Wavelets (eBook)

Front Cover 1
Wavelets: Theory, Algorithms, and Applications 5
Copyright Page 
6 
Table of 
7 
Contributors 11
Preface 15
Part I: Multiresolution and Multilevel Analyses 19
Chapter 1. Non-stationary Multiscale Analysis 21
§1 Introduction 21
§2 Regularity and time-frequency localization in the standard case 23
§3 Non-stationary wavelets 24
§4 Non-stationary wavelet packets 27
References 29
Chapetr 2. The Spectral Theory of Multiresolution Operators and Applications 31
§1 Introduction 31
§2 Wavelets and multiresolution operators 32
§3 The Lawton-Cohen-Gopinath Theorem on wavelet orthonormal bases 38
§4 Differentiability estimates via the spectral theory of the multiresolution operator 39
§5 Conclusion 46
References 47
Chapter 3. Multiresolution Analysis, Haar Bases and Wavelets on Riemannian Manifolds 51
§1 Introduction 51
§2 Construction of a multiresolution analysis 53
§3 Construction of a generalized Haar function 61
§4 Examples 65
§5 Wavelets 68
References 70
Chapter 4. Orthonormal Cardinal Functions 71
§1 Introduction 71
§2 Local orthogonal spline projectors with a given accuracy 72
§3 Orthonormal cardinal functions using cardinal splines 81
§4 Orthonormal refinable cardinal functions 91
References 105
Part II: Wavelet Transforms 107
Chapter 5. Some Remarks on Wavelet Representations and Geometric Aspects 109
§1 Introduction 109
§2 The group-theoretical picture 110
§3 Examples 114
§4 Algorithms from group theoretical considerations 120
§5 Approximate algorithms 125
References 132
Chapter 6. A Matrix Approach to Discrete Wavelets 135
§1 Introduction 135
§2 Towards matrices 136
§3 Factorizations of compactly supported orthogonal wavelets 138
§4 Biorthogonal wavelet transforms 140
§5 Vanishing moments and regularity of wavelet matrices 145
§6 Construction from the first row 148
§7 Symmetric orthogonal wavelets 150
References 152
Chapter 7. A Unified Approach to Periodic Wavelets 155
§1 Introduction 155
§2 Periodic shift-invariant spaces 156
§3 Periodic multiresolution 159
§4 Periodic wavelet spaces 162
§5 Decomposition and reconstruction algorithms 166
References 168
Part III: Spline Wavelets 171
Chapter 8. Spline Wavelets over R, Z, R/NZ, and Z/NZ 173
§1 Introduction 173
§2 Generalized cardinal B-splines over classical LCA groups 174
§3 Generalized Euler-Frobenius polynomials 177
§4 Multiresolution analysis 182
§5 Shifted spline wavelets 184
§6 Decomposition and reconstruction of functions 189
References 194
Chapter 9. A Practice of Data Smoothing by B-spline Wavelets 197
§1 Introduction 197
§2 Compactly supported B-spline wavelets 198
§3 The cubic cardinal B-spline wavelet analysis 202
§4 An application to mechanical vibration 209
§5 Conclusion 210
References 213
Chapter 10. L-Spline Wavelets 215
§1 Introduction 215
§2 L-splines 216
§3 A basis of locally supported splines 217
§4 L-spline wavelets 220
§5 The translation invariant case 225
§6 A multiresolution framework 227
§7 Examples 228
References 229
Chapter 11. Wavelets and Frames on the Four-Directional Mesh 231
§1 Introduction 231
§2 Four-directional box splines as scaling functions 233
§3 Four-directional wavelets 238
§4 Four-directional frames 244
References 247
Part IV: Other Mathematical Tools for Time-Frequency Analysis 249
Chapter 12. On Minimum Entropy Segmentation 251
§1 Introduction 251
§2 1-d Segmented wavelet transforms 254
§3 Adapting by minimum entropy 266
§4 Fast computation of all segmentations 274
§5 MES as an edge locator 277
§6 Multi-segmented analysis 280
§7 Discussion 285
References 285
Chapter 13. Adaptive Time-Frequency Approximations with Matching Pursuits 289
§1 Introduction 289
§2 Optimal adaptive approximations in dictionaries 290
§3 Matching pursuit 292
§4 Back-projection and orthogonal pursuit 294
§5 Numerical implementations of matching pursuits 298
§6 Matching pursuit with time-frequency dictionaries 299
§7 Chaos in matching pursuit and noise removal 304
§8 Conclusion 308
References 311
Chapter 14. Getting Around the Balian-Low Theorem Using Generalized Malvar Wavelets 313
§1 Introduction 313
§2 Wilson bases and generalized Malvar wavelets 315
§3 Complex-valued Malvar wavelets 318
§4 Discrete time implementation of CGMWT 324
§5 Conclusions 325
References 326
Chapter 15. Time Scale Energetic Distribution 329
§1 Introduction 329
§2 Analytic signal and Hilbert space 330
§3 Analytic signal and wavelet representation 332
§4 The affine Wigner representation 333
§5 Time-scale affine representation of finite energy signals 336
§6 Computation of Pax(t,t0,.) 338
§7 Conclusion 339
References 339
Part V: Wavelets and Fractals 341
Chapter 16. Some Mathematical Results about the Multifractal Formalism for Functions 343
§1 Introduction 343
§2 Regularity, singularities, and two-microlocalization 353
§3 Some functional norm estimates 356
§4 Upper bounds and counterexamples 358
§5 Basic properties of selfsimilar functions 362
§6 The wavelet maxima method 368
§7 Riemann's function 371
§8 Some concluding remarks 376
References 377
Chapter 17. Fractal Wavelet Dimensions and Time Evolution 381
§1 Introduction 381
§2 Introduction to wavelet transforms 382
§3 The definition of the wavelet dimensions 386
§4 Time evolution and the dimension .(2) 391
§5 Appendix: some estimates and explicit formulas 393
References 398
Part VI: Numerical Methods and Algorithms 401
Chapter 18. Multiscale Methods for Pseudo-Differential Equations on Smooth Closed Manifolds 403
§1 Introduction 403
§2 Pseudo-differential equations on smooth manifolds 405
§3 Multiscale decompositions 407
§4 Galerkin scheme 425
§5 Optimal convergence order 429
§6 Optimal compression 435
References 440
Chapter 19. Wavelet Methods for the Numerical Solution of Boundary Value Problems on the Interval 443
§1 Introduction 443
§2 Galerkin approach 446
§3 Collocation approach 452
§4 Numerical results 460
References 465
Chapter 20. On the Nodal Values of the Franklin Analyzing Wavelet 467
§1 Introduction 467
§2 An alternative form for g(3/2) 468
§3 A lower bound for S 470
§4 Refinements of the estimates for S¯ and g(3/2)¯ 473
§5 An upper bound for S and g(3/2) 474
References 475
Chapter 21. Parallel Numerical Algorithms with Orthonormal Wavelet Packet Bases 477
§1 Introduction 477
§2 Orthogonal wavelet and wavelet-packet bases 479
§3 Linear operators in wavelet and wavelet-packet bases 491
§4 Parallel algorithms for matrix realization in wavelet packet basis 496
§5 Parallel algorithms for TWP compressed matrices 500
§6 Parallel preconditioned conjugate gradient method 503
§7 Conclusion 510
References 510
Chapter 22. Representation of the Atomic Hartree-Fock Equations in a Wavelet Basis by Means of the BCR Algorithm 513
§1 Introduction 513
§2 Description of the BCR algorithm 514
§3 Non-Standard form of the Hartree-Fock operator 518
§4 Numerical applications 521
References 524
Part VII: Applications 525
Chapter 23. Efficiency Comparison of Wavelet Packet and Adapted Local Cosine Bases for Compression of a Two-dimensional Turbulent Flow 527
§ 1 Background 527
§2 Methods 529
§3 Results 534
§4 Perspective 547
References 547
Chapter 24. Wavelet Spectra of Buoyant Atmospheric Turbulence 551
§1 Introduction 551
§2 Summary of definitions 552
§3 Data acquisition and sampling 554
§4 Results 554
§5 Conclusions 564
References 566
Chapter 25. Experimental Study of Inhomogeneous Turbulence in the Lower Troposphere by Wavelet Analysis 569
§1 Introduction 569
§2 Airborne measurements 570
§3 The "reference" case: homogeneous turbulence 571
§4 Inhomogeneous samples 573
§5 Lee waves and turbulence 575
§6 Local winds 578
§7 Conclusion 582
References 584
Chapter 26. Applications of Wavelet Transform for Seismic Activity Monitoring 587
§1 The seismic monitoring automation 587
§2 The preprocessing phase 588
§3 Applications of WT to geophysical signals: state of the art 590
§4 The WT de-noising capability 590
§5 WT effectiveness for arrival times estimation: synthetic data 591
§6 WT effectiveness for arrival times estimation: seismic data 595
§7 Conclusion 597
References 597
Chapter 27. Mean Value Jump Detection: A Survey of Conventional and Wavelet Based Methods 599
§1 Detection problems 599
§2 Overview of the detection strategies 600
§3 A comparison of the general methods 603
§4 Simulation results 606
§5 Conclusion 609
References 610
Chapter 28. Comparison of Picture Compression Methods: Wavelet, Wavelet Packet, and Local Cosine Transform Coding 611
§1 Introduction 611
§2 Relevant notions from mathematics 611
§3 Transform coding methods 615
§4 Wavelet and wavelet packet methods 618
§5 How to compare coding methods 632
§6 Transforming compressed pictures 636
§7 Source programs 643
References 646
Subject Index 649
WAVELET ANALYSIS AND ITS APPLICATIONS 655