Montefusco / Puccio Wavelets

Theory, Algorithms, and Applications
1. Auflage 2014
ISBN: 978-0-08-052084-1
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band Volume 5, 627 Seiten, Web PDF

Reihe: Wavelet Analysis and Its Applications

Wavelets: Theory, Algorithms, and Applications is the fifth volume in the highly respected series, WAVELET ANALYSIS AND ITS APPLICATIONS. This volume shows why wavelet analysis has become a tool of choice infields ranging from image compression, to signal detection and analysis in electrical engineering and geophysics, to analysis of turbulent or intermittent processes. The 28 papers comprising this volume are organized into seven subject areas: multiresolution analysis, wavelet transforms, tools for time-frequency analysis, wavelets and fractals, numerical methods and algorithms, and applications. More than 135 figures supplement the text.Features theory, techniques, and applicationsPresents alternative theoretical approaches including multiresolution analysis, splines, minimum entropy, and fractal aspectsContributors cover a broad range of approaches and applications

Autoren/Hrsg.

Montefusco, Laura

Puccio, Luigia

Inhaltsverzeichnis

1;Front Cover;1
2;Wavelets: Theory, Algorithms, and Applications;5
3;Copyright Page
;6
4;Table of
Contents;7
5;Contributors;11
6;Preface;15
7;Part I: Multiresolution and Multilevel Analyses;19
7.1;Chapter 1. Non-stationary Multiscale Analysis;21
7.1.1;§1 Introduction;21
7.1.2;§2 Regularity and time-frequency localization in the standard case;23
7.1.3;§3 Non-stationary wavelets;24
7.1.4;§4 Non-stationary wavelet packets;27
7.1.5;References;29
7.2;Chapetr 2. The Spectral Theory of Multiresolution Operators and Applications;31
7.2.1;§1 Introduction;31
7.2.2;§2 Wavelets and multiresolution operators;32
7.2.3;§3 The Lawton-Cohen-Gopinath Theorem on wavelet orthonormal bases;38
7.2.4;§4 Differentiability estimates via the spectral theory of the multiresolution operator;39
7.2.5;§5 Conclusion;46
7.2.6;References;47
7.3;Chapter 3. Multiresolution Analysis, Haar Bases and Wavelets on Riemannian Manifolds;51
7.3.1;§1 Introduction;51
7.3.2;§2 Construction of a multiresolution analysis;53
7.3.3;§3 Construction of a generalized Haar function;61
7.3.4;§4 Examples;65
7.3.5;§5 Wavelets;68
7.3.6;References;70
7.4;Chapter 4. Orthonormal Cardinal Functions;71
7.4.1;§1 Introduction;71
7.4.2;§2 Local orthogonal spline projectors with a given accuracy;72
7.4.3;§3 Orthonormal cardinal functions using cardinal splines;81
7.4.4;§4 Orthonormal refinable cardinal functions;91
7.4.5;References;105
8;Part II: Wavelet Transforms;107
8.1;Chapter 5. Some Remarks on Wavelet Representations and Geometric Aspects;109
8.1.1;§1 Introduction;109
8.1.2;§2 The group-theoretical picture;110
8.1.3;§3 Examples;114
8.1.4;§4 Algorithms from group theoretical considerations;120
8.1.5;§5 Approximate algorithms;125
8.1.6;References;132
8.2;Chapter 6. A Matrix Approach to Discrete Wavelets;135
8.2.1;§1 Introduction;135
8.2.2;§2 Towards matrices;136
8.2.3;§3 Factorizations of compactly supported orthogonal wavelets;138
8.2.4;§4 Biorthogonal wavelet transforms;140
8.2.5;§5 Vanishing moments and regularity of wavelet matrices;145
8.2.6;§6 Construction from the first row;148
8.2.7;§7 Symmetric orthogonal wavelets;150
8.2.8;References;152
8.3;Chapter 7. A Unified Approach to Periodic Wavelets;155
8.3.1;§1 Introduction;155
8.3.2;§2 Periodic shift-invariant spaces;156
8.3.3;§3 Periodic multiresolution;159
8.3.4;§4 Periodic wavelet spaces;162
8.3.5;§5 Decomposition and reconstruction algorithms;166
8.3.6;References;168
9;Part III: Spline Wavelets;171
9.1;Chapter 8. Spline Wavelets over R, Z, R/NZ, and Z/NZ;173
9.1.1;§1 Introduction;173
9.1.2;§2 Generalized cardinal B-splines over classical LCA groups;174
9.1.3;§3 Generalized Euler-Frobenius polynomials;177
9.1.4;§4 Multiresolution analysis;182
9.1.5;§5 Shifted spline wavelets;184
9.1.6;§6 Decomposition and reconstruction of functions;189
9.1.7;References;194
9.2;Chapter 9. A Practice of Data Smoothing by B-spline Wavelets;197
9.2.1;§1 Introduction;197
9.2.2;§2 Compactly supported B-spline wavelets;198
9.2.3;§3 The cubic cardinal B-spline wavelet analysis;202
9.2.4;§4 An application to mechanical vibration;209
9.2.5;§5 Conclusion;210
9.2.6;References;213
9.3;Chapter 10. L-Spline Wavelets;215
9.3.1;§1 Introduction;215
9.3.2;§2 L-splines;216
9.3.3;§3 A basis of locally supported splines;217
9.3.4;§4 L-spline wavelets;220
9.3.5;§5 The translation invariant case;225
9.3.6;§6 A multiresolution framework;227
9.3.7;§7 Examples;228
9.3.8;References;229
9.4;Chapter 11. Wavelets and Frames on the Four-Directional Mesh;231
9.4.1;§1 Introduction;231
9.4.2;§2 Four-directional box splines as scaling functions;233
9.4.3;§3 Four-directional wavelets;238
9.4.4;§4 Four-directional frames;244
9.4.5;References;247
10;Part IV: Other Mathematical Tools for Time-Frequency Analysis;249
10.1;Chapter 12. On Minimum Entropy Segmentation;251
10.1.1;§1 Introduction;251
10.1.2;§2 1-d Segmented wavelet transforms;254
10.1.3;§3 Adapting by minimum entropy;266
10.1.4;§4 Fast computation of all segmentations;274
10.1.5;§5 MES as an edge locator;277
10.1.6;§6 Multi-segmented analysis;280
10.1.7;§7 Discussion;285
10.1.8;References;285
10.2;Chapter 13. Adaptive Time-Frequency Approximations with Matching Pursuits;289
10.2.1;§1 Introduction;289
10.2.2;§2 Optimal adaptive approximations in dictionaries;290
10.2.3;§3 Matching pursuit;292
10.2.4;§4 Back-projection and orthogonal pursuit;294
10.2.5;§5 Numerical implementations of matching pursuits;298
10.2.6;§6 Matching pursuit with time-frequency dictionaries;299
10.2.7;§7 Chaos in matching pursuit and noise removal;304
10.2.8;§8 Conclusion;308
10.2.9;References;311
10.3;Chapter 14. Getting Around the Balian-Low Theorem Using Generalized Malvar Wavelets;313
10.3.1;§1 Introduction;313
10.3.2;§2 Wilson bases and generalized Malvar wavelets;315
10.3.3;§3 Complex-valued Malvar wavelets;318
10.3.4;§4 Discrete time implementation of CGMWT;324
10.3.5;§5 Conclusions;325
10.3.6;References;326
10.4;Chapter 15. Time Scale Energetic Distribution;329
10.4.1;§1 Introduction;329
10.4.2;§2 Analytic signal and Hilbert space;330
10.4.3;§3 Analytic signal and wavelet representation;332
10.4.4;§4 The affine Wigner representation;333
10.4.5;§5 Time-scale affine representation of finite energy signals;336
10.4.6;§6 Computation of Pax(t,t0,.);338
10.4.7;§7 Conclusion;339
10.4.8;References;339
11;Part V: Wavelets and Fractals;341
11.1;Chapter 16. Some Mathematical Results about the Multifractal Formalism for Functions;343
11.1.1;§1 Introduction;343
11.1.2;§2 Regularity, singularities, and two-microlocalization;353
11.1.3;§3 Some functional norm estimates;356
11.1.4;§4 Upper bounds and counterexamples;358
11.1.5;§5 Basic properties of selfsimilar functions;362
11.1.6;§6 The wavelet maxima method;368
11.1.7;§7 Riemann's function;371
11.1.8;§8 Some concluding remarks;376
11.1.9;References;377
11.2;Chapter 17. Fractal Wavelet Dimensions and Time Evolution;381
11.2.1;§1 Introduction;381
11.2.2;§2 Introduction to wavelet transforms;382
11.2.3;§3 The definition of the wavelet dimensions;386
11.2.4;§4 Time evolution and the dimension .(2);391
11.2.5;§5 Appendix: some estimates and explicit formulas;393
11.2.6;References;398
12;Part VI: Numerical Methods and Algorithms;401
12.1;Chapter 18. Multiscale Methods for Pseudo-Differential Equations on Smooth Closed Manifolds;403
12.1.1;§1 Introduction;403
12.1.2;§2 Pseudo-differential equations on smooth manifolds;405
12.1.3;§3 Multiscale decompositions;407
12.1.4;§4 Galerkin scheme;425
12.1.5;§5 Optimal convergence order;429
12.1.6;§6 Optimal compression;435
12.1.7;References;440
12.2;Chapter 19. Wavelet Methods for the Numerical Solution of Boundary Value Problems on the Interval;443
12.2.1;§1 Introduction;443
12.2.2;§2 Galerkin approach;446
12.2.3;§3 Collocation approach;452
12.2.4;§4 Numerical results;460
12.2.5;References;465
12.3;Chapter 20. On the Nodal Values of the Franklin Analyzing Wavelet;467
12.3.1;§1 Introduction;467
12.3.2;§2 An alternative form for g(3/2);468
12.3.3;§3 A lower bound for S;470
12.3.4;§4 Refinements of the estimates for S¯ and g(3/2)¯;473
12.3.5;§5 An upper bound for S and g(3/2);474
12.3.6;References;475
12.4;Chapter 21. Parallel Numerical Algorithms with Orthonormal Wavelet Packet Bases;477
12.4.1;§1 Introduction;477
12.4.2;§2 Orthogonal wavelet and wavelet-packet bases;479
12.4.3;§3 Linear operators in wavelet and wavelet-packet bases;491
12.4.4;§4 Parallel algorithms for matrix realization in wavelet packet basis;496
12.4.5;§5 Parallel algorithms for TWP compressed matrices;500
12.4.6;§6 Parallel preconditioned conjugate gradient method;503
12.4.7;§7 Conclusion;510
12.4.8;References;510
12.5;Chapter 22. Representation of the Atomic Hartree-Fock Equations in a Wavelet Basis by Means of the BCR Algorithm;513
12.5.1;§1 Introduction;513
12.5.2;§2 Description of the BCR algorithm;514
12.5.3;§3 Non-Standard form of the Hartree-Fock operator;518
12.5.4;§4 Numerical applications;521
12.5.5;References;524
13;Part VII: Applications;525
13.1;Chapter 23. Efficiency Comparison of Wavelet Packet and Adapted Local Cosine Bases for Compression of a Two-dimensional Turbulent Flow;527
13.1.1;§ 1 Background;527
13.1.2;§2 Methods;529
13.1.3;§3 Results;534
13.1.4;§4 Perspective;547
13.1.5;References;547
13.2;Chapter 24. Wavelet Spectra of Buoyant Atmospheric Turbulence;551
13.2.1;§1 Introduction;551
13.2.2;§2 Summary of definitions;552
13.2.3;§3 Data acquisition and sampling;554
13.2.4;§4 Results;554
13.2.5;§5 Conclusions;564
13.2.6;References;566
13.3;Chapter 25. Experimental Study of Inhomogeneous Turbulence in the Lower Troposphere by Wavelet Analysis;569
13.3.1;§1 Introduction;569
13.3.2;§2 Airborne measurements;570
13.3.3;§3 The "reference" case: homogeneous turbulence;571
13.3.4;§4 Inhomogeneous samples;573
13.3.5;§5 Lee waves and turbulence;575
13.3.6;§6 Local winds;578
13.3.7;§7 Conclusion;582
13.3.8;References;584
13.4;Chapter 26. Applications of Wavelet Transform for Seismic Activity Monitoring;587
13.4.1;§1 The seismic monitoring automation;587
13.4.2;§2 The preprocessing phase;588
13.4.3;§3 Applications of WT to geophysical signals: state of the art;590
13.4.4;§4 The WT de-noising capability;590
13.4.5;§5 WT effectiveness for arrival times estimation: synthetic data;591
13.4.6;§6 WT effectiveness for arrival times estimation: seismic data;595
13.4.7;§7 Conclusion;597
13.4.8;References;597
13.5;Chapter 27. Mean Value Jump Detection: A Survey of Conventional and Wavelet Based Methods;599
13.5.1;§1 Detection problems;599
13.5.2;§2 Overview of the detection strategies;600
13.5.3;§3 A comparison of the general methods;603
13.5.4;§4 Simulation results;606
13.5.5;§5 Conclusion;609
13.5.6;References;610
13.6;Chapter 28. Comparison of Picture Compression Methods: Wavelet, Wavelet Packet, and Local Cosine Transform Coding;611
13.6.1;§1 Introduction;611
13.6.2;§2 Relevant notions from mathematics;611
13.6.3;§3 Transform coding methods;615
13.6.4;§4 Wavelet and wavelet packet methods;618
13.6.5;§5 How to compare coding methods;632
13.6.6;§6 Transforming compressed pictures;636
13.6.7;§7 Source programs;643
13.6.8;References;646
14;Subject Index;649
15;WAVELET ANALYSIS AND ITS APPLICATIONS;655