Wavelet Analysis

by Resnikoff, Howard L.; Wells, R. O.

• ISBN13: 9780387983837
• ISBN10: 038798383X
• Format: Hardcover
• Copyright: 1998-09-01
• Publisher: Springer Verlag

Summary

The past decade has witnessed the rapid development of a new mathematical tool, called wavlet analysis, for analyzing complex signals. It has begin to play a serious role in applications ranging from communications to geophysics, and from simulations to image processing. Like Fourier analysis (of which it is a generalization), or musical notation, wavelet analysis provides a method for representing a set of complex phenomena in a simpler, more compact, and thus more efficient manner. This text introduces the ideas and methods of wavelet analysis, relates them to previously known methods in mathematics and engineering, and shows how to apply wavelet analysis to digital signal processing. It begins by describing the multiscale (sometimes called "fractal") nature of information in many aspects of thereal world; it then turns to the algebra and analysis of wavelet matrices, scaling and wavelet functions, and the corresponding analysis of square-integrable functins on a space. The discussion then turns from the continuous to the discrete and shows how a properly selected set of wavelets can be used to represent -- and even differentiate -- a wide range of signls efficiently and effectively. The last part of the book presents a wide variety of applications of wavelets to probllems in data compression and telecommunications.

Table of Contents

Preface	vii
I The Scalable Structure of Information	3	(36)
1 The New Mathematical Engineering	3	(9)
1.1 Introduction	3	(3)
1.2 Trial and Error in the Twenty-First Century	6	(1)
1.3 Active Mathematics	6	(1)
1.4 The Three Types of Bandwidth	7	(3)
1.5 An Introduction to This Book	10	(2)
2 Good Approximations	12	(18)
2.1 Approximation and the Perception of Reality	12	(4)
2.2 Information Gained from Measurement	16	(9)
2.3 Functions and Their Representation	25	(5)
3 Wavelets: A Positional Notation for Functions	30	(9)
3.1 Multiresolution Representation	30	(2)
3.2 The Democratization of Arithmetic: Positional Notation for Numbers	32	(2)
3.3 Music Notation as a Metaphor for Wavelet Series	34	(1)
3.4 Wavelet Phase Space	35	(4)
II Wavelet Theory	39	(152)
4 Algebra and Geometry of Wavelet Matrices	39	(47)
4.1 Introduction	39	(2)
4.2 Wavelet Matrices	41	(6)
4.3 Haar Wavelet Matrices	47	(8)
4.4 The Algebraic and Geometric Structure of the Space of Wavelet Matrices	55	(25)
4.5 Wavelet Matrix Series and Discrete Orthonormal Expansions	80	(6)
5 One-Dimensional Wavelet Systems	86	(54)
5.1 Introduction	86	(1)
5.2 The Scaling Equation	86	(19)
5.3 Wavelet Systems	105	(32)
5.4 Recent Developments: Multiwavelets and Lifting	137	(3)
6 Examples of One-Dimensional Wavelet Systems	140	(25)
6.1 Introduction to the Examples	140	(1)
6.2 Universal Scaling Functions	141	(5)
6.3 Orthonormal Wavelet Systems	146	(10)
6.4 Flat Wavelets	156	(1)
6.5 Polynomial-Regular and Smooth Wavelets	157	(6)
6.6 Fourier-Polynomial Wavelet Matrices	163	(2)
7 Higher-Dimensional Wavelet Systems	165	(26)
7.1 Introduction	165	(9)
7.2 Scaling Functions	174	(7)
7.3 Scaling Tiles	181	(2)
7.4 Orthonormal Wavelet Bases	183	(8)
III Wavelet Approximation and Algorithms	191	(152)
8 The Mallat Algorithm	191	(11)
8.1 Introduction	191	(1)
8.2 Wavelet Series and the Mallat Algorithm	192	(4)
8.3 The Mallat Algorithm for Periodic Data	196	(6)
9 Wavelet Approximation	202	(34)
9.1 Introduction	202	(1)
9.2 Vanishing Moments of Wavelet Bases	203	(3)
9.3 Sampling, Reconstruction, and Approximation	206	(7)
9.4 Newton's Method and the Problem of Constructing the Orthogonal Coifman Scaling Function	213	(11)
9.5 Biorthogonal Coifman Wavelet Systems	224	(9)
9.6 Comparison with Daubechies Wavelet Systems	233	(3)
10 Wavelet Calculus and Connection Coefficients	236	(30)
10.1 An Introduction to Connection Coefficients	236	(3)
10.2 Fundamental Properties of Connection Coefficients for First-Order Differentiation	239	(10)
10.3 Wavelet Differentiation and Classical Finite Difference Operators	249	(8)
10.4 Algorithms for Computing Connection Coefficients	257	(9)
11 Multiscale Representation of Geometry	266	(14)
11.1 Introduction	266	(1)
11.2 Differential Forms and Distributions	267	(2)
11.3 A Multiresolution Representation of Boundary Integration	269	(2)
11.4 Elements of Geometric Measure Theory	271	(2)
11.5 The Wavelet Representation of Integration over Domains and Their Boundaries	273	(7)
12 Wavelet-Galerkin Solutions of Partial Differential Equations	280	(63)
12.1 Introduction	280	(3)
12.2 Estimates for Wavelet-Based Approximations to Elliptic Partial Differential Equations	283	(7)
12.3 The Dirichlet Problem	290	(9)
12.4 The Neumann Problem for Elliptic Operators: Variational Formulations	299	(9)
12.5 Iterative Multiscale Methods for Elliptic Boundary Value Problems	308	(18)
12.6 A Wavelet-Based Multigrid Iterative Method for an Anisotropic Partial Differential Equation	326	(17)
IV Wavelet Applications	343	(54)
13 Wavelet Data Compression	343	(23)
13.1 Understanding Compression	343	(2)
13.2 Image Compression	345	(3)
13.3 Transform Image Compression Systems	348	(2)
13.4 Wavelet Image Compression	350	(6)
13.5 Embedded Coding and the Wavelet-Difference-Reduction Compression Algorithm	356	(4)
13.6 Multiresolution Audio Compression	360	(4)
13.7 Denoising Algorithms	364	(2)
14 Modulation and Channel Coding	366	(31)
14.1 Understanding Channel Coding	366	(3)
14.2 Multicarrier Communication Systems	369	(1)
14.3 Wavelet Filter Design	370	(3)
14.4 Waveform Design	373	(2)
14.5 Wavelet Channel Coding	375	(1)
14.6 The Wavelet Channel Coding Algorithm	376	(7)
14.7 Wavelet Channel Coding and Digital Modulation Techniques	383	(1)
14.8 Performance of Wavelet Channel Coding	384	(2)
14.9 The DWMT Modem	386	(9)
14.10 Applications and Test Results	395	(2)
References	397	(16)
Index	413