Two-Dimensional Wavelets and their Relatives

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Prologue......Page 11
A historical note......Page 12
About the contents of the book......Page 13
Acknowledgements......Page 19
1.1 What is wavelet analysis?......Page 21
1.2 The continuous wavelet transform......Page 25
1.2.1 Examples......Page 28
1.3 Discretization of the CWT, frames......Page 30
1.4 Ridges and skeleton......Page 34
1.5 The discrete WT: orthonormal bases of wavelets......Page 39
1.6.1 Continuous wavelet packets......Page 44
1.6.2 Orthogonal or redundant wavelet expansions?......Page 46
1.7 Applications of the 1-D CWT......Page 49
2.1.1 Images and elementary operations on them......Page 52
2.1.2 Wavelets and continuous wavelet transform......Page 54
2.2 Basic properties of the 2-D CWT......Page 56
2.3.1 Interpretation of the CWT as a singularity scanner......Page 61
2.3.2 The CWT as a phase space representation......Page 65
2.3.3 Visualization of the CWT: the various representations......Page 66
2.3.4 Partial energy densities of the CWT......Page 68
2.3.5 Ridges in the 2-D CWT......Page 70
2.4.1 Generalities on frames......Page 74
2.4.2 Two-dimensional wavelet frames......Page 77
2.4.4 The dyadic wavelet transform......Page 84
2.5.1 Multiresolution analysis in 2-D and the 2-D DWT......Page 88
2.5.2.1 More isotropic 2-D wavelets......Page 90
2.5.2.2 Biorthogonal wavelet bases......Page 91
2.5.2.3 Wavelet packets and the Best Basis Algorithm......Page 93
2.5.2.4 The lifting scheme: second-generation wavelets......Page 94
2.6.1 Custom design of dyadic frames......Page 98
2.6.2 Example of a typical design......Page 101
2.6.3 Designing directional dyadic frames......Page 102
2.6.4 Implementation using approximate QMFs......Page 104
2.6.5 Some implementation issues......Page 110
2.7 Steerable filters......Page 113
2.8 Redundancy: plus and minus......Page 116
3.1 Which wavelets?......Page 117
3.2.1 The 2-D Mexican hat and its generalizations......Page 119
3.2.2 Difference wavelets......Page 121
3.3.1.1 Some precursors......Page 123
3.3.1.2 The concept of directional wavelets......Page 124
3.3.2 The 2-D Morlet wavelet......Page 125
3.3.3 End-stopped wavelets......Page 128
3.3.4 Conical wavelets......Page 130
3.3.5 Multidirectional wavelets......Page 137
3.4 Wavelet calibration: evaluating the performances of the CWT......Page 138
3.4.1 The scale and angle resolving power......Page 139
3.4.3 Calibration of a wavelet with benchmark signals......Page 140
4.1.1 The detection principle......Page 145
4.1.2 Application to character recognition......Page 150
4.2 Object detection and recognition in noisy images......Page 154
4.2.1 Principle of the ATR wavelet algorithm......Page 155
4.2.2 Application to infrared radar imagery: position features......Page 157
4.2.3 Scale-angle features and object recognition......Page 161
4.2.3.2 Application to character recognition......Page 163
4.2.3.3 Application to radar imagery......Page 164
4.3.1 The problem of content-based image retrieval......Page 165
4.3.2 Feature point detection using an end-stopped wavelet......Page 167
4.5.1 The tools for symmetry detection......Page 170
4.5.2.1 Geometric patterns......Page 172
4.5.2.2 Quasiperiodic point sets......Page 174
4.5.2.3 Other examples of aperiodic patterns......Page 177
4.5.2.4 Point sets generated from noncrystallographic Coxeter groups......Page 180
4.6 Image denoising......Page 182
4.7.1 Local contrast......Page 183
4.7.2 Watermarking of images......Page 190
5.1.1 Wavelets and astronomical images......Page 195
5.1.2 Structure of the Universe, cosmic microwave background (CMB) radiation......Page 196
5.1.3.1 Special coronal objects......Page 199
5.1.3.2 Distribution of small features......Page 200
5.1.3.3 Analysis of academic objects......Page 202
5.1.3.5 Conclusion and open questions......Page 204
5.1.4 Detection of gamma-ray sources in the Universe......Page 206
5.1.4.1 Sample data and the classical solutions......Page 207
5.1.4.2 Decision criteria and results......Page 209
5.2.1 Geology: fault detection......Page 212
5.2.2 Seismology......Page 214
5.2.3 Climatology......Page 216
5.3 Applications in fluid dynamics......Page 217
5.3.1 Detecting coherent structures in turbulent fluids......Page 218
5.3.2 Directional filtering......Page 219
5.3.3 Measuring a velocity field......Page 220
5.3.4 Disentangling of a wave train......Page 222
5.4.1 Analysis of 2-D fractals: the WTMM method......Page 225
5.4.2 Shape recognition and classification of patterns......Page 229
5.5 Texture analysis......Page 230
5.6 Applications of the DWT......Page 232
6.1 Group theory and matrix geometry of wavelets......Page 234
6.1.1 The 1-D CWT revisited......Page 235
6.1.2 The space of all wavelet transforms......Page 241
6.1.2.1 An intrinsic characterization of the space of wavelet transforms......Page 242
6.1.2.2 Decomposition of the space of all finite energy wavelet transforms......Page 244
6.1.3 Localization operators......Page 248
6.2 Phase space analysis......Page 250
6.2.1 Holomorphic wavelet transforms......Page 251
6.2.2 Matrix analysis of phase space......Page 254
6.3.1 Group theoretical analysis......Page 258
6.3.1.1 Holomorphic Gabor wavelets......Page 262
6.3.2 Phase space considerations......Page 265
7.1 A group-adapted wavelet analysis......Page 267
7.1.1 Some generalities......Page 268
7.1.2 Square integrability of representations......Page 270
7.1.3 Construction of generalized wavelet transforms......Page 271
7.1.4 Reproducing kernels, partial isometries and localization operators......Page 272
7.1.4.1 Partial isometries......Page 273
7.1.4.2 Left regular representation and localization operators......Page 275
7.1.5 Wavelet transforms on general quotient spaces......Page 277
7.2.1 The similitude group and 2-D wavelets......Page 279
7.2.3 Decomposition theory of 2-D wavelet transforms......Page 284
7.2.3.1 A concrete example......Page 286
7.2.3.2 Decomposition into orthogonal angular channels......Page 287
7.3.1 Lie algebra and orbits......Page 288
7.3.2 The coadjoint orbit … as a phase space......Page 293
7.4 The affine Poincaré group......Page 296
7.4.1 Group structure and representations......Page 297
7.4.2 Affine Poincaré wavelets......Page 299
8.1 Phase space distributions and minimal uncertainty gaborettes......Page 301
8.2 Minimal uncertainty wavelets......Page 304
8.3 Wigner functions......Page 307
8.4 Wigner functions for the wavelet groups......Page 311
8.4.1 Wigner functions for the affine group......Page 314
8.4.2 Wigner functions for the similitude group......Page 315
8.4.3 The Wigner function and the wavelet transform......Page 318
9.1.1 Constructing 3-D wavelets......Page 320
9.1.2 The 3-D continuous wavelet transform......Page 322
9.1.3 Extension to higher dimensions......Page 326
9.2.1 The problem......Page 328
9.2.2.1 Affine transformations on the sphere S2......Page 329
9.2.2.2 Spherical wavelets......Page 330
9.2.2.3 The spherical wavelet transform......Page 332
9.2.3 The Euclidean limit......Page 336
9.2.4.1 General remarks......Page 337
9.2.4.2 Estimating the angular selectivity of a wavelet......Page 339
9.2.4.3 Designing directional spherical wavelets......Page 340
9.2.4.4 Representation of object surfaces......Page 342
9.2.5.1 Discretization and algorithm......Page 343
9.2.5.2 Numerical criterion for the scale range......Page 345
9.2.5.3 Examples of spherical wavelet transforms......Page 348
9.2.6 Extension to other manifolds......Page 351
9.3 Wavelet approximations on the sphere......Page 352
10.1 Introduction......Page 363
10.2 Spatio-temporal signals and their transformations......Page 364
10.3 The transformation group and its representations......Page 368
10.4.1 Spatio-temporal wavelets: definition and examples......Page 372
10.4.2 The spatio-temporal wavelet transform......Page 373
10.4.3 An alternative: relativistic wavelets......Page 375
10.5.1 Partial energy densities......Page 376
10.5.2 Description of the algorithm......Page 378
10.5.2.1 Velocity update stage......Page 380
10.5.2.2 Position update stage......Page 381
10.5.3 Application of the algorithm: results......Page 382
10.5.4 Conclusions......Page 391
11 Beyond wavelets......Page 393
11.1.1 The ridgelet transform......Page 394
11.1.2 Links with the Radon and wavelet transforms......Page 396
11.1.3 Ridgelets and 2-D singularities......Page 399
11.1.4 The curvelet transform......Page 400
11.1.5 Other wavelet-like decompositions......Page 402
11.2 Rate-distortion analysis of anisotropic approximations......Page 403
11.2.1 An anisotropic quadtree......Page 404
11.2.1.1 Case d+ > 0 and d– > 0......Page 405
11.2.1.3 Case d– = d+ = 0......Page 407
11.3.1 Redundant approximations......Page 409
11.3.2.1 N-term approximations......Page 410
11.3.2.2 Greedy algorithms......Page 411
11.3.2.3 Basis pursuit......Page 412
11.3.3.1 Image compression using redundant dictionaries......Page 414
11.4 Algebraic wavelets......Page 418
11.4.1.1 Tau-wavelets of Haar......Page 419
11.4.1.2 Pisot wavelets, beta-spline wavelets, etc.......Page 422
11.4.2.1 Introduction......Page 423
11.4.2.2 Penrose–Robinson tiling of the plane......Page 424
11.4.2.3 Multiresolution scheme for stone-invariant tilings......Page 428
11.4.2.4 Wavelet bases for a stone-inflation invariant tiling......Page 430
Epilogue......Page 433
A.1.1 Definitions......Page 435
A.1.2 Examples......Page 437
A.1.3 Integration on groups......Page 440
A.1.4 Convolution on groups......Page 441
A.2.1 Definitions......Page 443
A.2.2 Examples......Page 444
A.2.3 Square integrable representations......Page 445
A.3 Lie groups and Lie algebras......Page 447
A.4 Some useful formulas from harmonic analysis on the sphere......Page 449
A. Books and Theses......Page 451
B. Articles......Page 455
Index......Page 475