### 基本信息

- 原书名：A Wavelet Tour of Signal Processing,Second Edition
- 原出版社： Elsevier

- 作者：
**（法）Stephane Mallat** - 丛书名：
**经典原版书库** - 出版社：机械工业出版社
- ISBN：
**9787111127680** - 上架时间：2003-9-29
- 出版日期：2003 年9月
- 开本：16开
- 页码：637
- 版次：2-1
- 所属分类：数学 > 分析 > 傅里叶分析与小波分析

教材

### 内容简介

数学书籍

本书取材于作者在多所国际知名大学讲授”小波信号处理”课程时的讲义，作者以信号处理的问题为背

景，用浅显的数学语言阐述小波理论及应用，使读者可以透过复杂的数学公式来窥探小波的精髓，但又不致

陷入小波纯数学理论的迷宫。

本书既可以让计算机及电子工程系的学生了解工程问题的数学描述．又可以让数学系的学生了解数学公

式的工程意义，是一本极有价值的教材及参考书。

作者简介：

Stephane Mallat是纽约大学Courant学院计算机科学系的教授，法国巴黎Ecole理工大学应用数学系的教授，并兼任麻省理工学院电子工程系以及特拉维夫大学应用数学系的客座教授。

Mallat博士还曾荣获

·1990年IEEE信号处理学会评选的论文奖。

·1993年“斯隆”数学研究基金。

·1997年SPIE光学仪器工程师学会评选的杰出成就奖。

·1997年法国科学院评选的应用数学方面的“Blaise帕斯卡”奖。

### 作译者

### 目录

PREFACE TO THE SECOND EDITION xviii

NOTATION xx

I

INTRODUCTION TO A TRANSIENT WORLD

1.1 Fourier Kingdom

1.2 Time-Frequency Wedding

1.2.1 Windowed Fourier Transform

1.2.2 Wavelet Transform

1.3 Bases of Time-Frequency Atoms

1.3.1 Wavelet Bases and Filter Banks

1.3.2 Tilings of Wavelet Packet and Local Cosine Bases

1.4 Bases for What?

1.4.1 Approximation

1.4.2 Estimation

1.4.3 Compression

1.5 Travel Guide

1.5.1 Reproducible Computational Science

1.5.2 Road Map

II

### 前言

Wavelets are not based on a "bright new idea", but on concepts that already existed under various forms in many different fields. The formalization and emergence of this "wavelet theory" is the result of a multidisciplinary effort that brought together mathematicians, physicists and engineers, who recognized that they were independently developing similar ideas. For signal processing, this connection has created a flow of ideas that goes well beyond the construction of new bases or transforms.

A Personal Experience At one point, you cannot avoid mentioning who did what.For wavelets, this is a particularly sensitive task, risking aggressive replies from forgotten scientific tribes arguing that such and such results originally belong to them. As I said, this wavelet theory is truly the result of a dialogue between scien-tists who often met by chance, and were ready to listen. From my totally subjective point of view, among the many researchers who made important contributions, I would like to single out one, Yves Meyer, whose deep scientific vision was a major ingredient sparking this catalysis. It is ironic to see a French pure mathematician,

raised in a Bourbakist culture where applied meant trivial, playing a central role along this wavelet bridge between engineers and scientists coming from different disciplines.

When beginning my Ph.D. in the U.S., the only project I had in mind was to travel, never become a researcher, and certainly never teach. I had clearly destined myself to come back to France, and quickly begin climbing the ladder of some big corporation. Ten years later, I was still in the U.S., the mind buried in the hole of some obscure scientific problem, while teaching in a university. So what went wrong? Probably the fact that I met scientists like Yves Meyer, whose ethic and creativity have given me a totally different view of research and teaching. Trying to communicate this flame was a central motivation for writing this book. I hope

that you will excuse me if my prose ends up too often in the no man's land of scientific neutrality.

A Few Ideas Beyond mathematics and algorithms, the book carries a few impor-tant ideas that I would like to emphasize.

Time-frequency wedding Important information often appears through a simultaneous analysis of the signal's time and frequency properties. This motivates decompositions over elementary "atoms" that are well concen-trated in time and frequency. It is therefore necessary to understand how the uncertainty principle limits the flexibility of time and frequency transforms.

· Scale for zooming Wavelets are scaled waveforms that measure signal vari-ations. By traveling through scales, zooming procedures provide powerful characterizations of signal structures such as singularities.

· More and more bases Many orthonormal bases can be designed with fast computational algorithms. The discovery of filter banks and wavelet bases has created a popular new sport of basis hunting. Families of orthogonal bases are created every day. This game may however become tedious if not motivated by applications.

Sparse representations An orthonormal basis is useful if it defines a representation where signals are well approximated with a few non-zero coef ficients. Applications to signal estimation in noise and image compression are closely related to approximation theory.

Try it non-linear and diagonal Linearity has long predominated because of its apparent simplicity. We are used to slogans that often hide the limitations of "optimal" linear procedures such as Wiener filtering or Karhunen-Loeve bases expansions. In sparse representations, simple non, linear diagonal operators can considerably outperform "optimal" linear procedures, and fast algorithms are available.

WAVELAB and LASTWAVE Toolboxes Numerical experimentations are necessary to fully understand the algorithms and theorems in this book. To avoid the painful programming of standard procedures, nearly all wavelet and time-frequency algo-rithms are available in the WAvELAB package, programmed in MATLAB. WAVELAB is a freeware software that can be retrieved through the Internet. The correspon-dence between algorithms and WAVELAB subroutines is explained in Appendix B.

All computational figures can be reproduced as demos in WAVELAB. LASTWAVE is a wavelet signal and image processing environment, written in C for Xl 1/Unix and Macintosh computers. This stand-alone freeware does not require any additional commercial package. It is also described in Appendix B.

Teaching This book is intended as a graduate textbook. It took form after teaching "wavelet signal processing" courses in electrical engineering depart, lents at MIT and Tel Aviv University, and in applied mathematics departments at the Courant Institute and Ecole Polytechnique (Paris).

In electrical engineering, students are often initially frightened by the use of vector space formalism as opposed to simple linear algebra. The predominance of linear time invariant systems has led many to think that convolutions and the Fourier transform are mathematically sufficient to handle all applications. Sadly enough, this is not the case. The mathematics used in the book are not motivated by theoretical beauty; they are truly necessary to face the complexity of transient signal processing. Discovering the use of higher level mathematics happens to

be an important pedagogical side-effect of this course. Numerical algorithms and figures escort most theorems. The use of WAvELAB makes it particularly easy to include numerical simulations in homework. Exercises and deeper problems for class projects are listed at the end of each chapter.

In applied mathematics, this course is an introduction to wavelets but also to signal processing. Signal processing is a newcomer on the stage of legitimate applied mathematics topics. Yet, it is spectacularly well adapted to illustrate the applied mathematics chain, from problem modeling to efficient calculations of solutions and theorem proving. Images and sounds give a sensual contact with theorems, that can wake up most students. For teaching, formatted overhead transparencies with enlarged figures are available on the Internet:

http: //www. cmap. polytechnique, fr/~mallat/Wavetour_fig/.Francois Chaplais also offers an introductory Web tour of basic concepts in the book at

http: //cas. ensmp, fr/~chaplais/Wavetour_presentation/.