概率小波信号分析与分形电子书详解

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《概率小波信号与分形分析》是一本深入研究概率小波理论的专业电子书，它属于 GraduateTexts in Mathematics 系列中的经典著作。该系列汇集了数学领域的高级教材，专为研究生和高级学者设计，涵盖了广泛的数学主题，从基础的集合论到复杂的代数、拓扑、函数分析和几何等领域。在这本书中，读者可以探索概率小波这一现代数学工具，它结合了概率论与小波分析，被广泛应用于信号处理、数据分析和随机过程等领域。小波分析是一种强大的工具，因为它能够捕捉信号在不同尺度下的局部特性，这对于时变信号的分析尤其重要。而将概率论引入其中，使得这个理论更具适应性和鲁棒性，特别是在面对不确定性或噪声时。书中可能会探讨以下几个关键知识点： 1. **概率小波基础**：介绍小波理论的基本概念，包括小波基的选择、正交性、尺度和频率分析，以及它们如何转化为概率框架中的概念。 2. **随机过程与分形**：概率小波如何与随机过程如随机游走、布朗运动等结合，用于描述和分析分形信号的统计特性，如自相似性和局部精细结构。 3. **信号处理应用**：讨论如何利用概率小波对信号进行去噪、特征提取、时频分析等实际应用，尤其是在通信、图像处理和金融时间序列分析中的作用。 4. **理论发展**：涉及概率小波理论的最新研究进展，可能包括新的构造方法、理论结果和计算技术，以及与其他数学分支（如泛函分析）的交叉融合。 5. **实例与案例研究**：书中可能会提供一系列实际案例，展示概率小波在解决实际问题中的效果，帮助读者理解和掌握该方法的实用技巧。 6. **数学背景**：为了深入理解概率小波，书中的内容可能会涉及一定的数学预备知识，如测度论、泛函分析、线性代数和抽象代数等。作为一本英文版的权威教材，这本书不仅适合正在攻读相关专业的研究生，也对于有深厚数学基础的研究人员和工程师来说，是一本极具价值的学习资料。通过阅读和学习《概率小波信号与分形分析》，读者将能够扩展其在复杂数据处理和信号分析方面的专业知识，并对未来的科研工作产生深远影响。

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xvi Getting started

scale-similarity can be exploited in basis computations that use tricks devised for

signal processing. Just open the book and glance at some of the wavelet fimctions.

At the same time, the key notion of self-similarity, such as the scale-similarity used

everywhere for wavelets, is essential to our understanding of fractals: Fern-like pic-

tures that look the same at small and at large scales. One problem in the generation of

wavelet bases is selecting the "nice" (here this means differentiable) wavelets among

huge families of fractal-looking (non-smooth, or singular) functions. Z^-fiinctions

can be very "bad" indeed!! Computers generate the good and the bad, and we are

left with the task of sorting them out and making selections. We will see (directly

from large libraries of pictures) that mathematical wavelet machines are more likely

to spit out bad fimctions unless they are told where to concentrate the search from

the intrinsic mathematics.

These wavelets, signals, and fractals are things that have caught our attention

in recent decades, but the mathematical part of this has roots back at least a hun-

dred years, for example, to Alfred Haar and to Oliver Heaviside at the turn of the

last century. From Haar we have the first wavelet basis, and with Heaviside we see

the beginning of signal analysis. It is unlikely that either one knew about the other.

Ironically, at the time (1909), Haar's paper had little impact and was hardly noticed,

even on the small scale of "notice" that is usually applied to mathematics papers.

Haar's wonderful wavelet only began to draw attention in the mid-nineteen-eighties

when the connections to modem signal processing became much better understood.

These connections certainly served as a main catalyst in what are now known as

wavelet tools in pure and applied mathematics. But at the outset, the pioneers in

wavelets had to "rediscover" a lot of stuff from signal processing: frequency bands,

high-pass, low-pass, analysis and synthesis using down-sampling, and up-sampling,

reconstruction of signals, resolution of images; all tools that have wonderfiil graphics

representations in the engineering literature.

But still, why would we think that Fourier's basis, and his lovely integral decom-

position, are not good enough? Many reasons: Fourier's method has computational

drawbacks. This was less evident before computers became common and began to

play important roles in applied and theoretical work. But expansion of functions or

signals into basis decompositions (called "analysis" in signal processing) involves

basis coefficients (Fourier coefficients, and so on), and if we are limited to Fourier

bases,

then the computation of the coefficients must by necessity rely on integration.

"Computers can't integrate!" Hmmm! Well, not directly. The problem must first be

discretized. And there is need for a more direct and algorithmic approach. Hence the

wavelet algorithm! In any case, algorithms are central in mathematics even if you

do not concern yourself with computers. And it is the engineering connections that

inspired the most successfiil algorithms in our subject.

Glossary xvii

Glossary

function, random variable, signal, state, sequence (incl. vector-valued), random

walk,

time-series, measurement, nested subspaces, refinement, multiresolution, scales of visual

resolutions,

operator,

process, black

box,

observable (if self

adjoint),

Fourier dual pair,

generating function, timefrequency, P/Q, convolution, filter, smearing, decomposition

(e.g.,

Fourier coefficients in a Fourier expansion),

analysis,

frequency components, integrate

(e.g.,

inverse Fourier transform), reconstruct, synthesis, superposition, subspace, resolution,

(signals in a) frequency

band,

Cuntz relations, perfect reconstruction from subbands,

subband decomposition, inner product, correlation, transition probability, probability of

transition from one state to

another,

/out = ^/in^ input/output, transformation of states,

fractal, conditional expectation, martingale, data mining (A translation guide!)

"The question is," said Alice, "whether you can make words mean so many

different things." —^Lewis Carroll

This glossary consists of a list of terms used inside the book in varied contexts of

mathematics, probability, engineering, and on occasion physics. To clarify the seem-

ingly confusing use of up to four different names for the same idea or concept, we

have further added informal explanations spelling out the reasons behind the differ-

ences in current terminology from neighboring fields.

When sorting through the disparate variations of lingo in the sciences, the more

mundane problems of plain and "ordinary" languages might seem minor: There is the

variety of differences from Lido-European to the schizophrenia of Finno-Hungarian.

And inside the same building on campus, there are even the variations in lingo that

separate areas of

mathematics:

If you are a mathematician doing analysis in "plain

English" you might well learn to understand the "Portuguese" spoken by your col-

leagues in algebra; but when it comes to the Hungarian of an engineer, or the Bantu

of some physicists, you can get lost or confused. A former student just wrote me

about the language gulf between domestic mathematics and the jungle of industry.

He now works in a company and is doing applied wavelets. And he writes that the

language of implementation is quite different from that of our standard mathematics

books.

DISCLAIMER: This glossary has the structure of four columns. A number of terms

are listed line by line, and each line is followed by explanation. Some "terms" have

up to four separate (yet commonly accepted) names. The last four terms in the list,

"fractal," "conditional expectation," "martingale," and "data mining," are the only

ones where I could think of only one name.

It should be added that my "descriptions" for the various terms are meant to stress

intuitive aspects, as opposed to mathematical definitions. One reason for stressing the

intuition behind the concepts is that there is not quite agreement about the precise

meaning of some of the terms in the four fields: in mathematics, in probability, in

engineering, and in physics.

xviii Getting started

MATHEMATICS PROBABILITY ENGINEERING PHYSICS

function random variable signal state

(measurable)

Mathematically, fiinctions may map between any two sets, say, from X to

but if X is a probability space (typically called ^), it comes with a cr-

algebra B of measurable sets, and probability measure P. Elements E inB

are called events, and P(E) the probability of E. Corresponding measurable

functions with values in a vector space are called random variables, a ter-

minology which suggests a stochastic viewpoint. The fimction values of a

random variable may represent the outcomes of an experiment, for example

"throwing of a die." hi this simplest experiment, the range of the random

variable is the set of integers from 1 to 6, the six possible measurements

resulting from the experiment.

Yet, fimction theory is widely used also in engineering where fiinctions

are typically thought of as signal. In this case, X may be the real line for

time,

or W^. And I noticed that engineers visualize fiinctions as signals. A

particular signal may have a stochastic component, and this feature simply

introduces an extra stochastic variable into the "signal," for example noise.

Turning to physics, in our present application, the physical fiinctions will

be typically be in some L^-space, and I^-fimctions with unit norm represent

quantum mechanical "states."

sequence (incl. random walk time-series measurement

vector-valued)

Mathematically, a sequence is a fimction defined on the integers Z or on

subsets of

for example the natural numbers N. Hence, if time is discrete,

this to the engineer represents a time series, such as a speech signal, or any

measurement which depends on time. But we will also allow fiinctions on

lattices such as Z^.

In the case c/ = 2, we may be considering the grayscale numbers which

represent exposure in a digital camera. In this case, the fimction (grayscale)

is defined on a subset of

Z^,

and is then simply a matrix.

A random walk on Z^ is an assignment of a sequential and random motion

as a fimction of

time.

The randomness presupposes assigned probabilities.

But we will use the term "random walk" also in connection with random

walks on combinatorial trees.

nested refinement multiresolution scales of visual

subspaces resolutions

While finite or infinite families of nested subspaces are ubiquitous in math-

ematics, and have been popular in Hilbert-space theory for generations (at

Glossary xix

MATHEMATICS PROBABILITY ENGINEERING PHYSICS

least since the 1930s), this idea was revived in a different guise in 1986 by

Stephane Mallat, then an engineering graduate student; see [Mal89]. In its

adaptation to wavelets, the idea is now referred to as the multiresolution

method.

What made the idea especially popular in the wavelet community was that

it offered a skeleton on which various discrete algorithms in applied mathe-

matics could be attached and turned into wavelet constructions in harmonic

analysis. In fact what we now call multiresolutions have come to signify

a crucial link between the world of discrete wavelet algorithms, which are

popular in computational mathematics and in engineering (signal/image pro-

cessing, data mining, etc.) on the one side, and on the other side continuous

wavelet bases in function spaces, especially in

1-^(1^.^).

Further, the mul-

tiresolution idea closely mimics how fractals are analyzed with the use of

finite fiinction systems.

But in mathematics, or more precisely in operator theory, the underlying

idea dates back to work of John von Neumann, Norbert Wiener, and Herman

Wold, where nested and closed subspaces in Hilbert space were used exten-

sively in an axiomatic approach to stationary processes, especially for time

series.

Wold proved that any (stationary) time series can be decomposed into

two different

parts:

The first (deterministic) part can be exactly described by

a linear combination of its own past, while the second part is the opposite

extreme; it is unitary, in the language of von Neumann.

von Neumann's version of the same theorem is a pillar in operator theory.

It states that every isometry in a Hilbert space H is the unique sum of a

shift isometry and a unitary operator, i.e., the initial Hilbert space H splits

canonically as an orthogonal sum of two subspaces Hs and Hu in H, one

which carries the shift operator, and the other Hu the unitary part. The shift

isometry is defined from a nested scale of closed spaces F„, such that the

intersection of these spaces is Hu.

However, Stephane Mallat was motivated instead by the notion of scales

of resolutions in the sense of optics. This in turn is based on a certain

"artificial-intelligence" approach to vision and optics, developed earlier by

David Marr at MIT, an approach which imitates the mechanism of vision in

the human eye.

The connection from these developments in the 1980s back to von Neu-

mann is this: Each of the closed subspaces Vn corresponds to a level of

resolution in such a way that a larger subspace represents a finer resolution.

Resolutions are relative, not absolute! In this view, the relative complement

of the smaller (or coarser) subspace in larger space then represents the visual

XX Getting started

MATHEMATICS PROBABILITY ENGINEERING PHYSICS

detail which is added in passing from a blurred image to a finer one, i.e., to

a finer visual resolution.

This view became an instant hit in the wavelet community, as it offered a

repository for the fimdamental father and the mother fimctions, also called

the scaling fimction

cp,

and the wavelet fimction y/. Via a system of transla-

tion and scaling operators, these fimctions then generate nested subspaces,

and we recover the scaling identities which initialize the appropriate al-

gorithms. What results is now called the family of pyramid algorithms in

wavelet analysis. The approach itself

called the multiresolution approach

(MRA) to wavelets. And in the meantime various generalizations (GMRAs)

have emerged.

In all of this, there was a second "accident" at play: As it turned out,

pyramid algorithms in wavelet analysis now lend themselves via multires-

olutions, or nested scales of closed subspaces, to an analysis based on fre-

quency bands. Here we refer to bands of frequencies as they have already

been used for a long time in signal processing.

Even though J. von Neumann and H. Wold had been using nested or

scaled families of closed subspaces in representing past and fiature for time

series,

S. Mallat found that this same idea applies successfiilly to the repre-

sentation of visual resolutions. And even more importantly, it offers a variety

of powerfiil algorithms for processing of digital images.

Now parallel to all of

this,

pioneers in probability theory had in fact devel-

oped versions of the same refinement analysis. For example, in the theory

of martingales, consistency relations may naturally be reformulated in the

language of nested subspaces in Hilbert space.

One reason for the success in varied disciplines of the same geometric

idea is perhaps that it is closely modeled on how we historically have repre-

sented numbers in the positional number system; see, e.g., [Knu81]. Analo-

gies to the Euclidean algorithm seem especially compelling.

operator process black box observable

(if selfadjoint)

In linear algebra students are familiar with the distinctions between (linear)

transformations T (here called "operators") and matrices. For a fixed opera-

tor T: V -^ W, there is a variety of matrices, one for each choice of basis in

V and inJV.ln many engineering applications, the transformations are not

restricted to be linear, but instead represent some experiment

("black

box,"

in Norbert Wiener's terminology), one with an input and an output, usually

fimctions of

time.

The input could be an external voltage fimction, the black

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