Ripples in Mathematics

The Discrete Wavelet Transform

By: A. Jensen, Anders la Cour-Harbo

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Paperback | 6 June 2001

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260 Pages

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22.86 x 15.24 x 1.27

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Yet another book on wavelets. There are many books on wavelets available, written for readers with different backgrounds. But the topic is becoming ever more important in mainstream signal processing, since the new JPEG2000 standard is based on wavelet techniques. Wavelet techniques are also impor- tant in the MPEG-4 standard. So we thought that there might be room for yet another book on wavelets. This one is limited in scope, since it only covers the discrete wavelet trans- form, which is central in modern digital signal processing. The presentation is based on the lifting technique discovered by W. Sweldens in 1994. Due to a result by I. Daubechies and W. Sweldens from 1996 this approach covers the same class of discrete wavelet transforms as the one based on two channel filter banks with perfect reconstruction. The goal of this book is to enable readers, with modest backgrounds in mathematics, signal analysis, and programming, to understand wavelet based techniques in signal analysis, and perhaps to enable them to apply such methods to real world problems. The book started as a set of lecture notes, written in Danish, for a group of teachers of signal analysis at Danish Engineering Colleges. The material has also been presented to groups of engineers working in industry, and used in mathematics courses at Aalborg University.

Industry Reviews

From the reviews:

"This book is a very well-written introduction to discrete wavelet transforms, very convenient for students in electrical engineering, computer science, and applied mathematics. It is based on the lifting approach to discrete wavelet transforms ... . MATLAB is used as the computational environment for examples and implementations of discrete wavelet transforms." (Manfred Tasche, Zentralblatt MATH, Vol. 989 (14), 2002)

"This is an algorithm based, completely elementary introduction to the discrete wavelet transform (DWT) and wavelet packet transform, easy to read and easy to understand, well suited for an introductory course on wavelets for undergraduate students of applied sciences or mathematics. ... Implementations and examples using basic Matlab (TM) as well as the public domain ubi-wave wavelet toolbox help to further a deeper understanding of the algorithms." (C. Cenker, Internationale Mathematische Nachrichten, Vol. 56 (189), 2002)

"This book gives an introduction to the discrete wavelet transform and some of its applications. It is based on a novel approach to discrete wavelets called lifting. ... MATLAB is used as the computational environment for examples and implementation of transforms. The book is well suited for undergraduate mathematics and electrical engineering students and engineers in industry." (ETDE Energy Database, December, 2001)

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You Can Find This Book In

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Introduction	p. 1
Prerequisites	p. 1
Guide to the Book	p. 2
Background Information	p. 3
A First Example	p. 7
The Example	p. 7
Generalizations	p. 10
Exercises	p. 10
The Discrete Wavelet Transform via Lifting	p. 11
The First Example Again	p. 11
De nition of Lifting	p. 13
A Second Example	p. 17
Lifting in General	p. 19
DWT in General	p. 21
Further Examples	p. 23
Exercises	p. 24
Analysis of Synthetic Signals	p. 25
The Haar Transform	p. 25
The CDF(2,2) Transform	p. 31
Exercises	p. 33
Interpretation	p. 37
The First Example	p. 37
Further Results on the Haar Transform	p. 40
Interpretation of General DWT	p. 45
Exercises	p. 50
Two Dimensional Transforms	p. 51
One Scale DWT in Two Dimensions	p. 51
Interpretation and Examples	p. 53
A 2D Transform Based on Lifting	p. 57
Exercises	p. 60
Lifting and Filters I	p. 61
Fourier Series and the z-Transform	p. 61
Lifting in the z-Transform Representation	p. 64
Two Channel Filter Banks	p. 69
Orthonormal and Biorthogonal Bases	p. 74
Two Channel Filter Banks in the Time Domain	p. 76
Summary of Results on Lifting and Filters	p. 79
Properties of Orthogonal Filters	p. 79
Some Examples	p. 82
Exercises	p. 86
Wavelet Packets	p. 87
From Wavelets to Wavelet Packets	p. 87
Choice of Basis	p. 90
Cost Functions	p. 96
Exercises	p. 98
The Time-Frequency Plane	p. 99
Sampling and Frequency Contents	p. 99
De nition of the Time-Frequency Plane	p. 102
Wavelet Packets and Frequency Contents	p. 107
More about Time-Frequency Planes	p. 111
More Fourier Analysis. The Spectrogram	p. 121
Exercises	p. 125
Finite Signals	p. 127
The Extent of the Boundary Problem	p. 127
DWT in Matrix Form	p. 130
Gram-Schmidt Boundary Filters	p. 134
Periodization	p. 140
Moment Preserving Boundary Filters	p. 144
Exercises	p. 148
Implementation	p. 151
Introduction to Software	p. 151
Implementing the Haar Transform Through Lifting	p. 152
Implementing the DWT Through Lifting	p. 155
The Real Time Method	p. 160
Filter Bank Implementation	p. 171
Construction of Boundary Filters	p. 175
Wavelet Packet Decomposition	p. 180
Wavelet Packet Bases	p. 181
Cost Functions	p. 185
Exercises	p. 185
Lifting and Filters II	p. 189
The Three Basic Representations	p. 189
From Matrix to Equation Form	p. 190
From Equation to Filter Form	p. 192
From Filters to Lifting Steps	p. 193
Factoring Daubechies 4 into Lifting Steps	p. 202
Factorizing Coi et 12 into Lifting Steps	p. 204
Exercises	p. 209
Wavelets in Matlab	p. 211
Multiresolution Analysis	p. 212
Frequency Properties of the Wavelet Transform	p. 216
Wavelet Packets Used for Denoising	p. 220
Best Basis Algorithm	p. 225
Some Commands in Uvi Wave	p. 230
Exercises	p. 232
Applications and Outlook	p. 233
Applications	p. 233
Outlook	p. 235
Some Web Sites	p. 237
References	p. 239
Index	p. 241
Table of Contents provided by Publisher. All Rights Reserved.

Ripples in Mathematics

The Discrete Wavelet Transform

At a Glance

Paperback

Industry Reviews

Shipping

How to return your order

Defective items

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