小波分析导论 英文PDF格式文档图书下载

Ⅰ Preliminaries 1

1 Functions and Convergence 3

1.1 Functions 3

1.1.1 Bounded（L∞）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

3.4 Convolution 68

3.5 Plancherel's Formula 72

3.6 The Fourier Transform for L2 Functions 75

3.7 Smoothness versus Decay 76

3.8 Dilation,Translation,and Modulation 79

3.9 Bandlimited Functions and the Sampling Formula 81

4 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

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

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 Momens 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

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.1 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.1 Other Wavelet Constructions 433

B.1.1 M-band Wavelets 433

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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