclusion......Page 533
Acknowledgments......Page 534
31.1 Introduction......Page 537
31.3 Boundary-based active contours for DTI segmentation......Page 539
31.4.1 An information theoretic diffusion tensor \"distance\"......Page 541
31.4.2 The DTI Segmentation Model......Page 543
31.4.3 The Piecewise Constant Model for DTI Segmentation......Page 544
31.4.4 The Piecewise Smooth DTI Segmentation Model......Page 545
31.4.5 Experimental Results......Page 547
Acknowledgment......Page 548
32.1 Introduction......Page 551
32.2.1 Data acquisition......Page 552
32.2.3 Variational estimation......Page 553
32.3 Regularization of Diffusion Tensor Images......Page 554
32.3.2 A fast isospectral method......Page 555
32.4 Segmentation of Diffusion Tensor Images......Page 556
32.4.2 Multivariate Gaussian distributions as a linear space......Page 557
32.4.3 Information-theoretic statistics between distributions......Page 558
32.4.4 A Riemannian approach to DTI segmentation......Page 561
32.5 Conclusion......Page 564
33.1 Introduction......Page 565
33.2 The Similarity Measures......Page 566
33.2.1 Maximum Likelihood......Page 567
33.2.2 Approximate Maximum Likelihood......Page 569
33.2.3 Kullback–Leibler Divergence......Page 571
33.2.4 Mutual Information and Joint Entropy......Page 573
Acknowledgment......Page 575
Bibliography......Page 577
Index......Page 629