Handbook of mathematical methods in imaging

Cover......Page 1
front-matter......Page 2
Handbook of Mathematical Methods in Imaging......Page 4
Preface......Page 6
Table of Contents......Page 8
About the Editor......Page 12
List of Contributors......Page 14
Linear Inverse Problems......Page 20
1.1 Introduction......Page 21
1.2 Background......Page 23
1.3.1 A Platonic Inverse Problem......Page 28
1.3.2 Cormack's Inverse Problem......Page 31
1.3.3 Forward and Reverse Diffusion......Page 32
1.3.4 Deblurring as an Inverse Problem......Page 33
1.3.5 Extrapolation of Band-Limited Signals......Page 35
1.3.6 PET......Page 36
1.3.7 Some Mathematics for Inverse Problems......Page 37
1.3.7.1 Weak Convergence......Page 39
1.3.7.2 Linear Operators......Page 40
1.3.7.3 Compact Operators and the SVD......Page 42
1.3.7.4 The Moore–Penrose Inverse......Page 44
1.3.7.5 Alternating Projection Theorem......Page 45
1.4.1 Tikhonov Regularization......Page 46
1.4.2 Iterative Regularization......Page 50
1.4.3 Discretization......Page 52
References and Further Reading......Page 56
Large-Scale Inverse Problems in Imaging......Page 60
2.1 Introduction......Page 61
2.2.1 Model Problems......Page 62
2.2.2.1 Image Deblurring and Deconvolution......Page 63
2.2.2.2 Multi-Frame Blind Deconvolution......Page 65
2.2.2.3 Tomosynthesis......Page 66
2.3 Mathematical Modelling and Analysis......Page 68
2.3.1.1 SVD Analysis......Page 69
2.3.1.2 Regularization by SVD Filtering......Page 70
2.3.1.3 Variational Regularization and Constraints......Page 71
2.3.1.4 Iterative Regularization......Page 72
2.3.1.5 Hybrid Iterative-Direct Regularization......Page 74
2.3.1.6 Choosing Regularization Parameters......Page 78
2.3.2 Separable Inverse Problems......Page 79
2.3.2.1 Fully Coupled Problem......Page 80
2.3.2.2 Decoupled Problem......Page 81
2.3.2.3 Variable Projection Method......Page 82
2.3.3 Nonlinear Inverse Problems......Page 83
2.4.1 Linear Example: Deconvolution......Page 86
2.4.2 Separable Example: Multi-Frame Blind Deconvolution......Page 90
2.4.3 Nonlinear Example: Tomosynthesis......Page 92
2.5 Conclusion......Page 97
2.6 Cross-References......Page 98
References and Further Reading......Page 99
Regularization Methods for Ill-Posed Problems......Page 104
3.1 Introduction......Page 105
3.2 Theory of Direct Regularization Methods......Page 106
3.2.1 Tikhonov Regularization in Hilbert Spaces with Quadratic Misfitand Penalty Terms......Page 108
3.2.2 Variational Regularization in Banach Spaces with Convex Penalty Term......Page 110
3.2.3 Extended Results for Hilbert Space Situations......Page 114
3.3 Examples......Page 116
3.5 Cross-References......Page 123
Distance Measures and Applications to Multi-Modal Variational Imaging......Page 128
4.1 Introduction......Page 129
4.2.1 Deterministic Pixel Measure......Page 130
4.2.2 Morphological Measures......Page 131
4.2.3 Statistical Distance Measures......Page 132
4.2.4 Statistical Distance Measures (Density Based)......Page 134
4.2.4.1 Density Estimation......Page 136
4.2.4.2 Csiszár-Divergences (f-Divergences)......Page 140
4.2.4.3 f-Information......Page 143
4.2.5 Distance Measures Including Statistical Prior Information......Page 147
4.3 Mathematical Models for Variational Imaging......Page 148
4.4 Registration......Page 149
4.5 Recommended Reading......Page 152
Acknowledgment......Page 153
References and Further Reading......Page 154
Energy Minimization Methods......Page 156
5.1 Introduction......Page 158
5.1.1 Background......Page 160
5.1.3 Organization of the Chapter......Page 162
5.2.2 Reminders and Definitions......Page 163
5.3.1 Some General Results......Page 167
5.3.2 Stability of the Minimizers of Energies with Possibly Nonconvex Priors......Page 168
5.3.2.1 Local Minimizers......Page 169
5.3.2.2 Global Minimizers of Energies with for Possibly Nonconvex Priors......Page 170
5.3.3 Nonasymptotic Bounds on Minimizers......Page 171
5.4.1 Motivation......Page 173
5.4.2 Assumptions on Potential Functions......Page 174
5.4.3 How It Works on bold0mu mumu RRRRRR......Page 175
5.4.4 Either Smoothing or Edge Enhancement......Page 177
5.4.5 Selection for the Global Minimum......Page 181
5.5 Minimizers Under Nonsmooth Regularization......Page 184
5.5.1 Main Theoretical Result......Page 185
5.5.2 The 1D TV Regularization......Page 187
5.5.3 An Application to Computed Tomography......Page 189
5.6.1 General Theory......Page 190
5.6.2 Applications......Page 196
5.6.3 The L1-TV Case......Page 198
References and Further Reading......Page 199
Compressive Sensing......Page 204
6.1 Introduction......Page 206
6.2.2 Sparse Approximation......Page 209
6.2.4 Compressive Sensing......Page 210
6.3 Mathematical Modeling and Analysis......Page 211
6.3.2 Sparsity and Compression......Page 212
6.3.3 Compressive Sensing......Page 214
6.3.4 The Null Space Property......Page 215
6.3.5 The Restricted Isometry Property......Page 217
6.3.6 Coherence......Page 219
6.3.7 RIP for Gaussian and Bernoulli Random Matrices......Page 220
6.3.8 Random Partial Fourier Matrices......Page 221
6.3.9 Compressive Sensing and Gelfand Widths......Page 223
6.4 Numerical Methods......Page 226
6.4.1 The Homotopy Method......Page 227
6.4.2 Iteratively Reweighted Least Squares......Page 230
6.4.2.2 An Iteratively Reweighted Least Squares Algorithm (IRLS)......Page 231
6.4.2.3 Convergence Properties......Page 232
6.4.2.4 Local Linear Rate of Convergence......Page 234
6.4.3 Numerical Experiments......Page 236
6.5.1 Deterministic Compressed Sensing Matrices......Page 239
6.6 Conclusions......Page 240
6.7 Cross-References......Page 241
Duality and Convex Programming......Page 246
7.1 Introduction......Page 247
7.1.1 Linear Inverse Problems with Convex Constraints......Page 250
7.1.2 Imaging with Missing Data......Page 251
7.1.3 Image Denoising and Deconvolution......Page 253
7.1.4 Inverse Scattering......Page 255
7.1.5 Fredholm Integral Equations......Page 256
7.2 Background......Page 258
7.2.1 Lipschitzian Properties......Page 259
7.2.2 Subdifferentials......Page 260
7.3.1 Fenchel Conjugation......Page 267
7.3.2 Fenchel Duality......Page 270
7.3.3 Applications......Page 272
7.3.4 Optimality and Lagrange Multipliers......Page 274
7.3.5 Variational Principles......Page 276
7.3.6 Fixed Point Theory and Monotone Operators......Page 277
7.4 Case Studies......Page 278
7.4.1 Linear Inverse Problems with Convex Constraints......Page 279
7.4.3 Inverse Scattering......Page 280
7.4.4 Fredholm Integral Equations......Page 281
7.7 Cross-References......Page 283
References and Further Reading......Page 284
EM Algorithms......Page 288
8.1 Maximum Likelihood Estimation......Page 290
8.2 The Kullback–Leibler Divergence......Page 292
8.3.1 The Maximum Likelihood Problem......Page 294
8.3.2 The Bare-Bones EM Algorithm......Page 295
8.3.3 The Bare-Bones EM Algorithm Fleshed Out......Page 296
8.3.4 The EM Algorithm Increases the Likelihood......Page 298
8.4.1 Mixtures of Known Densities......Page 299
8.4.2 A Deconvolution Problem......Page 301
8.4.3 The Deconvolution Problem with Binning......Page 305
8.4.4 Finite Mixtures of Unknown Distributions......Page 308
8.4.5 Empirical Bayes Estimation......Page 310
8.5.2 The Emission Tomography Experiment......Page 311
8.5.3 The Shepp–Vardi EM Algorithm for PET......Page 313
8.6.1 Imaging Macromolecular Assemblies......Page 316
8.6.2 The Maximum Likelihood Problem......Page 317
8.6.3 The EM Algorithm, up to a Point......Page 319
8.7.1 The Need for Regularization......Page 321
8.7.2 Smoothed EM Algorithms......Page 322
8.7.3 Good's Roughness Penalization......Page 323
8.7.4 Gibbs Smoothing......Page 325
8.8.1 The Two Monotonicity Properties......Page 327
8.8.2 Monotonicity of the Shepp–Vardi EM Algorithm......Page 329
8.8.3 Monotonicity for Mixtures......Page 330
8.8.4 Monotonicity of the Smoothed EM Algorithm......Page 332
8.8.5 Monotonicity for Exact Gibbs Smoothing......Page 336
8.9.1 Minimum Cross-Entropy Problems......Page 339
8.9.2 Nonnegative Least Squares......Page 342
8.9.3 Multiplicative Iterative Algorithms......Page 345
8.10.1 The Ordered Subset EM Algorithm......Page 346
8.10.2 The ART and Cimmino–Landweber Methods......Page 349
8.10.3 The MART and SMART Methods......Page 352
8.10.4 Row-Action and Block-Iterative EM Algorithms......Page 354
Iterative Solution Methods......Page 362
9.1.1 Conditions on F......Page 363
9.2 Gradient Methods......Page 365
9.2.1 Nonlinear Landweber Iteration......Page 366
9.2.2 Landweber Iteration in Hilbert Scales......Page 372
9.2.3 Steepest Descent and Minimal Error Method......Page 375
9.3 Newton Type Methods......Page 376
9.3.1 Levenberg-Marquardt and Inexact Newton Methods......Page 377
9.3.2 Further Literature on Inexact Newton Methods......Page 380
9.3.3 Iteratively Regularized Gauss–Newton Method......Page 381
9.3.4 Generalizations of the IRGNM......Page 384
9.3.4.1 Examples of Methods R......Page 388
9.3.5.3 Stochastic Noise Models......Page 390
9.4 Nonstandard Iterative Methods......Page 391
9.4.1 Kaczmarz and Splitting Methods......Page 392
9.4.2 EM Algorithms......Page 394
9.4.3 Bregman Iterations......Page 397
Level Set Methods for Structural Inversion and Image Reconstruction......Page 402
10.1.2 Images and Inverse Problems......Page 404
10.1.3 The Forward and the Inverse Problem......Page 406
10.2.1 Example 1: Microwave Breast Screening......Page 407
10.2.2 Example 2: History Matching in Petroleum Engineering......Page 409
10.2.3 Example 3: Crack Detection......Page 410
10.3 Level Set Representation of Images with Interfaces......Page 411
10.3.1 The Basic Level Set Formulation for Binary Media......Page 412
10.3.2.1 Different Levels of a Single Smooth Level Set Function......Page 413
10.3.2.3 Vector Level Set......Page 414
10.3.2.4 Color Level Set......Page 415
10.3.3.1 A Modification of Color Level Set for Tumor Detection......Page 416
10.3.3.2 A Modification of Color Level Set for Reservoir Characterization......Page 417
10.3.3.3 A Modification of the Classical Level Set Technique for Describing Cracksor Thin Shapes......Page 419
10.4.1 General Considerations......Page 421
10.4.3 Transformations and Velocity Flows......Page 422
10.4.4 Eulerian Derivatives of Shape Functionals......Page 423
10.4.5 The Material Derivative Method......Page 424
10.4.6 Some Useful Shape Functionals......Page 425
10.4.7 The Level Set Framework for Shape Evolution......Page 426
10.5 Shape Evolution Driven by Geometric Constraints......Page 427
10.5.1 Penalizing Total Length of Boundaries......Page 428
10.5.2 Penalizing Volume or Area of Shape......Page 429
10.6.1.1 Least Squares Cost Functionals and Gradient Directions......Page 430
10.6.1.2 Change of b due to Shape Deformations......Page 431
10.6.1.3 Variation of Cost due to Velocity Field v(x)......Page 432
10.6.1.4 Example: Shape Variation for TM-Waves......Page 433
10.6.1.5 Example: Evolution of Thin Shapes (Cracks)......Page 434
10.6.2 Shape Sensitivity Analysis and the Speed Method......Page 435
10.6.2.2 Shape Derivatives by a Min-Max Principle......Page 436
10.6.3 Formal Shape Evolution Using the Heaviside Function......Page 437
10.6.3.1 Example: Breast Screening–Smoothly Varying Internal Profiles......Page 438
10.6.3.2 Example: Reservoir Characterization–Parameterized Internal Profiles......Page 440
10.7.1 Regularization by Smoothed Level Set Updates......Page 441
10.7.3 Regularization by Smooth Velocity Fields......Page 444
10.8.1 Shape Evolution and Shape Optimization......Page 445
10.8.2 Some Remarks on Numerical Shape Evolution with Level Sets......Page 447
10.8.3 Speed of Convergence and Local Minima......Page 448
10.8.4 Topological Derivatives......Page 449
10.9.1 Case Study: Microwave Breast Screening......Page 451
10.9.2 Case Study: History Matching in Petroleum Engineering......Page 454
Acknowledgments......Page 457
Expansion Methods......Page 462
11.1 Introduction......Page 464
11.2.1 Physical Principles......Page 465
11.2.2 Mathematical Model......Page 466
11.2.3 Asymptotic Analysis of the Voltage Perturbations......Page 467
11.2.4.1 Detection of a Single Anomaly: A Projection-Type Algorithm......Page 469
11.2.5 Bibliography and Open Questions......Page 471
11.3.1 Physical Principles......Page 472
11.3.2 Asymptotic Formulas in the Frequency Domain......Page 473
11.3.3 Asymptotic Formulas in the Time Domain......Page 474
11.3.4.1 MUSIC-Type Imaging at a Single Frequency......Page 476
11.3.4.2 Backpropagation-Type Imaging at a Single Frequency......Page 477
11.3.4.3 Kirchhoff-Type Imaging Using a Broad Range of Frequencies......Page 478
11.3.4.4 Time-Reversal Imaging......Page 479
11.3.5 Bibliography and Open Questions......Page 481
11.4.2 Asymptotic Analysis of Temperature Perturbations......Page 482
11.4.3.1 Detection of a Single Anomaly......Page 484
11.4.3.2 Detection of Multiple Anomalies: A MUSIC-Type Algorithm......Page 485
11.4.4 Bibliography and Open Questions......Page 487
11.5.1 Physical Principles......Page 488
11.5.2 Mathematical Model......Page 489
11.5.3 Substitution Algorithm......Page 490
11.6 Magneto-Acoustic Imaging......Page 492
11.6.1.2 Mathematical Model......Page 493
11.6.1.3 Substitution Algorithm......Page 494
11.6.2.2 Mathematical Model......Page 496
11.6.2.3 Reconstruction Algorithm......Page 497
11.7.1 Physical Principles......Page 498
11.7.2 Mathematical Model......Page 499
11.7.3 Asymptotic Analysis of Displacement Fields......Page 501
11.7.4 Numerical Methods......Page 503
11.7.5 Bibliography and Open Questions......Page 504
11.8.2 Mathematical Model......Page 505
11.8.3.1 Determination of Location......Page 506
11.8.3.2 Estimation of Absorbing Energy......Page 507
11.8.3.3 Reconstruction of the Absorption Coefficient......Page 508
11.8.4 Bibliography and Open Questions......Page 509
11.10 Cross-References......Page 510
Sampling Methods......Page 516
12.1 Introduction and Historical Background......Page 517
12.2 The Factorization Method in Impedance Tomography......Page 519
12.2.1 Impedance Tomography in the Presence of Insulating Inclusions......Page 520
12.2.2 Conducting Obstacles......Page 527
12.2.3 Local Data......Page 533
12.2.4.1 The Half Space Problem......Page 534
12.2.4.2 The Crack Problem......Page 536
12.3 The Factorization Method in Inverse Scattering Theory......Page 537
12.3.1 Inverse Acoustic Scattering by a Sound-Soft Obstacle......Page 538
12.3.2 Inverse Electromagnetic Scattering by an Inhomogeneous Medium......Page 543
12.3.3 Historical Remarks and Open Questions......Page 548
12.4.1 The Linear Sampling Method......Page 549
12.4.2 MUSIC......Page 551
12.4.3 The Singular Sources Method......Page 555
12.4.4 The Probe Method......Page 557
12.5 Appendix......Page 559
12.6 Cross-References......Page 562
Inverse Scattering......Page 566
13.1 Introduction......Page 567
13.2.1 The Helmholtz Equation......Page 571
13.2.2 Obstacle Scattering......Page 573
13.2.3 Scattering by an Inhomogeneous Medium......Page 576
13.2.4 The Maxwell Equations......Page 577
13.2.5 Historical Remarks......Page 581
13.3.1 Scattering by an Obstacle......Page 582
13.3.2 Scattering by an Inhomogeneous Medium......Page 584
13.4.1 Newton Iterations in Inverse Obstacle Scattering......Page 586
13.4.2 Decomposition Methods......Page 589
13.4.3 Iterative Methods Based on Huygens' Principle......Page 591
13.4.4 Newton Iterations for the Inverse Medium Problem......Page 596
13.4.5 Least Squares Methods for the Inverse Medium Problem......Page 597
13.4.6 Born Approximation......Page 598
13.5.1 The Far Field Operator and Its Properties......Page 599
13.5.2 The Linear Sampling Method......Page 601
13.5.3 The Factorization Method......Page 604
13.5.4 Lower Bounds for the Surface Impedance......Page 605
13.5.5 Transmission Eigenvalues......Page 608
13.6 Cross-References......Page 609
Electrical Impedance Tomography......Page 614
14.1 Introduction......Page 616
14.1.1 Measurement Systems and Physical Derivation......Page 617
14.1.2 The Concentric Anomaly: A Simple Example......Page 621
14.1.3 Measurements with Electrodes......Page 623
14.2 Uniqueness of Solution......Page 627
14.2.1.1 Calderón's Paper......Page 628
14.2.1.3 Complex Geometrical Optics Solutions for the Schrödinger Equation......Page 631
14.2.1.4 Dirichlet-to-Neumann Map and Cauchy Data for the SchrödingerEquation......Page 633
14.2.1.5 Global Uniqueness for n3......Page 634
14.2.1.6 Global Uniqueness in the Two-Dimensional Case......Page 636
14.2.1.9 Global Stability for n3......Page 638
14.2.1.11 Some Open Problems for the Stability......Page 639
14.2.2.1 Non-uniqueness......Page 640
14.2.2.2 Uniqueness up to Diffeomorphism......Page 642
14.2.2.3 Anisotropy which is Partially a Priori Known......Page 645
14.2.3.1 EIT with Partial Data......Page 646
14.2.3.2 The Neumann-to-Dirichlet Map......Page 647
14.3.1 Locating Objects and Boundaries......Page 649
14.3.2 Forward Solution......Page 651
14.3.3 Regularized Linear Methods......Page 654
14.3.4 Regularized Iterative Nonlinear Methods......Page 655
14.3.5 Direct Nonlinear Solution......Page 661
14.4 Conclusions......Page 664
Synthetic Aperture Radar Imaging......Page 670
15.2 Historical Background......Page 672
15.3.2 Basic Facts About the Wave Equation......Page 674
15.3.3.1 The Lippmann–Schwinger Integral Equation......Page 675
15.3.3.3 The Born Approximation......Page 676
15.3.5 Model for the Scattered Field......Page 677
15.3.6 The Matched Filter......Page 678
The Effect of Matched Filtering on Radar Data......Page 679
15.3.8 The Range Profile......Page 680
Radar Data from Rotating Targets......Page 682
15.4.1.1 The Data Collection Manifold......Page 683
15.4.1.2 ISAR in the Time Domain......Page 684
15.4.2 Synthetic-Aperture Radar......Page 686
15.4.2.1 Spotlight SAR......Page 687
15.4.2.2 Stripmap SAR......Page 688
Other SAR Algorithms......Page 689
15.4.3 Resolution for ISAR and Spotlight SAR......Page 690
15.4.3.1 Down-Range Resolution in the Small-Angle Case......Page 691
15.4.3.2 Cross-Range Resolution in the Small-Angle Case......Page 692
Inversion by Filtered Backprojection......Page 693
15.5.2 Range Alignment*-24pt......Page 695
15.6.1 Problems Related to Unmodeled Motion......Page 698
15.6.2 Problems Related to Unmodeled Scattering Physics......Page 699
15.6.3 New Applications of Radar Imaging......Page 701
References and Further Reading......Page 702
Tomography......Page 706
16.1 Introduction......Page 707
16.2 Background......Page 708
16.3 Mathematical Modeling and Analysis......Page 709
16.4 Numerical Methods and Case Examples......Page 726
References and Further Reading......Page 746
Optical Imaging......Page 750
17.2 Background......Page 752
17.2.1 Spectroscopic Measurements......Page 753
17.2.2 Imaging Systems......Page 754
17.3.1 Radiative Transfer Equation......Page 755
17.3.2 Diffusion Approximation......Page 757
17.3.2.1 Boundary Conditions for the DA......Page 758
17.3.2.3 Validity of the DA......Page 760
17.3.3 Hybrid Approaches Utilizing the DA......Page 761
17.3.4 Green's Functions and the Robin to Neumann Map......Page 762
17.3.5 The Forward Problem......Page 763
17.3.6 Schrödinger Form......Page 764
17.3.7.1 Born Approximation......Page 765
17.3.7.2 Rytov Approximation......Page 766
17.3.8 Linearization......Page 768
17.3.8.1 Linear Approximations......Page 769
17.3.8.2 Sensitivity Functions......Page 770
17.3.9 Adjoint Field Method......Page 771
17.3.10 Light Propagation and Its Probabilistic Interpretation......Page 772
17.4.1 Image Reconstruction in Optical Tomography......Page 776
17.4.2.1 Bayesian Formulation for the Inverse Problem......Page 777
17.4.2.2 Inference......Page 778
17.4.2.3 Likelihood and Prior Models......Page 779
17.4.2.4 Nonstationary Problems......Page 780
17.4.2.5 Approximation Error Approach......Page 781
17.4.3 Experimental Results......Page 783
17.4.3.1 Experiment and Measurement Parameters......Page 784
17.4.3.2 Prior Model......Page 785
17.4.3.3 Selection of FEM Meshes and Discretization Accuracy......Page 786
17.4.3.4 Construction of Error Models......Page 787
17.4.3.5 Computation of the MAP Estimates......Page 788
References and Further Reading......Page 791
Photoacoustic and Thermoacoustic Tomography: Image Formation Principles......Page 796
18.1 Introduction......Page 798
18.2.1 The Thermoacoustic Effect and Signal Generation......Page 799
18.2.2 Image Contrast in Laser-Based PAT......Page 802
18.2.3 Image Contrast in RF-Based PAT......Page 803
18.2.4 Functional PAT......Page 804
18.3.1 PAT Imaging Models in Their Continuous Forms......Page 806
18.3.2 Universal Backprojection Algorithm......Page 807
18.3.3 The Fourier-Shell Identity......Page 808
18.3.3.1 Special Case: Planar Measurement Geometry......Page 809
18.3.4.1 Effects of Finite Transducer Bandwidth......Page 810
18.3.4.2 Effects of Non-Point-Like Transducers......Page 812
18.4.1 Frequency-Dependent Acoustic Attenuation......Page 813
18.4.2 Weak Variations in the Speed-of-Sound Distribution......Page 815
18.5.1 Data Redundancies......Page 816
18.5.2 Mitigation of Image Artifacts Due to Acoustic Heterogeneities......Page 817
18.6.1 Continuous-to-Discrete Imaging Models......Page 819
18.6.2 Finite-Dimensional Object Representations......Page 821
18.6.3 Discrete-to-Discrete Imaging Models......Page 822
18.6.3.1 Numerical Example: Impact of Representation Error on ComputedPressure Data......Page 823
18.6.4 Iterative Image Reconstruction......Page 824
18.6.4.1 Numerical Example: Influence of Representation Error onImage Accuracy......Page 825
18.8 Cross-References......Page 827
Mathematics of Photoacoustic and Thermoacoustic Tomography......Page 832
19.1 Introduction......Page 834
19.2.1 Point Detectors and the Wave Equation Model......Page 835
19.2.2 Acoustically Homogeneous Media and Spherical Means......Page 836
19.2.3 Main Mathematical Problems Arising in TAT......Page 837
19.2.4 Variations on the Theme: Planar, Linear, and CircularIntegrating Detectors......Page 839
19.3.1 Uniqueness of Reconstruction......Page 841
19.3.1.1 Acoustically Homogeneous Media......Page 842
General Acquisition Sets S......Page 843
Uniqueness Results for a Finite Observation Time......Page 845
Trapping and Non-trapping......Page 846
Uniqueness Results for Finite Observation Times......Page 847
19.3.2 Stability......Page 848
19.3.3 Incomplete Data......Page 849
Uniqueness for Acoustically Inhomogeneous Media......Page 850
19.3.3.2 ``Visible'' (``audible'') Singularities......Page 851
19.3.3.3 Stability of Reconstruction for Incomplete Data Problems......Page 853
19.3.4.2 Visibility for Acoustically Inhomogeneous Media......Page 854
19.3.5 Range Conditions......Page 855
19.3.5.1 The Range of the Spherical Mean Operator bold0mu mumu MMMMMM......Page 856
19.3.5.2 The Range of the Forward Operator bold0mu mumu WWWWWW......Page 857
19.3.6 Reconstruction of the Speed of Sound......Page 858
Series Solutions for Spherical Geometry......Page 860
Eigenfunction Expansions for a General Geometry......Page 861
Closed-Form Inversion Formulas......Page 863
Greens' Formula Approach and Some Symmetry Considerations......Page 865
Parametrix Approaches......Page 867
Numerical Implementation and Computational Examples......Page 868
Time Reversal......Page 869
Eigenfunction Expansions......Page 870
19.4.2 Partial (Incomplete) Data......Page 871
19.4.2.1 Constant Speed of Sound......Page 872
19.4.2.2 Variable Speed of Sound......Page 874
19.5 Final Remarks and Open Problems......Page 875
19.6 Cross-References......Page 876
Wave Phenomena......Page 882
20.1 Introduction......Page 883
20.2.1 Wave Imaging and Boundary Control Method......Page 884
20.2.2 Travel Times and Scattering Relation......Page 886
20.2.3 Curvelets and Wave Equations......Page 887
20.3.1.1 Inverse Problems on Riemannian Manifolds......Page 888
20.3.1.2 From Boundary Distance Functions to Riemannian Metric......Page 890
20.3.1.3 From Boundary Data to Inner Products of Waves......Page 900
20.3.1.4 From Inner Products of Waves to Boundary Distance Functions......Page 903
20.3.1.5 Alternative Reconstruction of Metric via Gaussian Beams......Page 905
20.3.2 Travel Times and Scattering Relation......Page 907
20.3.2.1 Geometrical Optics......Page 908
The Eikonal Equation......Page 909
The Transport Equation......Page 910
20.3.2.2 Scattering Relation......Page 911
20.3.3 Curvelets and Wave Equations......Page 912
20.3.3.1 Curvelet Decomposition......Page 913
20.3.3.2 Curvelets and Wave Equations......Page 915
20.3.3.3 Low Regularity Wave Speeds and Volterra Iteration......Page 918
20.4 Conclusion......Page 920
20.5 Cross-References......Page 921
Statistical Methods in Imaging......Page 926
21.1 Introduction......Page 927
21.2.2 Randomness, Distributions and Lack of Information......Page 928
21.2.3 Imaging Problems......Page 931
21.3.2 Accumulation of Information and Priors......Page 932
21.3.3 Likelihood: Forward Model and Statistical Properties of Noise......Page 936
21.3.4 Maximum Likelihood and Fisher Information......Page 939
21.3.5 Informative or Noninformative Priors?......Page 940
21.3.6 Adding Layers: Hierarchical Models......Page 941
21.4.1 Estimators......Page 943
21.4.1.2 Maximum Likelihood and Maximum A Posteriori......Page 944
21.4.1.3 Conditional Means......Page 947
21.4.2 Algorithms......Page 948
21.4.2.1 Iterative Linear Least Squares Solvers......Page 950
21.4.2.3 EM Algorithm......Page 951
21.4.2.4 Markov Chain Monte Carlo Sampling......Page 954
21.4.3.1 Beyond the Traditional Concept of Noise......Page 961
21.4.3.2 Sparsity and Hypermodels......Page 965
21.5 Conclusion......Page 967
References and Further Reading......Page 968
Supervised Learning by Support Vector Machines......Page 972
22.1 Introduction......Page 973
22.2 Background......Page 975
22.3.1.1 Linear Support Vector Classification......Page 977
22.3.1.2 Linear Support Vector Regression......Page 982
22.3.1.3 Linear Least Squares Classification and Regression......Page 985
22.3.2 Nonlinear Learning......Page 988
22.3.2.1 Kernel Trick......Page 989
22.3.2.2 Support Vector Classification......Page 990
22.3.2.3 Support Vector Regression......Page 992
22.3.2.4 Relations to Sparse Approximation in RKHSs, Interpolation by RadialBasis Functions and Kriging......Page 993
22.3.2.5 Least Squares Classification and Regression......Page 996
22.3.2.6 Other Models......Page 997
22.3.2.7 Multi-class Classification and Multitask Learning......Page 998
22.3.2.8 Applications of SVMs......Page 1002
22.4.1 Reproducing Kernel Hilbert Spaces......Page 1005
22.4.2 Quadratic Optimization......Page 1011
22.4.3 Results from Generalization Theory......Page 1015
22.5 Numerical Methods......Page 1020
References and Further Reading......Page 1022
Total Variation in Imaging......Page 1028
23.1 Introduction......Page 1030
23.2.2 Sets of Finite Perimeter: The Coarea Formula......Page 1034
23.2.3 The Structure of the Derivative of a BV Function......Page 1035
23.3.1 The Discontinuities of Solutions of the TV Denoising Problem......Page 1036
23.3.2 Hölder Regularity Results......Page 1040
23.4 Mathematical Analysis: Some Explicit Solutions......Page 1041
23.5.1 Notation......Page 1044
23.5.2 Chambolle's Algorithm......Page 1045
23.5.3 Primal-Dual Approaches......Page 1046
23.6 Numerical Methods: Maximum Flow Methods......Page 1048
23.6.1 Discrete Perimeters and Discrete Total Variation......Page 1049
23.6.2 Graph Representation of Energies for Binary MRF......Page 1050
23.7.1 Global Solutions of Geometric Problems......Page 1053
23.7.2 A Convex Formulation of Continuous Multi-label Problems......Page 1056
23.8 Other Problems: Image Restoration......Page 1058
23.8.1 Some Restoration Experiments......Page 1061
23.8.2 The1pc Image Model*-24pt......Page 1062
23.9 Final Remarks: A Different Total Variation-Based Approachto Denoising......Page 1065
Acknowledgement......Page 1067
References and Further Reading......Page 1068
Numerical Methods and Applications in Total Variation Image Restoration......Page 1072
24.2 Background......Page 1074
24.3.1.1 Basic Definition......Page 1075
24.3.1.2 Multichannel TV......Page 1076
24.3.1.4 Discrete TV......Page 1077
24.3.1.5 Nonlocal TV......Page 1078
24.3.2.1 Inpainting in Transformed Domains......Page 1079
24.3.2.2 Superresolution......Page 1081
24.3.2.3 Image Segmentation......Page 1082
24.3.2.4 Diffusion Tensors Images......Page 1084
24.4 Numerical Methods and Case Examples......Page 1085
24.4.1.1 Chan–Golub–Mulet's Primal-Dual Method......Page 1086
24.4.1.2 Chambolle's Dual Method......Page 1087
24.4.1.3 Primal-Dual Hybrid Gradient Method......Page 1089
24.4.1.5 Primal-Dual Active-Set Method......Page 1090
24.4.2.1 Original Bregman Iteration......Page 1092
24.4.2.3 Split Bregman Iteration......Page 1093
24.4.2.4 Augmented Lagrangian Method-12pt......Page 1094
24.4.3 Graph Cut Methods......Page 1095
24.4.3.1 Leveling the Objective......Page 1096
24.4.3.2 Defining a Graph......Page 1097
24.4.4 Quadratic Programming......Page 1098
24.4.5 Second-Order Cone Programming......Page 1099
24.4.6 Majorization-Minimization......Page 1100
24.4.7 Splitting Methods......Page 1102
References and Further Reading......Page 1104
Mumford and Shah Model and its Applications to Image Segmentation and Image Restoration......Page 1108
25.1 Introduction: Description of the Mumford and Shah Model......Page 1110
25.2 Background: The First Variation......Page 1111
25.2.1 Minimizing in u with K Fixed......Page 1112
25.2.2 Minimizing in K......Page 1115
25.3 Mathematical Modeling and Analysis: The Weak Formulationof the Mumford and Shah Functional......Page 1117
25.4 Numerical Methods: Approximations to the Mumfordand Shah Functional......Page 1119
25.4.1.1 Approximations of the Perimeter by Elliptic Functionals......Page 1120
25.4.1.2 Ambrosio-Tortorelli Approximations......Page 1121
25.4.2 Level Set Formulations of the Mumford and Shah Functional......Page 1122
25.4.2.1 Piecewise-Constant Mumford and Shah Segmentation Using Level Sets......Page 1127
25.4.2.2 Piecewise-Smooth Mumford and Shah Segmentation Using Level Sets......Page 1132
25.4.2.3 Extension to Level Set Based Mumford–Shah Segmentationwith Open Edge Set K......Page 1136
25.5 Case Examples: Variational Image Restoration with Segmentation-BasedRegularization......Page 1141
25.5.1 Non-blind Restoration......Page 1143
25.5.2 Semi-Blind Restoration......Page 1144
25.5.3 Image Restoration with Impulsive Noise......Page 1146
25.5.4 Color Image Restoration......Page 1151
25.5.5 Space-Variant Restoration......Page 1152
25.5.6 Level Set Formulations for Joint Restorationand Segmentation......Page 1155
25.5.7 Image Restoration by Nonlocal Mumford–Shah Regularizers......Page 1158
25.6 Conclusion......Page 1166
References and Further Reading......Page 1167
Local Smoothing Neighborhood Filters......Page 1172
26.1 Introduction......Page 1173
26.2.1 Analysis of Neighborhood Filter as a Denoising Algorithm......Page 1179
26.2.2 Neighborhood Filter Extension: The NL-Means Algorithm......Page 1181
26.2.3 Extension to Movies......Page 1185
26.3.1 PDE Models and Local Smoothing Filters......Page 1188
26.3.2 Asymptotic Behavior of Neighborhood Filters (Dimension 1)......Page 1190
26.3.3 The Two-Dimensional Case......Page 1193
26.3.4 A Regression Correction of the Neighborhood Filter......Page 1196
26.3.5 The Vector-Valued Case......Page 1201
26.3.5.1 Interpretation......Page 1203
26.4 Variational and Linear Diffusion......Page 1204
26.4.1 Linear Diffusion: Seed Growing......Page 1205
26.4.2 Linear Diffusion: Histogram Concentration......Page 1207
References and Further Reading......Page 1211
Neighborhood Filters and the Recovery of 3D Information......Page 1216
27.1 Introduction......Page 1217
Glossary and notation......Page 1218
27.2.1 Bilateral Filter Definitions......Page 1219
27.2.2 Trilateral Filters......Page 1222
27.2.3 Similarity Filters......Page 1223
27.2.4 Summary of 3D Mesh Bilateral Filter Definitions......Page 1225
27.2.5 Comparison of Bilateral Filter and Mean Curvature Motion Filter onArtificial Shapes......Page 1226
27.3.1 Bilateral Filter for Improving the Depth Map Provided by Stereo MatchingAlgorithms......Page 1228
27.3.2 Bilateral Filter for Enhancing the Resolution of Low-QualityRange Images......Page 1233
27.3.3 Bilateral Filter for the Global Integration of Local Depth Information......Page 1236
References and Further Reading......Page 1240
Splines and Multiresolution Analysis......Page 1244
28.1 Introduction......Page 1245
28.3.1.1 Regularity and Decay Under the Fourier Transform......Page 1250
28.3.1.2 Criteria for Riesz Sequences and Multiresolution Analyses......Page 1252
28.3.1.4 Order of Approximation......Page 1254
28.3.1.5 Wavelets......Page 1255
28.3.2 B-Splines......Page 1258
28.3.3 Polyharmonic B-Splines......Page 1261
28.4.1 Schoenberg's B-Splines for Image Analysis – the Tensor ProductApproach......Page 1264
28.4.2 Fractional and Complex B-Splines......Page 1265
28.4.3 Polyharmonic B-Splines and Variants......Page 1268
28.4.4.1 Splines on the Quincunx Lattice......Page 1271
28.4.4.2 Splines on the Hexagonal Lattice......Page 1272
28.5 Numerical Methods......Page 1275
28.6 Open Questions......Page 1278
28.7 Conclusion......Page 1279
References and Further Reading......Page 1280
Gabor Analysis for Imaging......Page 1284
29.2.1 The Pseudo-Inverse Operator......Page 1285
29.2.2 Bessel Sequences in Hilbert Spaces......Page 1287
29.2.3 General Bases and Orthonormal Bases......Page 1288
29.2.4 Frames and Their Properties......Page 1289
29.3 Operators......Page 1290
29.3.1 The Fourier Transform......Page 1291
29.3.2 Translation and Modulation......Page 1292
29.3.4 The Short-Time Fourier Transform......Page 1293
29.4 Gabor Frames in Lnormalnormal2normalnormal(normalnormalRdnormalnormal)......Page 1296
29.5.1 Gabor Frames in 2(Z)......Page 1299
29.5.2 Finite Discrete Periodic Signals......Page 1300
29.5.3 Frames and Gabor Frames in CL......Page 1301
29.6 Image Representation by Gabor Expansion......Page 1303
29.6.1 2D Gabor Expansions......Page 1304
29.6.2 Separable Atoms on Fully Separable Lattices......Page 1306
29.6.3 Efficient Gabor Expansion by Sampled STFT......Page 1309
29.6.4 Visualizing a Sampled STFT of an Image......Page 1311
29.6.5 Non-Separable Atoms on Fully Separable Lattices......Page 1314
29.7 Historical Notes and Hint to the Literature......Page 1316
References and Further Reading......Page 1317
Shape Spaces......Page 1322
30.1 Introduction......Page 1324
30.2 Background......Page 1325
30.3.1 Some Notation......Page 1326
30.3.2.1 Interpolating Splines and RKHSs......Page 1327
30.3.2.2 Riemannian Structure......Page 1329
30.3.2.3 Geodesic Equation......Page 1330
30.3.2.4 Metric Distortion and Curvature......Page 1331
30.3.2.5 Invariance......Page 1332
30.3.3.1 General Principles......Page 1335
30.3.3.3 Momentum Map and Conserved Quantities......Page 1337
30.3.3.4 Euler–Poincaré Equation......Page 1339
30.3.3.6 Application to the Group of Diffeomorphisms......Page 1340
30.3.3.7 Reduction via a Submersion......Page 1343
30.3.3.8 Reduction: Quotient Spaces......Page 1345
30.3.3.9 Reduction: Transitive Group Action......Page 1346
30.3.4.1 Introduction and Notation......Page 1348
30.3.4.2 Some Simple Distances......Page 1349
30.3.4.3 Riemannian Metrics on Curves......Page 1353
30.3.4.4 Projecting the Action of 2D Diffeomorphisms......Page 1358
30.3.5 Extension to More General Shape Spaces......Page 1360
30.3.6 Applications to Statistics on Shape Spaces......Page 1362
30.4 Numerical Methods and Case Examples......Page 1363
30.4.1 Landmark Matching via Shooting......Page 1364
30.4.3 Computing Geodesics Between Curves......Page 1367
30.4.4.1 Inexact Matching......Page 1369
30.4.4.2 Optimal Control Formulation......Page 1370
30.4.4.3 Gradient w.r.t. the Control......Page 1371
30.6 Cross-References......Page 1372
References and Further Reading......Page 1373
Variational Methods in Shape Analysis......Page 1376
31.2 Background......Page 1377
31.3.1 Recalling the Finite-Dimensional Case......Page 1381
31.3.2.1 Path-Based, Viscous Riemannian Setup......Page 1384
31.3.2.2 State-Based, Path-Independent Elastic Setup......Page 1387
31.3.2.3 Conceptual Differences Between the Path- and State-Based DissimilarityMeasures......Page 1390
31.4.1 Elasticity-Based Shape Space......Page 1391
31.4.1.1 Elastic Shape Averaging......Page 1392
31.4.1.2 Elasticity-Based PCA......Page 1394
31.4.2 Viscous Fluid-Based Shape Space......Page 1399
31.4.3 A Collection of Computational Tools......Page 1406
31.4.3.1 Shapes Described by Level Set Functions......Page 1407
31.4.3.2 Shapes Described via Phase Fields......Page 1408
31.4.3.3 Multi-Scale Finite Element Approximation......Page 1409
31.5 Conclusion......Page 1410
References and Further Reading......Page 1411
Manifold Intrinsic Similarity......Page 1416
32.1.1 Problems......Page 1419
32.1.2 Methods......Page 1420
32.2.1.1 Topological Spaces......Page 1421
32.2.2 Euclidean Geometry......Page 1422
32.2.3.2 Differential Structures......Page 1423
32.2.3.4 Embedded Manifolds......Page 1424
32.2.4.1 Diffusion Operators......Page 1425
32.2.5 Diffusion Distances......Page 1427
32.3.1 Sampling......Page 1428
32.3.1.1 Farthest Point Sampling......Page 1429
32.3.2.1 Simplicial Complexes......Page 1430
32.3.2.3 Implicit Surfaces......Page 1431
32.4.1.1 Dijkstra's Algorithm......Page 1432
32.4.2.1 Eikonal Equation......Page 1433
32.4.2.2 Triangular Meshes......Page 1435
32.4.2.3 Parametric Surfaces......Page 1436
32.4.2.5 Implicit Surfaces and Point Clouds......Page 1437
32.4.3 Diffusion Distance......Page 1438
32.4.3.2 Computation of Eigenfunctions and Eigenvalues......Page 1439
32.5 Invariant Shape Similarity......Page 1440
32.5.1.1 Hausdorff Distance......Page 1441
32.5.1.2 Iterative Closest Point Algorithms......Page 1442
32.5.1.4 Wasserstein Distances......Page 1443
32.5.2 Canonical Forms......Page 1444
32.5.2.1 Multidimensional Scaling......Page 1445
32.5.3 Gromov–Hausdorff Distance......Page 1446
32.5.3.1 Generalized Multidimensional Scaling......Page 1447
32.5.4.1 Probabilistic Gromov–Hausdorff Distance......Page 1449
32.5.5.1 Numerical Computation......Page 1450
32.6.1 Significance......Page 1451
32.6.2 Regularity......Page 1452
32.6.3 Partial Similarity Criterion......Page 1453
32.7 Self-Similarity and Symmetry......Page 1454
32.7.2 Intrinsic Symmetry......Page 1455
32.7.5 Repeating Structure......Page 1456
32.8.1.2 Feature Description......Page 1457
32.8.1.4 Scale-Invariant Heat Kernel Signatures......Page 1458
32.8.3 Combining Global and Local Information......Page 1459
References and Further Reading......Page 1460
Image Segmentation with Shape Priors: Explicit Versus Implicit Representations......Page 1466
33.1.1 Image Analysis and Prior Knowledge......Page 1467
33.1.2 Explicit Versus Implicit Shape Representation......Page 1468
33.2 Image Segmentation via Bayesian Inference......Page 1471
33.3 Statistical Shape Priors for Parametric Shape Representations......Page 1472
33.3.1 Linear Gaussian Shape Priors......Page 1473
33.3.2 Nonlinear Statistical Shape Priors......Page 1474
33.4 Statistical Priors for Level Set Representations......Page 1478
33.4.1 Shape Distances for Level Sets......Page 1479
33.4.2 Invariance by Intrinsic Alignment......Page 1480
33.4.2.2 Translation and Scale Invariance via Alignment......Page 1481
33.4.3 Kernel Density Estimation in the Level Set Domain......Page 1482
33.4.4 Gradient Descent Evolution for the Kernel Density Estimator......Page 1485
33.4.5 Nonlinear Shape Priors for Tracking a Walking Person......Page 1486
33.5.2 Level Set Based Tracking via Bayesian Inference......Page 1488
33.5.3 Linear Dynamical Models for Implicit Shapes......Page 1490
33.5.4 Variational Segmentation with Dynamical Shape Priors......Page 1491
33.6 Parametric Representations Revisited: Combinatorial Solutionsfor Segmentation with Shape Priors......Page 1493
33.7 Conclusion......Page 1495
Starlet Transform in Astronomical Data Processing......Page 1502
34.1 Introduction......Page 1504
34.1.1 Source Detection......Page 1505
34.2.1 The Traditional Data Model......Page 1506
34.2.3 Background Estimation......Page 1507
34.2.5 Detection......Page 1508
34.2.7.2 Star–Galaxy Separation......Page 1509
34.2.7.3 Galaxy Morphology Classification......Page 1510
34.3.1 Sparsity Data Model......Page 1511
34.3.2 The Starlet Transform......Page 1512
34.3.3 The Starlet Reconstruction......Page 1514
34.3.4 Starlet Transform: Second Generation......Page 1516
34.3.5 Sparse Modeling of Astronomical Images......Page 1518
34.3.5.1 Selection of Significant Coefficients Through Noise Modeling......Page 1519
34.3.6 Sparse Positive Decomposition......Page 1520
34.3.6.2 Example 2: Sparse positive starlet decomposition of a simulated image......Page 1522
34.4 Source Detection Using a Sparsity Model......Page 1523
34.4.1 Detection Through Wavelet Denoising......Page 1524
34.4.2.1 Introduction......Page 1525
Multiresolution Support Segmentation......Page 1526
Interscale Connectivity Graph......Page 1527
Object Identification......Page 1528
34.4.3.1 Band Extraction......Page 1529
34.4.3.3 Galaxy Nucleus Extraction......Page 1531
34.5 Deconvolution......Page 1532
34.5.1 Statistical Approach to Deconvolution......Page 1533
34.5.3 Deconvolution with a Sparsity Prior......Page 1536
34.5.3.1 Constraints in the Object or Image Domains......Page 1538
34.5.4.1 Object Reconstruction Using the PSF......Page 1539
34.5.4.3 Space-Variant PSF......Page 1540
34.5.4.5 Example: Application to Abell 1689 ISOCAM Data......Page 1541
34.7 Cross-References......Page 1542
Differential Methods for Multi-Dimensional Visual Data Analysis......Page 1546
35.1 Introduction......Page 1548
35.2 Modeling Data via Fiber Bundles......Page 1550
35.2.1.1 Tangential Vectors......Page 1551
35.2.1.3 Tensors......Page 1552
35.2.1.4 Exterior Product......Page 1554
35.2.1.5 Visualizing Exterior Products......Page 1555
35.2.1.7 Vector and Fiber Bundles......Page 1557
35.2.2 Topology: Discretized Manifolds......Page 1558
35.2.3 Ontological Scheme and Seven-Level Hierarchy......Page 1559
35.2.3.1 Field Properties......Page 1562
35.2.3.2 Topological Skeletons......Page 1563
35.2.3.3 Non-topological Representations......Page 1565
35.3.1 Differential Forms......Page 1566
35.3.1.1 Chains......Page 1568
35.3.1.2 Cochains......Page 1571
35.3.1.3 Duality between Chains and Cochains......Page 1573
35.3.2 Homology and Cohomology......Page 1575
35.3.3 Topology......Page 1577
35.4 Geometric Algebra Computing......Page 1579
35.4.1.1 Unification of Mathematical Systems......Page 1580
35.4.1.2 Uniform Handling of Different Geometric Primitives......Page 1581
35.4.1.3 Simplified Geometric Operations......Page 1582
35.4.2 Conformal Geometric Algebra......Page 1583
35.4.3 Computational Efficiency of Geometric Algebra Using Gaalop......Page 1585
35.5.1 Characteristic Curves of Vector Fields......Page 1587
35.5.2 Derived Measures of Vector Fields......Page 1590
35.5.3.1 Critical Points......Page 1592
3D Vector Fields......Page 1594
35.5.3.3 Application......Page 1595
35.6.1 Regularization PDE's : A review......Page 1596
35.6.1.1 Local Multi-valued Geometry and Diffusion Tensors......Page 1597
35.6.1.2 Divergence-based PDE's......Page 1598
35.6.1.3 Trace-based PDE's......Page 1599
35.6.1.4 Curvature-Preserving PDE's......Page 1600
35.6.2 Applications......Page 1602
35.6.2.1 Color Image Denoising......Page 1603
35.6.2.2 Color Image Inpainting......Page 1604
35.6.2.3 Visualization of Vector and Tensor Fields......Page 1605
Index......Page 1610