小波分析入门：十一章详解图解

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《小波分析第一课（英文第二版）》是一本针对初学者编写的入门书籍，由CRC Press LLC于1996年出版，是Studies in Advanced Mathematics系列的一部分，系列编辑为Steven G. Krantz，来自华盛顿大学圣路易斯分校。本书共有十一章，以其深入浅出的讲解和丰富的图表，引导读者逐步理解小波分析这一复杂的数学概念。小波分析是信号处理、图像处理、数据分析等领域的重要工具，它将信号分解为一系列不同尺度和频率的局部特征，从而提供了比传统傅立叶变换更精细的频域解析。小波分析的特点在于它能够捕捉到数据在不同时间尺度下的局部特性，这对于解决时变信号处理问题非常有效。书中可能涵盖了以下几个核心知识点： 1. **小波基础知识**：介绍了小波函数的基本概念，包括母波、基函数和多分辨分析，帮助读者理解小波是如何作为时间-频率分析的窗口函数。 2. **小波变换**：详细解释了如何通过小波函数对信号进行离散或连续的小波变换，以及这些变换如何揭示信号的局部特征。 3. **小波基的选择**：探讨了不同种类的小波基，如Haar小波、Daubechies小波、Morlet小波等，及其各自的适用场景。 4. **多尺度分析**：介绍多尺度分解技术，这是小波分析的核心，通过递归地应用小波变换来获取信号的不同层次信息。 5. **小波包分析**：如果本书涉及，还可能涵盖了小波包，这是一个扩展版本的小波分析，能提供更丰富的频率分辨率。 6. **应用实例**：书中可能包含实际问题的案例研究，展示小波分析在图像压缩、信号去噪、模式识别等领域的应用。 7. **理论背景**：书中可能介绍了相关的数学基础，如傅立叶变换、复分析和泛函分析，以便读者更好地理解和应用小波理论。 8. **软件工具和计算方法**：可能会介绍一些常用的小波分析软件包，以及如何运用计算机编程实现小波分析算法。通过阅读这本书，读者不仅可以掌握小波分析的基本原理，还能了解到如何将其理论应用于实践，对于希望进入信号处理或机器学习领域的专业人士来说，这是一本不可或缺的参考资料。

three-volume set ([Me1], [Me2], and [CM1] – the third one is co-authored

with R. Coifman) and I. Daubechies “Ten lectures on wavelets” ([Da1]).

Both are excellent presentations, and we recommend them with enthusiasm.

They are more advanced than this book and cover much more material.

Since they were written, however, the theory has advanced considerably

(partly due to their contributions). Some of the original constructions have

been simpliﬁed and extended. We hope that this book can serve as an

introduction to these two treatises. The book by C. Chui ([Chu]) should

also be mentioned. It is a good complement to the ones by Daubechies and

Meyer (as, we hope, is ours). We cite it often, particularly when we discuss

spline wavelets.

Description of the book

It is, perhaps, useful to describe this book in more detail and give some

advice about how to read it. The ﬁrst four chapters, together with Chap-

ter 7, make up a “natural” inter-related group. They are devoted to the

construction of wavelets. We feel that Chapter 7 is the most important one

in the book. There are two simple equations that completely characterize

all orthonormal wavelets. They are



j∈Z



ψ(2

ξ)



= 1 for a.e. ξ ∈ R, (1)

and for every odd integer m,

∞



j=0

ψ(2

ξ)

ψ(2

(ξ +2mπ)) = 0 for a.e. ξ ∈ R. (2)

More precisely, ψ ∈ L

(R) is an orthonormal wavelet if and only if ψ sat-

isﬁes (1) and (2), provided ψ

= 1. The proof of this is elementary but

it is not simple, and we present it in the seventh chapter. These equa-

tions are known and have been used by many investigators working with

wavelets. The proof of this characterization in full generality, however,

did not appear in the published literature until recently. It can be found

in a paper by G. Gripenberg ([Gri1]), the Ph.D. thesis of one of our stu-

dents, X. Wang ([Wan]), and will appear in an expository article we wrote

with him ([HWW3]). It has been one of our goals to study the properties

of wavelets by examining their Fourier transforms. One of the principal

 1996 by CRC Press LLC

features of this book, in fact, is the important role played by the Fourier

transform.

The ﬁrst four chapters are devoted to diﬀerent ways of constructing

wavelets. Chapter 1 deals with the local sine and cosine bases that were

discovered by R. Coifman and Y. Meyer. We show how they lead us to

bases for L

(R) that have the important features described in the begin-

ning of this introduction; that is, they enjoy particularly simple relations

with the basic operators: translations, dilations and the Fourier transform.

We use these bases to construct the wavelets of Lemari´e and Meyer, the

ﬁrst class of orthonormal wavelets that were introduced and that includes

ones such that they and their Fourier transform are smooth.

In the second chapter we develop a general method that was intro-

duced by Mallat and Meyer for constructing wavelets: the multiresolu-

tion analysis (MRA). We apply this method to obtain the compactly sup-

ported wavelets introduced by Daubechies. The third chapter is devoted to

the “band-limited” wavelets (the ones having compactly supported Fourier

transforms). We show that the elements of this class have some surprising

properties; for example, their Fourier transforms vanish in a neighborhood

of the origin. Perhaps one of the best reasons for studying this class sep-

arately is that the basic equations (1) and (2) are particularly easy to

study. Among other things, the series involved have only a ﬁnite number

of non-zero terms and we do not need to worry about their convergence.

This allows us to pave the way for the technically more diﬃcult analysis

involved in the seventh chapter. The fourth chapter introduces the reader

to the “spline wavelets.” This class appears to be particularly important in

the various applications of wavelet theory to signal and image analyses. We

also explain in this fourth chapter how one can construct periodic wavelets.

By the end of the ﬁrst four chapters we have enough examples and have

obtained suﬃciently many properties of wavelets to introduce the reader to

some of the uses of wavelets and their connection to other parts of Analysis.

We therefore interrupt our program of characterizing all wavelets in terms

of their Fourier transform and show how they provide us with tools for the

study of the important scales of function spaces we mentioned above. In

addition to providing us with orthonormal bases for the Hilbert space L

(R),

some wavelets give us natural bases for these other topological linear spaces

as well. Let us illustrate this with the Lebesgue spaces L

(R), 1 <p<∞,

of all those measurable functions f such that







∞

−∞



f(x)





< ∞. (3)

When p = 2 the ﬁniteness of this norm f 

is equivalent to the ﬁniteness

 1996 by CRC Press LLC

of the norm







j∈Z



k∈Z

j,k



of the coeﬃcient sequence

c = {c

j,k

} =



f,ψ

j,k





that provides us with

the representation

f =



j∈Z



k∈Z

j,k

Thus, L

(R) can be represented as the space

(Z × Z) of all sequences

c such that c

< ∞. Appropriate wavelet bases provide us with the

characterization of L

(R) in terms of a sequence space for the other indices

p ∈ (1, ∞). It can be shown that f belongs to L

(R) if and only if



c

(p)



j,k

}



(p)





∞

−∞





j∈Z



k∈Z

j,k

(x)





< ∞,

where χ

j,k

is the characteristic function of the interval [2

−j

k, 2

−j

(k + 1)]

and

c is the sequence of coeﬃcients of f associated with {ψ

j,k

}. Observe

that the ﬁniteness of 

c

(p)

is a condition on the size (or absolute value)

of the coeﬃcients c

j,k

. This provides us with the ability to study L

(R)

in terms of a corresponding sequence space in a way that is analogous to

the reduction of properties of L

(R) to properties of

(Z × Z). Note that



c

(2)

= c

. Thus, an operator that is diagonalized by the basis {ψ

j,k

}

can be analyzed in terms of its proper values, as is the case in Hilbert space

theory. Many important operators are “essentially” diagonalized by wavelet

bases. It is this circle of ideas that is presented in Chapter 5 and Chap-

ter 6. More speciﬁcally, we present a brief treatment of bases in Banach

spaces, with an emphasis on the notion of unconditionality, in Chapter 5.

In Chapter 6 we give the characterizations described above. This treatment

uses properties of Calder´on-Zygmund operators; consequently, we have

an opportunity to see how wavelets are associated with the study of these

important operators.

In Chapter 7 we resume the study of wavelets in L

(R). We not only

give a characterization of all wavelets, as described above, but we also

characterize all wavelets that arise from an MRA and the basic functions

(the scaling functions and low-pass ﬁlters) involved in this method. These

characterizations allow us to construct several other classes of wavelets as

well.

Though most of the bases discussed in the ﬁrst seven chapters are or-

thonormal, we do mention some other types of bases. In Chapter 8 we

 1996 by CRC Press LLC

present a more thorough treatment of systems that are more general, with

particular attention to frames and their importance to wavelets. We pay

special attention to the way they can be used to analyze and reconstruct

functions; we also extend the Balian-Low theorem to frames.

The last chapter is devoted to certain topics that are important and

relevant to the applications of the theory of wavelets. We indicate how

the mathematical theory is transformed when it is applied to “discrete”

signals. We develop the Discrete Fourier (and Cosine) Transform in what

is, probably, a manner that is diﬀerent from the usual treatment but has

some features that are adaptable to programming for computers. We also

describe the decomposition and reconstruction algorithms for wavelets and

we end the chapter with a treatment of “wavelet packets.”

One of our colleagues, M.V. Wickerhauser, has recently written a book,

“Adapted wavelet analysis from theory to software” ([Wi2]), that treats the

subject we just mentioned, and many more applications, in great detail.

We believe that his book will prove to be most useful. We found no need,

therefore, to go further than we did in this direction. As we stated about

the books by Daubechies and Meyer, we hope that our book makes a good

companion to, and complements, the book by Wickerhauser.

Some advice to the reader

The background we assume of the reader is a “good undergraduate”

preparation in mathematics. We do use the language of measure theory;

for example, we talk about “measurable functions.” One should not be dis-

couraged if he/she only knows the ordinary Riemann integral. Substituting

the Lebesgue integral for the latter will, in general, not aﬀect the meaning

or the validity of most statements. Some notions in elementary functional

analysis are used; again, the results or statements involving these notions

can almost always be understood by ignoring unfamiliar language.

It is our hope that graduate students in mathematics, the sciences and

engineering can proﬁt from our presentation. We advise the reader not

to be discouraged by the few somewhat technical notions we introduce at

times (distributions, maximal functions, vector-valued inequalities, etc). If

it’s “too much,” just skip it at ﬁrst; there is enough material that can be

understood with the background mentioned in the previous paragraph. It

is our experience that even those whose main interest is in the applications

can proﬁt by learning about the theory we present.

 1996 by CRC Press LLC

In each section we number results consecutively; that is, we do not form

separate lists of theorems, propositions, corollaries, lemmas, formulae and

inequalities. These items are listed as ordered pairs n.m, where “n” denotes

the section (in the chapter) and “m” the m

item so numbered in the

section. If we need to refer to something in another chapter, we mention

the chapter and the relevant ordered pair. The sections in each chapter are

also assigned an ordered pair, n.m; in this case, “n” denotes the number of

the chapter and “m” the m

section.

We do not present a list of exercises at the end of each chapter. In many

cases we leave certain calculations to be worked out by the reader. This is

particularly true of the comments made in the last section of each chapter,

which is labelled “Notes and references.”

We also feel that we should state quite clearly that, though the bibliogra-

phy we include is quite large, it is far from a list that comes anywhere close

to exhausting what has been published in the theory of wavelets during the

relatively short period of its existence. We have tried to give proper credits;

however, since some of the material we discuss is quite new, we realize that

it is very likely we omitted some references that should have been included.

Acknowledgments

The idea of writing a book on wavelets was suggested by F. Soria when he

asked the second author to give a course in this subject in the Universidad

Aut´onoma de Madrid during the spring semester of 1993. This course was

repeated and extended considerably during the academic year 1994 – 95

when both authors were at Washington University.

Many individuals helped us in the preparation of these two courses, be-

fore and during the time when the lectures were given, and while we were

writing this book: P. Auscher, A. Bonami, M.J. Carro, J. Dziubaˇnski,

X. Fang, G. Garrig´os, Y. Han, A. Ho, A. Hulanicki, E. Laeng, Y. Meyer,

M. Paluszyˇnski, P.M. Soardi, F. Soria, J. Soria, M. Taibleson, A. Trgo,

X. Wang and D. Weiland. Some on this list were particularly important to

us and we feel that special thanks are in order.

Y. Meyer, during many years, continuously provided us with the many

manuscripts he wrote, not only for publication, but for private circula-

tion among his many students and friends. We learned a lot from him.

P. Auscher, A. Bonami and F. Soria were collaborators on several projects

 1996 by CRC Press LLC

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