Shearlets : multiscale analysis for multivariate data

Cover......Page 1
Shearlets......Page 4
ANHA Series Preface......Page 6
Preface......Page 10
Contents......Page 14
Contributors......Page 20
Image Processing Using Shearlets......Page 22
2.1 The Role of Applied Harmonic Analysis......Page 24
2.2 Wavelets and Beyond......Page 25
3.1 Fourier Analysis......Page 27
3.2 Modeling of Signal Classes......Page 28
3.3 Frame Theory......Page 30
3.4 Wavelets......Page 31
3.5 Wavelets for Multivariate Data and Their Limitations......Page 34
4 Continuous Shearlet Systems......Page 36
4.2 The Continuous Shearlet Transform......Page 38
4.3 Cone-Adapted Continuous Shearlet Systems......Page 41
4.4 The Cone-Adapted Continuous Shearlet Transform......Page 42
4.5 Microlocal Properties and Characterization of Singularities......Page 43
5 Discrete Shearlet Systems......Page 44
5.1 Discrete Shearlet Systems and Transforms......Page 45
5.2 Cone-Adapted Discrete Shearlet Systems and Transforms......Page 46
5.3 Compactly Supported Shearlets......Page 49
5.4 Sparse Approximations by Shearlets......Page 51
5.6 Extensions and Generalizations......Page 53
6 Algorithmic Implementations of the Shearlet Transform......Page 54
6.2 Spatial-Domain-Based Implementations......Page 55
7 Shearlets in Applications......Page 56
References......Page 57
1 Introduction......Page 60
1.1 Notation......Page 61
1.2 Getting to Know the Wavefront Set......Page 62
1.2.1 Shearlets and the wavefront set......Page 65
1.2.2 Wavelets and the wavefront set......Page 68
1.4 Other Ways to Characterize the Wavefront Set......Page 70
2 Reproduction Formulas......Page 71
3.1 A Direct Theorem......Page 76
3.2 Properties of the Wavefront Set......Page 80
3.3 Proof of the Main Result......Page 82
References......Page 88
1 Introduction......Page 89
1.1 Example: Line Singularity......Page 90
1.2 General Singularities......Page 95
2 Analysis of Step Singularities (2D)......Page 96
2.1 Shearlet Analysis of Circular Edges......Page 98
2.2 General 2D Boundaries......Page 101
2.3 Proofs of Theorems 2 and 3......Page 103
2.4 Extensions and Generalizations......Page 117
3 Extension to Higher Dimensions......Page 118
3.1 3D Continuous Shearlet Transform......Page 119
3.2 Characterization of 3D Boundaries......Page 120
References......Page 123
1 Introduction......Page 125
2.1 Unitary Representations of the Shearlet Group......Page 126
2.2 Square Integrable Representations of the Shearlet Group......Page 129
3 General Concept of Coorbit Space Theory......Page 132
3.1 General Coorbit Spaces......Page 134
3.2 Atomic Decompositions and Banach Frames......Page 135
4.1 Shearlet Coorbit Spaces......Page 136
4.2 Shearlet Atomic Decompositions and Shearlet Banach Frames......Page 137
4.3 Nonlinear Approximation......Page 138
5 Structure of Shearlet Coorbit Spaces......Page 140
5.1 Atomic Decomposition of Besov Spaces......Page 144
5.2 A Density Result......Page 145
5.3 Traces on the Real Axes......Page 146
5.4 Embedding Results......Page 150
6.1 Hyperplane Singularities......Page 153
6.2 Tetrahedron Singularities......Page 156
References......Page 161
1.1 Choice of Model for Anisotropic Features......Page 165
1.3 Why is 3D the Crucial Dimension?......Page 166
1.5 Band-Limited Versus Compactly Supported Systems......Page 167
2 Cartoon-Like Image Class......Page 168
3.1 (Nonlinear) N-term Approximations......Page 170
3.1.1 Orthonormal bases......Page 171
3.1.3 General Frames......Page 172
3.2 A Notion of Optimality......Page 174
3.3.1 Fourier series......Page 177
3.3.2 Wavelets......Page 178
3.3.3 Key problem......Page 179
4 Pyramid-Adapted Shearlet Systems......Page 180
4.1 General Definition......Page 181
4.2 Band-Limited 3D Shearlets......Page 183
4.3 Compactly Supported 3D Shearlets......Page 185
4.4 Some Remarks on Construction Issues......Page 188
5.1 Optimal Sparse Approximations in 2D......Page 189
5.1.1 A heuristic analysis......Page 190
5.1.2 Required hypotheses......Page 191
5.1.3 Main result......Page 194
5.1.4 Band-limitedness vs. compactly supportedness......Page 195
5.1.5 Proof for band-limited shearlets for L=1......Page 197
5.1.6 Proof for compactly supported shearlets for L=1......Page 199
5.1.7 The case L =1......Page 207
5.2.1 A heuristic analysis......Page 210
5.2.2 Main result......Page 211
5.2.3 Sketch of proof of Theorem 11......Page 212
5.2.4 Some extensions......Page 214
References......Page 215
1 Introduction......Page 217
2.1 Filterbanks......Page 219
2.2 Symbols and Transforms......Page 222
2.3 Filterbanks by Matrix Completion......Page 226
2.4 Subbands and Multiresolution......Page 228
3.1 Convergence and Basic Properties......Page 230
3.2 Interpolatory Subdivision and Filterbanks......Page 232
3.3 Multiresolution......Page 234
4 Multiple Subdivision and Multiple Refinability......Page 236
4.1 Basic Properties......Page 237
4.2 The Multiple MRA......Page 239
4.3 Filterbanks, Cascades, Trees......Page 241
4.4 Things Work Along Trees......Page 244
4.5 A Canonical Interpolatory Construction......Page 245
5 Shearlet Subdivision and Multiresolution......Page 247
5.1 Shears and Scaling......Page 248
5.2 Shears of Codimension 1: Hyperplane Shearlets......Page 249
5.3 Orthogonal Shearlets by Tensor Product......Page 251
5.4 Implementation......Page 252
References......Page 254
1.1 A Unified Framework for the Continuum and Digital World......Page 257
1.2 Band-Limited vs. Compactly Supported Shearlet Transforms......Page 258
1.4 Framework for Quantifying Performance......Page 259
1.6 Outline......Page 260
2 Digital Shearlet Transform Using Band-Limited Shearlets......Page 261
2.1.1 Pseudo-polar grids with oversampling......Page 262
2.1.2 Fast PPFT......Page 263
2.2.1 A Plancherel theorem for the PPFT......Page 265
2.2.2 Relaxed form of weight functions......Page 266
2.2.3 Computation of the weighting......Page 267
2.3.1 Preparation for faithful digitization......Page 268
2.3.2 Subband windows on the pseudo-polar grid......Page 269
2.3.3 Range of parameters......Page 270
2.3.4 Support size of shearlets......Page 271
2.3.5 Digitization of the exponential term......Page 272
2.3.6 Digital shearlets......Page 273
2.4 Algorithmic Realization of the FDST......Page 276
2.4.2 Adjoint FDST......Page 277
3 Digital Shearlet Transform Using Compactly Supported Shearlets......Page 278
3.1.1 Faithful digitization of the compactly supported shearlet transform......Page 279
3.1.2 Algorithmic realization......Page 283
3.1.3 Digital realization of directionality......Page 284
3.1.4 Redundancy......Page 285
3.1.5 Computational complexity......Page 286
3.2.1 Shearlet generators......Page 287
3.2.2 Algorithmic realization......Page 288
4 Framework for Quantifying Performance......Page 289
4.2 Isometry of Pseudo-Polar Transform......Page 290
4.3 Parseval Frame Property......Page 291
4.4 Space-Frequency-Localization......Page 292
4.5 Shear Invariance......Page 294
4.6 Speed......Page 295
4.7 Geometric Exactness......Page 296
4.8 Stability......Page 297
References......Page 299
1 Introduction......Page 300
2 Image Denoising......Page 301
2.1 Discrete Shearlet Transform......Page 303
2.2 Shearlet Thresholding......Page 306
2.3 Denoising Using Shearlet-Based Total Variation Regularization......Page 309
2.4 Complex-Valued Denoising......Page 311
2.5 Other Shearlet-Based Denoising Techniques......Page 312
3.1 Inverting the Radon Transform......Page 313
3.2 Deconvolution......Page 316
3.3 Inverse-Halftoning......Page 320
4 Image Enhancement......Page 322
5 Edge Analysis and Detection......Page 326
5.1 Edge Analysis Using Shearlets......Page 327
5.2 Edge Detection Using Shearlets......Page 328
5.3 Edge Analysis Using Shearlets......Page 330
6.1 Image Model......Page 332
6.2 Geometric Separation Algorithm......Page 333
7 Shearlets Analysis of 3D Data......Page 335
8 Additional Applications......Page 337
References......Page 338
Index......Page 343