Handbook of Mathematical Models in Computer Vision (eBook)

Contents 5
Preface 19
List of Contributors 23
Part I Image Reconstruction 35
Chapter 1Diffusion Filters and Wavelets: What Can They Learn from Each Other? 36
1.1 Introduction 36
1.2 Basic Methods 37
1.3 Relations for Space-Discrete Diffusion 39
1.4 Relations for Fully Discrete Diffusion 42
1.5 Wavelets with Higher Vanishing Moments 46
1.6 Summary 49
Chapter 2 Total Variation Image Restoration: Overview and Recent Developments 50
2.1 Introduction 50
2.2 Properties and Extensions 52
2.3 Caveats 54
2.4 Variants 55
2.5 Further Applications to Image Reconstruction 59
2.6 Numerical Methods 62
Chapter 3 PDE-Based Image and Surface Inpainting 66
3.1 Introduction 66
3.2 Inpainting by Propagation of Information 69
3.3 Variational Models for Filling-In 75
3.4 Surface Reconstruction: The Laplace and the Absolute Minimizing Lipschitz Extension Interpolation 85
3.5 Dealing with texture 88
3.6 Other Approaches 91
3.7 Concluding Remarks 93
3.9 Acknowledgments 94
3.8 Appendix 93
3.9 Acknowledgments 94
Part II Boundary Extraction, Segmentation and Grouping 96
Chapter 4 Levelings: Theory and Practice 98
4.1 Introduction 98
4.2 Binary connected operators 99
4.3 Flat grey-tone connected operators 100
4.4 Extended connected operators 101
4.5 Levelings for image simplification 104
4.6 Conclusion 110
Chapter 5 Graph Cuts in Vision and Graphics: Theories and Applications 112
5.1 Introduction 112
5.2 Graph Cuts Basics 113
5.3 Graph Cuts for Binary Optimization 115
5.4 Graph Cuts as Hypersurfaces 117
5.5 Generalizing Graph Cuts for Multi- Label Problems 125
Chapter 6 Minimal Paths and Fast Marching Methods for Image Analysis 130
6.1 Introduction 130
6.2 Minimal Paths 131
6.3 Minimal paths from a set of endpoints pk 138
6.4 Multiple minimal paths between regions Rk 140
6.5 Segmentation by Fast Marching 141
6.6 Centered Minimal Paths and virtual endoscopy 143
6.7 Conclusion 144
Chapter 7 Integrating Shape and Texture in Deformable Models: from Hybrid Methods to Metamorphs 146
7.1 Introduction 146
7.2 Hybrid Segmentation Method 149
7.3 Metamorphs: Deformable Shape and Texture Models 153
7.4 Conclusions 161
Chapter 8 Variational Segmentation with Shape Priors 164
8.1 Introduction 164
8.2 Shape Representation 166
8.3 Learning Shape Statistics 169
8.4 Variational Segmentation and Shape Priors 172
8.5 Conclusion and Further Work 176
Chapter 9 Curve Propagation, Level Set Methods and Grouping 178
9.1 Introduction 178
9.2 On the Propagation of Curves 179
9.3 Data-driven Segmentation 184
9.4 Prior Knowledge 187
9.5 Discussion 192
Chapter 10 On a Stochastic Model of Geometric Snakes 194
10.1 Introduction 194
10.2 Overview of Geodesic Snake Models 196
10.3 Birth and Death Zero Range Particle Systems 196
10.4 Poisson System Simulation 197
10.5 Choosing a Random Event 199
10.6 Similarity Invariant Flows 201
10.7 Stochastic Snakes 204
10.8 Experimental Results 206
10.9 Conclusions and Future Research 207
Part III Shape Modeling & Registration
Chapter 11 Invariant Processing and Occlusion Resistant Recognition of Planar Shapes 210
11.1 Introduction 210
11.2 Invariant Point Locations and Displacements 211
11.3 Invariant Boundary Signatures for Recognition under Partial Occlusions 215
11.4 Invariant Processing of Planar Shapes 217
11.5 Concluding Remarks 221
Chapter 12 Planar Shape Analysis and Its Applications in Image-Based Inferences 222
12.1 Introduction 222
12.2 A Framework for Planar Shape Analysis 224
12.3 Clustering of Shapes 227
12.4 Interpolation of Shapes in Echocardiographic Image- Sequences 229
12.5 Study of Human Silhouettes in Infrared Images 233
12.6 Summary & Discussion
Chapter 13 Diffeomorphic Point Matching 238
13.1 Introduction 238
13.2 Diffeomorphic Landmark Matching 239
13.3 Diffeomorphic Point Shape Matching 247
13.4 Discussion 252
Chapter 14 Uncertainty-Driven, Point-Based Image Registration 254
14.1 Introduction 254
14.2 Objective Function, ICP and Normal Distances 256
14.3 Parameter Estimates and Covariance Matrices 259
14.4 Stable Sampling of ICP Constraints 261
14.5 Dual-Bootstrap ICP 263
14.6 Discussion and Conclusion 267
Part IV Motion Analysis, Optical Flow & Tracking
Chapter 15 Optical Flow Estimation 272
15.1 Introduction 272
15.2 Basic Gradient-Based Estimation 273
15.3 Iterative Optical Flow Estimation 276
15.4 Robust Motion Estimation 279
15.5 Motion Models 280
15.6 Global Smoothing 282
15.7 Conservation Assumptions 283
15.8 Probabilistic Formulations 285
Chapter 16 From Bayes to PDEs in Image Warping 292
16.1 Motivation and problem statement 292
16.2 Admissible warps 293
16.3 Bayesian formulation of warp estimation 295
16.4 Likelihood: Matching criteria 297
16.5 Prior: Smoothness criteria 299
16.6 Warp time and Computing time 302
16.7 From fluid registration to diffeomorphic minimizers 303
16.8 Discussion and open problems 304
Chapter 17 Image Alignment and Stitching 306
17.1 Introduction 306
17.2 Motion models 307
17.3 Direct and feature-based alignment 310
17.4 Global registration 316
17.5 Choosing a compositing surface 319
17.6 Seam selection and pixel blending 320
17.7 Extensions and open issues 324
Chapter 18 Visual Tracking: A Short Research Roadmap 326
18.1 Introduction 326
18.2 Simple appearance models 327
18.3 Active contours 329
18.4 Spatio-temporal filtering 334
18.5 Further topics 339
Chapter 19 Shape Gradient for Image and Video Segmentation 342
19.1 Introduction 342
19.2 Problem Statement 343
19.3 From shape derivation tools towards region-based active contours models 345
19.4 Segmentation using Statistical Region-dependent descriptors 350
19.5 Discussion 355
Chapter 20 Model-Based Human Motion Capture 358
20.1 Introduction 358
20.2 Methods 360
20.3 Results 367
20.4 Discussion 371
Chapter 21 Modeling Dynamic Scenes: An Overview of Dynamic Textures 374
21.1 Introduction 374
21.2 Representation of dynamic textures 377
21.3 Leaming dynamic textures 377
21.4 Model Validation 380
21.5 Recognition 382
21.6 Segmentation 384
21.7 Discussion 388
PartV 3D from Images, Projective Geometry & Stereo Reconstruction
Chapter 22 Differential Geometry from the Frenet Point of View: Boundary Detection, Stereo^ Texture and Color 392
22.1 Introduction 392
22.2 Introduction to Frenet-Serret 394
22.3 Co-Circularity in R^ x S 1 396
22.4 Stereo: Inferring Frenet 3-Frames from 2-Frames 398
22.5 Covariant Derivatives, Oriented Textures, and Color 400
22.6 Discussion 405
Chapter 23 Shape From Shading 408
23.1 Introduction 408
23.2 Mathematical formulation of the SFS problem 410
23.3 Mathematical study of the SFS problem 412
23.4 Numerical Solutions by "Propagation and PDEs methods" 415
23.5 Examples of numerical results 418
23.6 Conclusion 421
Chapter 24 3D from Image Sequences: Calibration, Motion and Shape Recovery 422
24.1 Introduction 422
24.2 Relating Images 425
24.3 Structure and motion recovery 426
24.4 Dense surface estimation 431
24.5 3D surface reconstruction 433
24.6 Conclusion 435
Chapter 25 Multi-view Reconstruction of Static and Dynamic Scenes 438
25.1 Introduction 438
25.2 Reconstruction of Static Scenes 439
25.3 Reconstraction of Dynamic Scenes 449
25.4 Sensor Planning 452
25.5 Conclusion 454
Chapter 26 Graph Cut Algorithms for Binocular Stereo with Occlusions 456
26.1 Traditional stereo methods 456
26.2 Stereo with occlusions 459
26.3 Voxel labeling algorithm 462
26.4 Pixel labeling algorithm 463
26.5 Minimizing the energy 464
26.6 Experimental results 465
26.7 Conclusions 467
Chapter 27 Modelling Non-Rigid Dynamic Scenes from Multi-View Image Sequences 472
27.1 Introduction 472
27.2 Previous Work 473
27.3 The Prediction Error as a New Metrie for Stereovision and Scene Flow Estimation 476
27.4 Experimental Results 481
27.5 Conclusion and Future Work 484
Part VI Applications: Medical Image Analysis 487
Chapter 28 Interactive Graph-Based Segmentation Methods in Cardiovascular Imaging 488
28.1 Introduction 488
28.2 Characteristic Behaviors of the Algorithms 489
28.3 Applications on CT Cardiovascular data 492
28.4 Conclusions 502
Chapter 29 3D Active Shape and Appearance Models in Cardiac Image Analysis 504
29.1 Introduction 504
29.2 Methods 508
29.3 Discussion and Conclusion 517
Chapter 30 Characterization of Diffusion Anisotropy in DWI 520
30.1 Introduction 520
30.2 EstimationofPDF 522
30.3 Estimation of ADC profiles 526
30.4 Conclusion 532
Chapter 31 Segmentation of Diffusion Tensor Images 536
31.1 Introduction 536
31.2 K-means for DTI segmentation 538
31.3 Boundary-based active contours for DTI segmentation 538
31.4 Region- based active contour for DTI segmentation 540
31.5 Conclusion 547
Chapter 32 Variational Approaches to the Estimation^ Regularization and Segmentation of Diffusion Tensor Images 550
32.1 Introduction 550
32.2 Estimation of Diffusion Tensor Images 551
32.3 Regularization of Diffusion Tensor Images 553
32.4 Segmentation of Diffusion Tensor Images 555
32.5 Conclusion 563
Chapter 33 An Introduction to Statistical Methods of Medical Image Registration 564
33.1 Introduction 564
33.2 The Similarity Measures 565
33.3 Conclusion 574
Bibliography 576