小波变换深入探索：Stéphane Mallat的信号处理指南

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"Stéphane Mallat 著的《信号处理的小波导引》" 本书是小波变换领域的经典之作，由Stéphane Mallat撰写，他在法国巴黎综合理工学院和纽约大学Courant研究所任职。第二版的《信号处理的小波导引》深入浅出地介绍了小波分析在信号处理中的应用。小波变换是一种数学工具，它将信号在时间和频率上同时进行局部化分析，解决了传统傅立叶变换在时频分析中的局限性。Mallat的著作中详细阐述了小波理论的基础，包括小波函数的构造、多分辨率分析、以及小波变换的计算方法。通过小波变换，可以对非平稳信号进行精确分析，这对于音频、图像和工程信号的处理具有重要意义。书中不仅涵盖了理论知识，还提供了实际应用案例，如图像压缩、噪声消除、信号检测和故障诊断等。此外，Mallat还讨论了小波在信号去噪、模式识别、信号重构等多个领域的前沿研究进展，展示了小波分析在实际问题解决中的强大能力。小波变换的关键概念包括： 1. 小波基：一组具有有限支持或快速衰减的函数，可以形成一个完备的空间，用于表示各种类型的信号。 2. 多分辨率分析：小波分析的核心思想，通过不同尺度和位置的小波基函数来逐步逼近信号，实现精细到粗略的层次分析。 3. 小波分解与重构：将信号分解为一系列小波系数，这些系数对应于不同频率和时间的信息，然后可以通过逆变换将信号重新构造出来。 4. 小波包变换：扩展了经典小波变换，提供更全面的频率分辨率，允许在更多的频率子带内进行分析。 Mallat的这本书不仅是学术研究者的参考书，也是工程师和技术人员理解并应用小波变换的宝贵资源。通过阅读此书，读者可以深入了解小波理论，并掌握如何在实际信号处理任务中有效地利用小波变换。版权信息提示，本书的所有权利归Elsevier所有，未经许可，不得复制或传播。对于需要使用书中的内容，应向出版商申请正式授权。同时，读者也可以通过Elsevier的网站在线获取权限请求的相关信息。

Preface

Facing the unusual popularity of wavelets in sciences, I began to wonder whether

this was just another fashion that would fade away with time. After several years of

research and teaching on this topic, and surviving the painful experience of writing

a book, you may rightly expect that I have calmed my anguish. This might be the

natural self-delusion affecting any researcher studying his comer of the world, but

there might be more.

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 emer-

gence 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

xvi PREFACE

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.

9 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.

9 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.

9 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.

9 Sparse representations An orthonormal basis is useful if it defines a rep-

resentation 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.

9 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-Lobve

bases expansions. In sparse representations, simple non-linear diagonal

operators can considerably outperform "optimal" linear procedures, and

fast algorithms are available.

PREFACE

XVII

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 X1 i/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 departments at MIT

and Tel Aviv University, and in applied mathematics departments at the Courant

Institute and l~cole 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/.

Not all theorems of the book are proved in detail, but the important techniques

are included. I hope that the reader will excuse the lack of mathematical rigor in

the many instances where I have privileged ideas over details. Few proofs are long;

they are concentrated to avoid diluting the mathematics into many intermediate

results, which would obscure the text.

xviii

PREFACE

Course Design

Level numbers explained in Section 1.5.2 can help in designing

an introductory or a more advanced course. Beginning with a review of the Fourier

transform is often necessary. Although most applied mathematics students have

already seen the Fourier transform, they have rarely had the time to understand

it well. A non-technical review can stress applications, including the sampling

theorem. Refreshing basic mathematical results is also needed for electrical en-

gineering students. A mathematically oriented review of time-invariant signal

processing in Chapters 2 and 3 is the occasion to remind the student of elementary

properties of linear operators, projectors and vector spaces, which can be found

in Appendix A. For a course of a single semester, one can follow several paths,

oriented by different themes. Here are a few possibilities.

One can teach a course that surveys the key ideas previously outlined. Chapter

4 is particularly important in introducing the concept of local time-frequency de-

compositions. Section 4.4 on instantaneous frequencies illustrates the limitations

of time-frequency resolution. Chapter 6 gives a different perspective on the wavelet

transform, by relating the local regularity of a signal to the decay of its wavelet

coefficients across scales. It is useful to stress the importance of the wavelet van-

ishing moments. The course can continue with the presentation of wavelet bases

in Chapter 7, and concentrate on Sections 7.1-7.3 on orthogonal bases, multireso-

lution approximations and filter bank algorithms in one dimension. Linear and

non-linear approximations in wavelet bases are covered in Chapter 9. Depending

upon students' backgrounds and interests, the course can finish in Chapter 10 with

an application to signal estimation with wavelet thresholding, or in Chapter 11 by

presenting image transform codes in wavelet bases.

A different course may study the construction of new orthogonal bases and

their applications. Beginning with the wavelet basis, Chapter 7 also gives an in-

troduction to filter banks. Continuing with Chapter 8 on wavelet packet and local

cosine bases introduces different orthogonal tilings of the time-frequency plane.

It explains the main ideas of time-frequency decompositions. Chapter 9 on linear

and non-linear approximation is then particularly important for understanding how

to measure the efficiency of these bases, and for studying best bases search proce-

dures. To illustrate the differences between linear and non-linear approximation

procedures, one can compare the linear and non-linear thresholding estimations

studied in Chapter 10.

The course can also concentrate on the construction of sparse representations

with orthonormal bases, and study applications of non-linear diagonal operators in

these bases. It may start in Chapter 10 with a comparison of linear and non-linear

operators used to estimate piecewise regular signals contaminated by a white noise.

A quick excursion in Chapter 9 introduces linear and non-linear approximations

to explain what is a sparse representation. Wavelet orthonormal bases are then

presented in Chapter 7, with special emphasis on their non-linear approximation

properties for piecewise regular signals. An application of non-linear diagonal op-

erators to image compression or to thresholding estimation should then be studied

in some detail, to motivate the use of modem mathematics for understanding these

problems.

PREFACE

xix

A more advanced course can emphasize non-linear and adaptive signal pro-

cessing. Chapter 5 on frames introduces flexible tools that are useful in analyzing

the properties of non-linear representations such as irregularly sampled transforms.

The dyadic wavelet maxima representation illustrates the frame theory, with ap-

plications to multiscale edge detection. To study applications of adaptive repre-

sentations with orthonormal bases, one might start with non-linear and adaptive

approximations, introduced in Chapter 9. Best bases, basis pursuit or matching

pursuit algorithms are examples of adaptive transforms that construct sparse rep-

resentations for complex signals. A central issue is to understand to what extent

adaptivity improves applications such as noise removal or signal compression,

depending on the signal properties.

Responsibilities

This book was a one-year project that ended up in a never will

finish nightmare. Ruzena Bajcsy bears a major responsibility for not encourag-

ing me to choose another profession, while guiding my first research steps. Her

profound scientific intuition opened my eyes to and well beyond computer vision.

Then of course, are all the collaborators who could have done a much better job

of showing me that science is a selfish world where only competition counts. The

wavelet story was initiated by remarkable scientists like Alex Grossmann, whose

modesty created a warm atmosphere of collaboration, where strange new ideas

and ingenuity were welcome as elements of creativity.

I am also grateful to the few people who have been willing to work with me.

Some have less merit because they had to finish their degree but others did it on

a voluntary basis. I am thinking of Amir Averbuch, Emmanuel Bacry, Francois

Bergeaud, Geoff Davis, Davi Geiger, Fr6d6ric Falzon, Wen Liang Hwang, Hamid

Krim, George Papanicolaou, Jean-Jacques Slotine, Alan Willsky, Zifeng Zhang

and Sifen Zhong. Their patience will certainly be rewarded in a future life.

Although the reproduction of these 600 pages will probably not kill many

trees, I do not want to bear the responsibility alone. After four years writing and

rewriting each chapter, I first saw the end of the tunnel during a working retreat

at the Fondation des Treilles, which offers an exceptional environment to think,

write and eat in Provence. With WAVELAB, David Donoho saved me from spending

the second half of my life programming wavelet algorithms. This opportunity was

beautifully implemented by Maureen Clerc and J6r6me Kalifa, who made all the

figures and found many more mistakes than I dare say. Dear reader, you should

thank Barbara Burke Hubbard, who corrected my Franglais (remaining errors are

mine), and forced me to modify many notations and explanations. I thank her for

doing it with tact and humor. My editor, Chuck Glaser, had the patience to wait

but I appreciate even more his wisdom to let me think that I would finish in a year.

She will not read this book, yet my deepest gratitude goes to Branka with

whom life has nothing to do with wavelets.

St6phane Mallat

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