Wavelets and Hilbert Bases

Chapters in this book

1. Frontmatter i
2. Contents v
3. Contributor Affiliations ix
4. Preface xiii
5. Foreword xv
6. Introduction 1
7. Section I. Precursors in Signal Processing
8. Introduction 23
9. The Laplacian Pyramid as a Compact Image Code 28
10. Digital Coding of Speech in Sub-bands 37
11. Application of quadrature mirror filters to split-band voice coding schemes 54
12. Procedure for designing exact reconstruction filter banks for tree-structured subband coders 59
13. Filters for distortion-free two-band multirate filter banks 63
14. Filter banks allowing perfect reconstruction 68
15. Theory and design of M-channel maximally decimated quadrature mirror filters with arbitrary M, having the perfect-reconstruction property 94
16. SECTION II. Precursors in Physics: Affine Coherent States
17. Introduction 113
18. Continuous representation theory using the affine group 117
19. Decomposition of Hardy functions into square integrable wavelets of constant shape 126
20. Transforms associated to square integrable group representations I General results 140
21. Section III. Precursors in Mathematics: Early Wavelet Bases
22. Introduction 149
23. On the Theory of Orthogonal Function Systems 155
24. A set of continuous orthogonal functions 189
25. A modified Franklin system and higher-order spline systems on Rⁿ as unconditional bases for Hardy spaces 197
26. Uncertainty Principle, Hilbert Bases and Algebras of Operators 216
27. Wavelets and Hilbert Bases 229
28. A block spin construction of Ondelettes. Part i: Lemarié Functions 245
29. SECTION IV. Precursors and Development in Mathematics: Atom and Frame Decompositions 261
30. Introduction 263
31. A Class of Nonharmonic Fourier Series 269
32. Extensions of Hardy Spaces and Their Use in Analysis 295
33. Painless Nonorthogonal Expansions 372
34. Decomposition of Besov Spaces 385
35. Banach Spaces Related to Integrable Group Representations and Their Atomic Decompositions, I 408
36. The Wavelet Transform, Time-Frequency Localization And Signal Analysis 442
37. Section V. Multiresolution Analysis
38. Introduction 489
39. A Theory for Multiresolution Signal Decomposition: The Wavelet Representation 494
40. Wavelets with Compact Support 514
41. Approximations and Wavelet Orthonormal Bases of L<Sup>2</Sup>(R) 524
42. Wavelets, Multiresolution Analysis, and Quadrature Mirror F 543
43. Tight frames of compactly supported affine wavelets 560
44. Orthonormal Bases of Compactly Supported Wavelets 564
45. SECTION VI. Multidimensional Wavelets
46. Introduction 655
47. Wavelets, Spline Functions, and Multiresolution Analysis 659
48. Multiscale Analyses and Wavelet Bases 690
49. Nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for Rⁿ 694
50. Multiresolution analysis, Haar bases and self-similar tilings of Rⁿ 717
51. SECTION VII. Selected Applications
52. Introduction 733
53. Fast wavelet transforms and numerical algorithms 741
54. Compression of wavelet decompositions 784
55. Adapting to unknown smoothness by wavelet shrinkage 833
56. Hölder Exponents at Given Points and Wavelet Coefficients 858
57. Embedded image coding using zerotrees of wavelet coefficients 861