An introduction to wavelet analysis:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Boston [u.a.]
Birkhäuser
2004
|
| Ausgabe: | 2. print. with corr. |
| Schriftenreihe: | Applied and numerical harmonic analysis
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Literaturverz. S. 441 - 444 |
| Beschreibung: | XVII, 449 S. Ill., graph. Darst. |
| ISBN: | 3764339624 0817639624 9780817639624 9781461265672 |
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| 100 | 1 | |a Walnut, David F. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a An introduction to wavelet analysis |c David F. Walnut |
| 250 | |a 2. print. with corr. | ||
| 264 | 1 | |a Boston [u.a.] |b Birkhäuser |c 2004 | |
| 300 | |a XVII, 449 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Applied and numerical harmonic analysis | |
| 500 | |a Literaturverz. S. 441 - 444 | ||
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Datensatz im Suchindex
| DE-BY-862_location | 2000 |
|---|---|
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| adam_text | Titel: An introduction to wavelet analysis
Autor: Walnut, David F
Jahr: 2004
Contents
Preface xiii
I Preliminaries 1
1 Functions and Convergence 3
1.1 Functions............................ 3
1.1.1 Bounded (I°°) Functions............... 3
1.1.2 Integrable (L1) Functions............... 3
1.1.3 Square Integrable (L2) Functions........... 6
1.1.4 Differentiable (Cn) Functions............. 9
1.2 Convergence of Sequences of Functions............ 11
1.2.1 Numerical Convergence................ 11
1.2.2 Pointwise Convergence................. 13
1.2.3 Uniform (L°°) Convergence.............. 14
1.2.4 Mean (L1) Convergence................ 17
1.2.5 Mean-square (L2) Convergence............ 19
1.2.6 Interchange of Limits and Integrals.......... 21
2 Fourier Series 27
2.1 Trigonometric Series...................... 27
2.1.1 Periodic Functions................... 27
2.1.2 The Trigonometric System .............. 28
2.1.3 The Fourier Coefficients................ 30
2.1.4 Convergence of Fourier Series............. 32
2.2 Approximate Identities..................... 37
2.2.1 Motivation from Fourier Series............ 38
2.2.2 Definition and Examples................ 40
2.2.3 Convergence Theorems................. 42
2.3 Generalized Fourier Series................... 47
2.3.1 Orthogonality...................... 47
2.3.2 Generalized Fourier Series............... 49
2.3.3 Completeness...................... 52
3 The Fourier Transform 59
3.1 Motivation and Definition................... 59
3.2 Basic Properties of the Fourier Transform.......... 63
3.3 Fourier Inversion........................ 65
i Contents
3.4 Convolution........................... 68
3.5 Plancherel s Formula...................... 72
3.6 The Fourier Transform for I? Functions........... 75
3.7 Smoothness versus Decay................... 76
3.8 Dilation, Translation, and Modulation............ 79
3.9 Bandlimited Functions and the Sampling Formula..... 81
Signals and Systems 87
4.1 Signals.............................. 88
4.2 Systems............................. 90
4.2.1 Causality and Stability ................ 95
4.3 Periodic Signals and the Discrete Fourier Transform .... 101
4.3.1 The Discrete Fourier Transform............ 102
4.4 The Fast Fourier Transform.................. 107
4.5 L2 Fourier Series........................ 109
II The Haar System 113
5 The Haar System 115
5.1 Dyadic Step Functions..................... 115
5.1.1 The Dyadic Intervals.................. 115
5.1.2 The Scale j Dyadic Step Functions.......... 116
5.2 The Haar System........................ 117
5.2.1 The Haar Scaling Functions and the
Haar Functions..................... 117
5.2.2 Orthogonality of the Haar System.......... 118
5.2.3 The Splitting Lemma ................. 120
5.3 Haar Bases on [0,1]....................... 122
5.4 Comparison of Haar Series with Fourier Series........ 127
5.4.1 Representation of Functions with Small Support . . 128
5.4.2 Behavior of Haar Coefficients Near
Jump Discontinuities.................. 130
5.4.3 Haar Coefficients and Global Smoothness...... 132
5.5 Haar Bases on R........................ 133
5.5.1 The Approximation and Detail Operators...... 134
5.5.2 The Scale J Haar System on R............ 138
5.5.3 The Haar System on R................ 138
6 The Discrete Haar Transform 141
6.1 Motivation ........................... 141
6.1.1 The Discrete Haar Transform (DHT)......... 142
6.2 The DHT in Two Dimensions................. 146
6.2.1 The Row-wise and Column-wise Approximations
and Details....................... 146
Contents ix
6.2.2 The DHT for Matrices................. 147
6.3 Image Analysis with the DHT................. 150
6.3.1 Approximation and Blurring ............. 151
6.3.2 Horizontal, Vertical, and Diagonal Edges ...... 153
6.3.3 Naive Image Compression.............. 154
III Orthonormal Wavelet Bases 161
7 Multiresolution Analysis 163
7.1 Orthonormal Systems of Translates.............. 164
7.2 Definition of Multiresolution Analysis ............ 169
7.2.1 Some Basic Properties of MRAs ........... 170
7.3 Examples of Multiresolution Analysis............. 174
7.3.1 The Haar MRA..................... 174
7.3.2 The Piecewise Linear MRA.............. 174
7.3.3 The Bandlimited MRA ................ 179
7.3.4 The Meyer MRA.................... 180
7.4 Construction and Examples of Orthonormal
Wavelet Bases.......................... 185
7.4.1 Examples of Wavelet Bases.............. 186
7.4.2 Wavelets in Two Dimensions............. 190
7.4.3 Localization of Wavelet Bases............. 193
7.5 Proof of Theorem 7.35..................... 196
7.5.1 Sufficient Conditions for a Wavelet Basis....... 197
7.5.2 Proof of Theorem 7.35................. 199
7.6 Necessary Properties of the Scaling Function ........ 203
7.7 General Spline Wavelets.................... 206
7.7.1 Basic Properties of Spline Functions......... 206
7.7.2 Spline Multiresolution Analyses............ 208
8 The Discrete Wavelet Transform 215
8.1 Motivation: From MRA to a Discrete Transform...... 215
8.2 The Quadrature Mirror Filter Conditions.......... 218
8.2.1 Motivation from MRA................. 218
8.2.2 The Approximation and Detail Operators and
Their Adjoints..................... 221
8.2.3 The Quadrature Mirror Filter (QMF) Conditions . . 223
8.3 The Discrete Wavelet Transform (DWT)........... 231
8.3.1 The DWT for Signals................. 231
8.3.2 The DWT for Finite Signals.............. 231
8.3.3 The DWT as an Orthogonal Transformation .... 232
8.4 Scaling Functions from Scaling Sequences.......... 236
8.4.1 The Infinite Product Formula............. 237
8.4.2 The Cascade Algorithm................ 243
x Contents
8.4.3 The Support of the Scaling Function......... 245
9 Smooth, Compactly Supported Wavelets 249
9.1 Vanishing Moments....................... 249
9.1.1 Vanishing Moments and Smoothness......... 250
9.1.2 Vanishing Moments and Approximation....... 254
9.1.3 Vanishing Moments and the Reproduction
of Polynomials..................... 257
9.1.4 Equivalent Conditions for Vanishing Moments .... 260
9.2 The Daubechies Wavelets................... 264
9.2.1 The Daubechies Polynomials............. 264
9.2.2 Spectral Factorization................. 269
9.3 Image Analysis with Smooth Wavelets............ 277
9.3.1 Approximation and Blurring............. 278
9.3.2 Naive Image Compression with
Smooth Wavelets.................... 278
IV Other Wavelet Constructions 287
10 Biorthogonal Wavelets 289
10.1 Linear Independence and Biorthogonality.......... 289
10.2 Riesz Bases and the Frame Condition ............ 290
10.3 Riesz Bases of Translates ................... 293
10.4 Generalized Multiresolution Analysis (GMRA)....... 300
10.4.1 Basic Properties of GMRA.............. 301
10.4.2 Dual GMRA and Riesz Bases of Wavelets...... 302
10.5 Riesz Bases Orthogonal Across Scales............ 311
10.5.1 Example: The Piecewise Linear GMRA....... 313
10.6 A Discrete Transform for Biorthogonal Wavelets...... 315
10.6.1 Motivation from GMRA................ 315
10.6.2 The QMF Conditions................. 317
10.7 Compactly Supported Biorthogonal Wavelets........ 319
10.7.1 Compactly Supported Spline Wavelets........ 320
10.7.2 Symmetric Biorthogonal Wavelets.......... 324
10.7.3 Using Symmetry in the DWT............. 328
11 Wavelet Packets 335
11.1 Motivation: Completing the Wavelet Tree.......... 335
11.2 Localization of Wavelet Packets................ 337
11.2.1 Time/Spatial Localization............... 337
11.2.2 Frequency Localization ................ 338
11.3 Orthogonality and Completeness Properties of
Wavelet Packets ........................ 346
11.3.1 Wavelet Packet Bases with a Fixed Scale...... 347
Contents xi
11.3.2 Wavelet Packets with Mixed Scales.......... 350
11.4 The Discrete Wavelet Packet Transform (DWPT)...... 354
11.4.1 The DWPT for Signals ................ 354
11.4.2 The DWPT for Finite Signals............. 354
11.5 The Best-Basis Algorithm................... 357
11.5.1 The Discrete Wavelet Packet Library......... 357
11.5.2 The Idea of the Best Basis............... 360
11.5.3 Description of the Algorithm............. 363
V Applications 369
12 Image Compression 371
12.1 The Transform Step...................... 372
12.1.1 Wavelets or Wavelet Packets?............. 372
12.1.2 Choosing a Filter.................... 373
12.2 The Quantization Step..................... 373
12.3 The Coding Step........................ 375
12.3.1 Sources and Codes................... 376
12.3.2 Entropy and Information ............... 378
12.3.3 Coding and Compression ............... 380
12.4 The Binary Huffman Code................... 385
12.5 A Model Wavelet Transform Image Coder.......... 387
12.5.1 Examples........................ 388
13 Integral Operators 397
13.1 Examples of Integral Operators ................ 397
13.1.1 Sturm-Liouville Boundary Value Problems...... 397
13.1.2 The Hilbert Transform................. 402
13.1.3 The Radon Transform................. 406
13.2 The BCR Algorithm...................... 414
13.2.1 The Scale j Approximation to T........... 415
13.2.2 Description of the Algorithm............. 418
VI Appendixes 423
A Review of Advanced Calculus and Linear Algebra 425
A.l Glossary of Basic Terms from Advanced Calculus and
Linear Algebra......................... 425
A.2 Basic Theorems from Advanced Calculus .......... 431
B Excursions in Wavelet Theory 433
B.l Other Wavelet Constructions................. 433
B.l.l M-band Wavelets ................... 433
xii Contents
B.1.2 Wavelets with Rational Noninteger
Dilation Factors .................... 434
B.1.3 Local Cosine Bases................... 434
B.1.4 The Continuous Wavelet Transform......... 435
B.1.5 Non-MRA Wavelets.................. 436
B.1.6 Multiwavelets...................... 436
B.2 Wavelets in Other Domains.................. 437
B.2.1 Wavelets on Intervals ................. 437
B.2.2 Wavelets in Higher Dimensions............ 438
B.2.3 The Lifting Scheme .................. 438
B.3 Applications of Wavelets.................... 439
B.3.1 Wavelet Denoising................... 439
B.3.2 Multiscale Edge Detection............... 439
B.3.3 The FBI Fingerprint Compression Standard..... 439
C References Cited in the Text 441
Index 445
|
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| title | An introduction to wavelet analysis |
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| title_full | An introduction to wavelet analysis David F. Walnut |
| title_fullStr | An introduction to wavelet analysis David F. Walnut |
| title_full_unstemmed | An introduction to wavelet analysis David F. Walnut |
| title_short | An introduction to wavelet analysis |
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