Introduction to Wavelet Transforms
Introduction to Wavelet Transforms provides the basics of wavelet transforms in a self-contained manner. Applications of wavelet transform theory permeate our daily lives. Therefore, it is imperative to have a strong foundation for this subject. Features No prior knowledge of the subject is assumed....
Saved in:
| Main Author: | |
|---|---|
| Format: | eBook Book |
| Language: | English |
| Published: |
Boca Raton
CRC Press
2020
CRC Press LLC Chapman & Hall |
| Edition: | 1 |
| Subjects: | |
| ISBN: | 9780367438791, 0367438798, 9781032174839, 1032174838 |
| Online Access: | Get full text |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
Table of Contents:
- Cover -- Half Title -- Title Page -- Copyright Page -- Dedication -- Table of Contents -- Preface -- List of Symbols -- Greek Symbols -- Part I: Basics of Wavelet Transforms -- 1. Introduction to Wavelets -- 1.1 Introduction -- 1.2 Representation of Functions -- 1.2.1 Basis Representation -- 1.2.2 Representation via Frames -- 1.2.3 Riesz Basis Representation -- 1.2.4 Multiscale Representation -- 1.2.5 Representation via Dictionaries -- 1.2.6 Redundancy in Representation -- 1.3 Fourier Analysis -- 1.3.1 Fourier Series -- 1.3.2 Fourier Transform and Spectral Analysis -- 1.4 Wavelet Analysis -- 1.5 Why Use Wavelets? -- 1.6 Story of Wavelets -- 1.7 Applications -- Problems -- 2. Continuous Wavelet Transform -- 2.1 Introduction -- 2.2 Basics of Continuous Wavelet Transform -- 2.3 Properties of Continuous Wavelet Transform -- 2.4 Examples -- 2.4.1 Wavelets -- 2.4.2 Continuous Wavelet Transforms -- 2.5 Regularity of Wavelets -- Problems -- 3. Discrete Wavelet Transform -- 3.1 Introduction -- 3.2 Basics of Discrete Wavelet Transform -- 3.3 Multiresolution Analysis -- 3.4 Scaling Function -- 3.5 Characterization of the Wj Spaces -- 3.6 Expansions and Transformations -- 3.6.1 Coefficient Relationships Between Different Scales -- 3.6.2 Pyramid Algorithm -- 3.7 Digital Filter Interpretation -- 3.8 Computation of the Scaling Function -- 3.9 An Alternate Multiresolution Analysis -- Problems -- 4. Daubechies Wavelets -- 4.1 Introduction -- 4.2 Regularity and Moments -- 4.2.1 Regularity -- 4.2.2 Moments -- 4.3 Compactness -- 4.4 Construction of Daubechies Scaling Coefficients -- 4.5 Computation of Scaling and Mother Wavelet Functions -- Problems -- 5. Some Examples of Wavelets -- 5.1 Introduction -- 5.2 Shannon Wavelets -- 5.3 Meyer Wavelets -- 5.4 Splines -- 5.4.1 Properties of B-Splines -- 5.4.2 Examples of B-Splines -- 5.4.3 Orthogonalization of B-Splines
- 10.5.1 Lifting Technique Via Polyphase Matrix -- 10.5.2 Polyphase Matrix Factorization -- 10.5.3 Examples -- 10.6 Second-Generation Wavelets -- Problems -- 11. Wavelet Packets -- 11.1 Introduction -- 11.2 Elements of Graph Theory -- 11.3 Elementary Properties of Wavelet Packets -- 11.3.1 Basic Wavelet Packets -- 11.3.2 General Wavelet Packets -- 11.4 Wavelet Packet Transformation -- 11.5 Best Basis Selection Algorithm -- 11.5.1 Cost Function and Measures -- 11.5.2 Characteristics of Wavelet Packet Trees -- 11.5.3 Algorithm for Selection of Best Basis -- Problems -- 12. Lapped Orthogonal Transform -- 12.1 Introduction -- 12.2 Orthogonal Transforms -- 12.3 Transform Efficiency -- 12.3.1 Covariance Matrices -- 12.3.2 Transform Metrics -- 12.4 AR(1) Process -- 12.5 Karhunen-Loéve Transform -- 12.5.1 KLT Matrix -- 12.5.2 Properties of the KLT Matrix -- 12.5.3 Karhunen-Loéve Transform of Vector x -- 12.6 Discrete Cosine Transform -- 12.6.1 Basics of the DCT -- 12.6.2 Computation of the DCT -- 12.6.3 DCT Basis Vectors as Eigenvectors of Special Matrices -- 12.7 Lapped Transform -- Problems -- Part III: Signal Processing -- 13. Discrete Fourier Transform -- 13.1 Introduction -- 13.2 Elements of the DFT -- 13.2.1 Properties of the DFT -- 13.2.2 Computation of the DFT -- 13.3 DFT Computation for Ramanujan Numbers -- 13.3.1 Ramanujan Numbers -- 13.3.2 Recursive Computations -- 13.3.3 Discrete Fourier Transform Computation -- Problems -- 14. The z-Transform and Discrete-Time Fourier Transform -- 14.1 Introduction -- 14.2 z-Transform -- 14.2.1 Properties -- 14.2.2 Down-Sampled and Up-Sampled Sequences -- 14.2.3 Inversion -- 14.3 Discrete-Time Fourier Transform -- Problems -- 15. Elements of Continuous-Time Signal Processing -- 15.1 Introduction -- 15.2 Continuous-Time Signal Processing -- Problems -- 16. Elements of Discrete-Time Signal Processing
- 16.1 Introduction -- 16.2 Discrete-Time Signal Processing -- 16.3 z-Transform Analysis of a Discrete-Time Linear System -- 16.4 Special Filters -- 16.4.1 Linear Phase Filter -- 16.4.2 All-Pass Filter -- 16.4.3 Minimum-Phase Filter -- 16.4.4 Subband Coding -- Problems -- Part IV: Mathematical Concepts -- 17. Set-Theoretic Concepts and Number Theory -- 17.1 Introduction -- 17.2 Sets -- 17.2.1 Set Operations -- 17.2.2 Interval Notation -- 17.3 Functions and Sequences -- 17.3.1 Sequences -- 17.4 Elementary Number-Theoretic Concepts -- 17.4.1 Countability -- 17.4.2 Divisibility -- 17.4.3 Prime Numbers -- 17.4.4 Greatest Common Divisor -- 17.4.5 Polynomials -- 17.5 Congruence Arithmetic -- Problems -- 18. Matrices and Determinants -- 18.1 Introduction -- 18.2 Elements of Matrix Theory -- 18.2.1 Basic Matrix Operations -- 18.2.2 Different Types of Matrices -- 18.2.3 Matrix Norm -- 18.3 Determinants -- 18.4 More Matrix Theory -- 18.4.1 Rank of a Matrix -- 18.4.2 Matrices as Linear Transformations -- 18.5 Spectral Analysis of Matrices -- Problems -- 19. Applied Analysis -- 19.1 Introduction -- 19.2 Basic Concepts -- 19.2.1 Point Sets -- 19.2.2 Limits, Continuity, Derivatives, and Monotonicity -- 19.2.3 Partial Derivatives -- 19.2.4 Singularity and Related Topics -- 19.3 Complex Analysis -- 19.3.1 De Moivre and Euler Identities -- 19.3.2 Limits, Continuity, Derivatives, and Analyticity -- 19.3.3 Contours or Curves -- 19.3.4 Integration -- 19.3.5 Infinite Series -- 19.4 Asymptotics -- 19.5 Fields -- 19.6 Vector Spaces over Fields -- 19.7 Linear Mappings -- 19.8 Tensor Products -- 19.9 Vector Algebra -- 19.10 Vector Spaces Revisited -- 19.10.1 Normed Vector Space -- 19.10.2 Complete Vector Space and Compactness -- 19.10.3 Inner Product Space -- 19.10.4 Orthogonality -- 19.10.5 Gram-Schmidt Orthogonalization Process -- 19.11 More Hilbert Spaces
- Problems -- 6. Applications -- 6.1 Introduction -- 6.2 Signal Denoising via Wavelets -- 6.3 Image Compression -- 6.4 Wavelet Neural Networks -- 6.4.1 Artificial Neural Network -- 6.4.2 Gradient Descent -- 6.4.3 Wavelets and Neural Networks -- 6.4.4 Learning Algorithm -- 6.4.5 Wavelons with Vector Inputs -- Problems -- Part II: Intermediate Topics -- 7. Periodic Wavelet Transform -- 7.1 Introduction -- 7.2 Periodization of a Function -- 7.3 Periodization of Scaling and Wavelet Functions -- 7.4 Periodic Multiresolution Analysis -- 7.5 Periodic Series Expansions -- 7.6 Fast Periodic Wavelet Transform -- 7.6.1 Computational Complexity -- 7.6.2 A Matrix Formulation -- Problems -- 8. Biorthogonal Wavelet Transform -- 8.1 Introduction -- 8.2 Biorthogonal Representations of a Function -- 8.3 Biorthogonal Wavelets -- 8.3.1 Motivation for the Use of Biorthogonal Wavelet Bases -- 8.3.2 Biorthogonal Spaces -- 8.3.3 Biorthogonal Space Bases -- 8.3.4 Biorthogonal Scaling Functions and Dual Wavelets -- 8.3.5 Biorthogonal Relationships in the Frequency Domain -- 8.3.6 Relationships between Scaling Coefficients -- 8.3.7 Support Values -- 8.4 Decomposition and Reconstruction of Functions -- 8.4.1 Basics -- 8.4.2 Digital Filter Interpretation -- 8.4.3 Symmetric h(n)'s and eh(n)'s -- 8.4.4 Moments -- 8.5 Construction of Biorthogonal Scaling Coefficients -- 8.6 B-Spline-Based Biorthogonal Wavelets -- 8.7 Semi-Orthogonal Wavelets -- Problems -- 9. Coiflets -- 9.1 Introduction -- 9.2 Preliminaries -- 9.3 Construction of Coiflets -- Problems -- 10. The Lifting Technique -- 10.1 Introduction -- 10.2 Laurent Polynomials -- 10.3 Greatest Common Divisor of Two Laurent Polynomials -- 10.4 Biorthogonal Wavelet Transform -- 10.4.1 Perfect Deconstruction and Reconstruction -- 10.4.2 Single-Stage Deconstruction and Reconstruction -- 10.5 The Lifting Technique
- 19.11.1 Non-Orthogonal Expansion -- 19.11.2 Biorthogonal Bases -- Problems -- 20. Fourier Theory -- 20.1 Introduction -- 20.2 Fourier Series -- 20.2.1 Generalized Functions -- 20.2.2 Conditions for the Existence of Fourier Series -- 20.2.3 Complex Fourier Series -- 20.2.4 Trigonometric Fourier Series -- 20.2.5 Generalized Fourier Series -- 20.3 Transform Techniques -- 20.3.1 Fourier Transform -- 20.3.2 Short-Time Fourier Transform -- 20.3.3 Wigner-Ville Transform -- Problems -- 21. Probability Theory and Stochastic Processes -- 21.1 Introduction -- 21.2 Postulates of Probability Theory -- 21.3 Random Variables -- 21.4 Average Measures -- 21.4.1 Expectation -- 21.4.2 Second-Order Expectations -- 21.5 Independent Random Variables -- 21.6 Moment-Generating Function -- 21.7 Examples of Some Distributions -- 21.7.1 Discrete Distributions -- 21.7.2 Continuous Distributions -- 21.7.3 Multivariate Gaussian Distribution -- 21.8 Stochastic Processes -- Problems -- References -- Index