小波的理论与应用

PrefAce

ChApter 1 Overview of Fourier AnAlysis 1

Introduction 1

1.1

1.2 FourierseriespreliminAry 3

Continuous Fourier trAnsform 15

1.3

15

1.3.1 Concept And bAsic properties of continuous Fourier trAnsform Fourier trAnsform And lineAr .lter 23

1.3.2

1.4 SAmpling theorem And uncertAinty principle 27

Discrete Fourier trAnsform 32

1.5 ChApter 2 MAthemAticAl FoundAtion 35

2.1 EuclideAn Algorithm And lifting scheme 35

2.2 Hilbert spAce 41

2.2.1 OrthogonAlity And orthogonAl complement 41

2.2.2 OptimAl ApproximAtion in closed convex set And orthogonAl decomposition 42

OrthogonAl fAmily {.(x . k)|k ∈ Z} in L2(R)spAce 47

2.3

2.4 FrAmes in Hilbert spAce 51

ChApter 3 HAAr WAvelet AnAlysis 58

Short-time Fourier trAnsform 58

HAAr wAvelet 63

3.1

3.2

3.3 Decomposition And reconstruction Algorithms of signAls bAsed on HAAr wAvelet 68

ChApter 4 Multiresolution AnAlysis And WAvelets Design 78

The multiresolution frAmework 78

4.1

4.2 The MAllAt Algorithm for signAl decomposition And reconstruction 87

4.3 ImplementAtion of MAllAt Algorithm 93

InitiAlizAtion 93

4.3.1

4.3.2 BoundAry extension 96

4.4 WAvelet pAckets 99

4.5 ComputAtionofscAlingfunction 102

4.5.1 CAscAde Algorithm 102

4.5.2 MAtrix equAtion Algorithm 104

4.6 DAubechiesorthogonAlcompActlysupportedwAvelets 105

4.7 RAtionAlized compActly supported orthogonAl wAvelets 118

4.8 BiorthogonAlmultiresolutionAnAlysis 121

4.9 DesignofcompActlysupportedbiorthogonAlwAvelets 123

4.10 Design of perfect reconstruction .lters And biorthogonAl wAvelets with rAtionAlized coe.cients 131

4.10.1 Decomposition 131

Reconstruction 132

4.10.2

4.10.3 Discussion About the pArAmeters 134

ChApter 5 M-bAnd WAvelets And MultiwAvelets 138

Introduction 138

5.1

5.2 FundAmentAls of multirAte signAl processing 139

5.3 The properties of the perfect reconstruction .lter (PRF) bAnks 141

5.3.1 The two chAnnel quArdure mirror .lter (QMF) bAnks 145

5.3.2 The two chAnnel conjugAted quArdure .lter (CQF) bAnks 146

5.3.3 The design of M-chAnnel PR QMF bAnks 146

5.4 The block And lApped trAnsform bAsed on triAngulAr bAsis functions 148

PR .lter bAnks And M-bAnd wAvelets 154

5.5

5.5.1 The construction of M-bAnd orthogonAl wAvelets 156

5.5.2 The construction of M-bAnd biorthogonAl wAvelets 164

5.5.3 Construct M-bAnd wAvelets bAsed on cosine modulAtion 173

Multi-FB And multiwAvelets 180

5.6

5.7 MultiwAvelet MRA And discrete multiwAvelet trAnsform 183

5.8 The bAsic pricinple for the construction of multiwAvelet 186

5.8.1 The conditions in time domAin for orthogonAl .lters 186

5.8.2 The conditions in frequency domAin for orthogonAl .lters 187

5.9 TheconstructionoforthogonAlmultiwAvelets 187

5.9.1 Construct multiwAvelets bAsed on ApproximAtion orders And regulArity orders 188

5.9.2 Construct orthogonAl multiwAvelet bAsed on OPTER 201

ChApter 6 The WAvelet BAsed on the Lifting Scheme And Integer Discrete TrAnsform 207

Introduction 207

6.1

6.2 The design of the wAvelet trAnsform bAsed on the lifting scheme 207

6.2.1 Perfect reconstruction .lters And lifting decomposition 208

6.2.2 The lifting fActorizAtion of the symmetric biorthogonAl wAvelet 213

6.3 Integer DCTs And fAst Algorithms 218

6.3.1 Integer DCTs And Algorithms 218

6.3.2 The design of scAled DCT-II 224

6.4 Integer implement of the lApped biorthogonAl trAnsform 227

ChApter 7 The WAvelet-bAsed ImAge Compression 231

7.1 Introduction of imAge compression 231

7.1.1 BAsic concept 231

7.1.2 Coding model 232

7.2 ProgressiveimAgecoding 234

7.2.1 BAsic conceptions in progressive imAge coding 234

7.2.2 Embedded zerotree wAvelet (EZW) coding 238

7.2.3 Set pArtitioning in hierArchicAl trees (SPIHT) 240

7.2.4 Set pArtitioning embedded block (SPECK) coding 245

7.3 LinebAsedimAgecompression 246

7.3.1 Principle of low memory imAge compression 246

7.3.2 Line bAsed entropy coding 247

7.3.3 Experiments And results 249

7.4 Embedded block coding with optimAl truncAtion (EBCOT) 251

Code-block 252

7.4.1

7.4.2 QuAlity lAyers 252

7.4.3 Bit plAn coding 253

7.5 FAst wAvelet trAnsform in imAge compression 254

7.5.1 VAnishing moment And high frequency coe.cients 254

7.5.2 The lifting scheme in imAge compression 257

ChApter 8 WAvelet BAsed ImAge Denoising And EnhAncement 261

8.1 The singulArity detection of signAl And wAvelet trAnsform modulus mAximA 262

8.2 The thresholding methods for denoising 263

The selection of threshold function 264

8.2.1 The estimAtion of threshold 265

8.2.2

8.3 ScAlefActorshrinkingmethodfordenoising 273

8.4 CorrelAtion bAsed denoising methods 276

8.5 ImAgedenoisingbAsedonDTCWsAndmultiwAvelets 280

8.5.1 DTCWT bAsed imAge denoising 280

8.5.2 MultiwAvelet bAsed denosing 286

8.6 ImAgeenhAncementbAsedonthemultiscAletrAnsform 288

8.6.1 The bAsic concepts of imAge enhAncement 288

8.6.2 ImAge enhAncement techniques bAsed on multiscAle method 289

ChApter 9 Ridgelets And Its ApplicAtions 303

Introduction 303

9.1

9.2 The ridgelet trAnsform 304

9.2.1 Ridglet And continuous ridgelet trAnsform 304

9.2.2 Discrete trAnsform: ridgelet frAmes 308

9.2.3 MonoscAle ridgelets 311

Curvelet 313

9.2.4

9.3 The ApplicAtion of ridgelet trAnsform in signAl processing 315

9.3.1 ImAge enhAncement 315

Noise AttenuAtion 317

9.3.2

9.3.3 ImAge reconstruction 321

ChApter 10 ApplicAtion of WAvelet TrAnsform in the DigitAl WAtermArking 323 Introduction 323

10.1

10.2 DigitAl wAtermArking method bAsed on .oAt wAvelet trAnsforms 331

10.3 FrAgile digitAl wAtermArking method of integer wAvelet trAnsforms 335

10.3.1 The construction of HAsh function through RijndAel encrypted Algorithm 335

10.3.2 The insertion And test of wAtermArk to pictures 337

10.3.3 Experiments And compArisons 339

10.4 The technology of the visible digitAl wAtermArking bAsed on integrAl wAvelet trAnsform with pArAmeters 341

10.4.1 RijndAel code with di.erent type builds HAsh function 343

10.4.2 Inserting And removing of visible digitAl wAtermArk 344

10.4.3 Experiments And compArisons 347

10.5 The trAnslucent digitAl wAtermArking technique bAsed on integrAl wAvelet trAnsform with pArAmeters 350

10.5.1 The visuAl weight AnAlysis bAsed on wAvelet domAin quAntizAtion 350

10.5.2 TrAnslucent wAtermArking Algorithm reAlizAtion 351

10.5.3 ExperimentAl results 354

10.6 SimultAneous embedding of vArious kinds of wAtermArks bAsed on pArAmetric integer wAvelet trAnsforms 358

ChApter 11 The Solution of PDE BAsed on WAvelets 366

Introduction 366

11.1

11.2 The representAtion of operAtor T by wAveletbAses 367

11.2.1 The nonstAndArd form of representAtion 368

StAndArd form 377

11.2.2

11.2.3 The compression of operAtor by wAvelet bAses 379

11.3 Solving PDE bAsed on wAvelets 380

11.3.1 A clAssic exAmple 380

11.3.2 The periodizAtion of the originAl problem 381

11.3.3 Construct the inverse of periodized di.erentiAl operAtor 384

11.4 Computing the inverse of ellipse di.erentiAl operAtor bAsed on multiscAle method 385

ChApter 12 The Solution of Ill-conditioned Symmetric Toeplitz Systems viA Two-grid And WAvelet Methods 390 Introduction 390

12.1

12.2 Multigrid method 391

12.3 The solution of ill-conditioned Toeplitz systems bAsed on wAvelets And MGM 395

12.3.1 Toeplitz systems 395

12.3.2 Solving Toeplitz system by TGM 396

12.3.3 Solving Toeplitz system by MGM 399

NumericAl results 401

12.4

This book could be divided into two parts i.e. fundamental wavelet transform theory and method and some important applications of wavelet transform. In the first part, as preliminary knowledge, the Fourier analysis, inner product space, the characteristics of Haar functions, and concepts of multi-resolution analysis, are introduced followed by a description on how to construct wavelet functions both multi-band and multi wavelets, and finally introduces the design of integer wavelets via lifting schemes and its application to integer transform algorithm. In the second part, many applications are discussed in the field of image and signal processing by introducing other wavelet variants such as complex wavelets, ridgelets, and curvelets. Important application examples include image compression, image denoising/restoration, image enhancement, digital watermarking, numerical solution of partial differential equations, and solving ill- conditioned Toeplitz system. The book is intended for senior undergraduate students and graduate students.