小波分析导论 英文PDF|Epub|txt|kindle电子书版本网盘下载

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Ⅰ Preliminaries1

1 Functions and Convergence3

1.1 Functions3

1.1.1 Bounded（L∞）Functions3

1.1.2 Integrable(L1)Functions3

1.1.3 Square Integrable(L2)Functions6

1.1.4 Differentiable(Cn)Functions9

1.2 Convergence of Sequences of Functions11

1.2.1 Numerical Convergence11

1.2.2 Pointwise Convergence13

1.2.3 Uniform（L∞）Convergence14

1.2.4 Mean(L1)Convergence17

1.2.5 Mean-square(L2)Convergence19

1.2.6 Interchange of Limits and Integrals21

2 Fourier Series27

2.1 Trigonometric Series27

2.1.1 Periodic Functions27

2.1.2 The Trigonometric System28

2.1.3 The Fourier Coefficients30

2.1.4 Convergence of Fourier Series32

2.2 Approximate Identities37

2.2.1 Motivation from Fourier Series38

2.2.2 Definition and Examples40

2.2.3 Convergence Theorems42

2.3 Generalized Fourier Series47

2.3.1 Orthogonality47

2.3.2 Generalized Fourier Series49

2.3.3 Completeness52

3 The Fourier Transform59

3.1 Motivation and Definition59

3.2 Basic Properties of the Fourier Transform63

3.3 Fourier Inversion65

3.4 Convolution68

3.5 Plancherel's Formula72

3.6 The Fourier Transform for L2 Functions75

3.7 Smoothness versus Decay76

3.8 Dilation,Translation,and Modulation79

3.9 Bandlimited Functions and the Sampling Formula81

4 Signals and Systems87

4.1 Signals88

4.2 Systems90

4.2.1 Causality and Stability95

4.3 Periodic Signals and the Discrete Fourier Transform101

4.3.1 The Discrete Fourier Transform102

4.4 The Fast Fourier Transform107

4.5 L2 Fourier Series109

Ⅱ The Haar System113

5 The Haar System115

5.1 Dyadic Step Functions115

5.1.1 The Dyadic Intervals115

5.1.2 The Scale j Dyadic Step Functions116

5.2 The Haar System117

5.2.1 The Haar Scaling Functions and the Haar Functions117

5.2.2 Orthogonality of the Haar System118

5.2.3 The Splitting Lemma120

5.3 Haar Bases on[0,1]122

5.4 Comparison of Haar Series with Fourier Series127

5.4.1 Representation of Functions with Small Support128

5.4.2 Behavior of Haar Coefficients Near Jump Discontinuities130

5.4.3 Haar Coefficients and Global Smoothness132

5.5 Haar Bases on R133

5.5.1 The Approximation and Detail Operators134

5.5.2 The Scale J Haar System on R138

5.5.3 The Haar System on R138

6 The Discrete Haar Transform141

6.1 Motivation141

6.1.1 The Discrete Haar Transform(DHT)142

6.2 The DHT in Two Dimensions146

6.2.1 The Row-wise and Column-wise Approximations and Details146

6.2.2 The DHT for Matrices147

6.3 Image Analysis with the DHT150

6.3.1 Approximation and Blurring151

6.3.2 Horizontal,Vertical,and Diagonal Edges153

6.3.3 "Naive"Image Compression154

Ⅲ Orthonormal Wavelet Bases161

7 Multiresolution Analysis163

7.1 Orthonormal Systems of Translates164

7.2 Definition of Multiresolution Analysis169

7.2.1 Some Basic Properties of MRAs170

7.3 Examples of Multiresolution Analysis174

7.3.1 The Haar MRA174

7.3.2 The Piecewise Linear MRA174

7.3.3 The Bandlimited MRA179

7.3.4 The Meyer MRA180

7.4 Construction and Examples of Orthonormal Wavelet Bases185

7.4.1 Examples of Wavelet Bases186

7.4.2 Wavelets in Two Dimensions190

7.4.3 Localization of Wavelet Bases193

7.5 Proof of Theorem 7.35196

7.5.1 Sufficient Conditions for a Wavelet Basis197

7.5.2 Proof of Theorem 7.35199

7.6 Necessary Properties of the Scaling Function203

7.7 General Spline Wavelets206

7.7.1 Basic Properties of Spline Functions206

7.7.2 Spline Multiresolution Analyses208

8 The Discrete Wavelet Transform215

8.1 Motivation:From MRA to a Discrete Transform215

8.2 The Quadrature Mirror Filter Conditions218

8.2.1 Motivation from MRA218

8.2.2 The Approximation and Detail Operators and Their Adjoints221

8.2.3 The Quadrature Mirror Filter(QMF)Conditions223

8.3 The Discrete Wavelet Transform(DWT)231

8.3.1 The DWT for Signals231

8.3.2 The DWT for Finite Signals231

8.3.3 The DWT as an Orthogonal Transformation232

8.4 Scaling Functions from Scaling Sequences236

8.4.1 The Infinite Product Formula237

8.4.2 The Cascade Algorithm243

8.4.3 The Support of the Scaling Function245

9 Smooth,Compactly Supported Wavelets249

9.1 Vanishing Moments249

9.1.1 Vanishing Moments and Smoothness250

9.1.2 Vanishing Momens and Approximation254

9.1.3 Vanishing Moments and the Reproduction of Polynomials257

9.1.4 Equivalent Conditions for Vanishing Moments260

9.2 The Daubechies Wavelets264

9.2.1 The Daubechies Polynomials264

9.2.2 Spectral Factorization269

9.3 Image Analysis with Smooth Wavelets277

9.3.1 Approximation and Blurring278

9.3.2 "Naive"Image Compression with Smooth Wavelets278

Ⅳ Other Wavelet Constructions287

10 Biorthogonal Wavelets289

10.1 Linear Independence and Biorthogonality289

10.2 Riesz Bases and the Frame Condition290

10.3 Riesz Bases of Translates293

10.4 Generalized Multiresolution Analysis(GMRA)300

10.4.1 Basic Properties of GMRA301

10.4.2 Dual GMRA and Riesz Bases of Wavelets302

10.5 Riesz Bases Orthogonal Across Scales311

10.5.1 Example:The Piecewise Linear GMRA313

10.6 A Discrete Transform for Biorthogonal Wavelets315

10.6.1 Motivation from GMRA315

10.6.2 The QMF Conditions317

10.7 Compactly Supported Biorthogonal Wavelets319

10.7.1 Compactly Supported Spline Wavelets320

10.7.2 Symmetric Biorthogonal Wavelets324

10.7.3 Using Symmetry in the DWT328

11 Wavelet Packets335

11.1 Motivation:Completing the Wavelet Tree335

11.2 Localization of Wavelet Packets337

11.2.1 Time/Spatial Localization337

11.2.2 Frequency Localization338

11.3 Orthogonality and Completeness Properties of Wavelet Packets346

11.3.1 Wavelet Packet Bases with a Fixed Scale347

11.3.2 Wavelet Packets with Mixed Scales350

11.4 The Discrete Wavelet Packet Transform(DWPT)354

11.4.1 The DWPT for Signals354

11.4.2 The DWPT for Finite Signals354

11.5 The Best-Basis Algorithm357

11.5.1 The Discrete Wavelet Packet Library357

11.5.2 The Idea of the Best Basis360

11.5.3 Description of the Algorithm363

Ⅴ Applications369

12 Image Compression371

12.1 The Transform Step372

12.1.1 Wavelets or Wavelet Packets?372

12.1.2 Choosing a Filter373

12.2 The Quantization Step373

12.3 The Coding Step375

12.3.1 Sources and Codes376

12.3.2 Entropy and Information378

12.3.3 Coding and Compression380

12.4 The Binary Huffman Code385

12.5 A Model Wavelet Transform Image Coder387

12.5.1 Examples388

13 Integral Operators397

13.1 Examples of Integral Operators397

13.1.1 Sturm-Liouville Boundary Value Problems397

13.1.2 The Hilbert Transform402

13.1.3 The Radon Transform406

13.2 The BCR Algorithm414

13.2.1 The Scale j Approximation to T415

13.2.2 Description of the Algorithm418

Ⅵ Appendixes423

A Review of Advanced Calculus and Linear Algebra425

A.1 Glossary of Basic Terms from Advanced Calculus and Linear Algebra425

A.2 Basic Theorems from Advanced Calculus431

B Excursions in Wavelet Theory433

B.1 Other Wavelet Constructions433

B.1.1 M-band Wavelets433

B.1.2 Wavelets with Rational Noninteger Dilation Factors434

B.1.3 Local Cosine Bases434

B.1.4 The Continuous Wavelet Transform435

B.1.5 Non-MRA Wavelets436

B.1.6 Multiwavelets436

B.2 Wavelets in Other Domains437

B.2.1 Wavelets on Intervals437

B.2.2 Wavelets in Higher Dimensions438

B.2.3 The Lifting Scheme438

B.3 Applications of Wavelets439

B.3.1 Wavelet Denoising439

B.3.2 Multiscale Edge Detection439

B.3.3 The FBI Fingerprint Compression Standard439

C References Cited in the Text441

Index445