Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, Second Edition (Applied Mathematical Sciences)

Contents......Page 16
Foreword......Page 7
Preface to the Second Edition......Page 10
Preface to the First Edition......Page 13
Guide to the Main Mathematical Concepts and Their Application......Page 21
Notation and Symbols......Page 23
1.1 The Image Society......Page 28
1.2 What Is a Digital Image?......Page 30
1.4 Detailed Plan......Page 32
How to Read This Chapter......Page 55
2.1.1 Topologies on Banach Spaces......Page 56
2.1.2 Convexity and Lower Semicontinuity......Page 58
2.1.3 Relaxation......Page 63
2.1.4 About Γ-Convergence......Page 66
2.2 The Space of Functions of Bounded Variation......Page 68
2.2.1 Basic Definitions on Measures......Page 69
2.2.2 Definition of BV(Ω)......Page 71
2.2.3 Properties of BV(Ω)......Page 72
2.3.1 About the Eikonal Equation......Page 76
2.3.2 Definition of Viscosity Solutions......Page 78
2.3.3 About the Existence......Page 80
2.3.4 About the Uniqueness......Page 81
2.4 Elements of Differential Geometry: Curvature......Page 83
2.4.2 Curves as Isolevel of a Function u......Page 84
2.4.3 Images as Surfaces......Page 85
2.5.1 Inequalities......Page 86
2.5.3 About Convolution and Smoothing......Page 88
2.5.4 Uniform Convergence......Page 89
2.5.6 Well-Posed Problems......Page 90
How to Read This Chapter......Page 91
3.1 Image Degradation......Page 92
3.2.1 An Inverse Problem......Page 94
3.2.2 Regularization of the Problem......Page 95
3.2.3 Existence and Uniqueness of a Solution for the Minimization Problem......Page 98
3.2.4 Toward the Numerical Approximation......Page 102
3.2.5 Some Invariances and the Role of λ......Page 113
3.2.6 Some Remarks on the Nonconvex Case......Page 116
3.3 PDE-Based Methods......Page 120
3.3.1 Smoothing PDEs......Page 121
3.3.2 Smoothing–Enhancing PDEs......Page 147
3.3.3 Enhancing PDEs......Page 154
3.3.4 Neighborhood Filters, Nonlocal Means Algorithm, and PDEs......Page 163
How to Read This Chapter......Page 174
4.1 Definition and Objectives......Page 175
4.2.1 A Minimization Problem......Page 178
4.2.2 The Mathematical Framework for the Existence of a Solution......Page 179
4.2.3 Regularity of the Edge Set......Page 187
4.2.4 Approximations of the Mumford and Shah Functional......Page 191
4.2.5 Experimental Results......Page 196
4.3.1 The Kass–Witkin–Terzopoulos model......Page 198
4.3.2 The Geodesic Active Contours Model......Page 200
4.3.3 The Level-Set Method......Page 207
4.3.4 The Reinitialization Equation......Page 219
4.3.5 Experimental Results......Page 231
4.3.6 About Some Recent Advances......Page 233
How to Read This Chapter......Page 237
5.1.1 Introduction......Page 239
5.1.2 Variational Models......Page 240
5.1.3 PDE-Based Approaches......Page 246
5.1.4 Discussion......Page 249
5.2.1 Introduction......Page 252
5.2.2 A Space for Modeling Oscillating Patterns......Page 253
5.2.3 Meyer's Model......Page 256
5.2.4 An Algorithm to Solve Meyer's Model......Page 257
5.2.5 Experimental Results......Page 269
5.2.6 About Some Recent Advances......Page 272
5.3.1 Introduction......Page 273
5.3.2 The Optical Flow: An Apparent Motion......Page 274
5.3.3 Sequence Segmentation......Page 285
5.3.4 Sequence Restoration......Page 295
5.4.1 Introduction......Page 305
5.4.2 A Level-Set Approach for Image Classification......Page 306
5.4.3 A Variational Model for Image Classification and Restoration......Page 314
5.5.1 Introduction......Page 323
5.5.3 The Energy Method......Page 324
5.5.4 PDE-Based Methods......Page 326
How to Read This Chapter......Page 330
A.1.1 Getting Started......Page 331
A.1.2 Convergence......Page 334
A.1.4 Consistency......Page 336
A.1.5 Stability......Page 338
A.2 Hyperbolic Equations......Page 343
A.3.1 Getting Started......Page 352
A.3.2 Image Restoration by Energy Minimization......Page 356
A.3.3 Image Enhancement by the Osher and Rudin Shock Filters......Page 359
A.3.4 Curve Evolution with the Level-Set Method......Page 361
How to Read This Chapter......Page 366
B.2 What Is Available Online?......Page 367
References......Page 371
C......Page 394
I......Page 395
P......Page 396
U......Page 397
W......Page 398