小波分析经典教材：《A Wavelet Tour of Signal Processing》

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"A Wavelet Tour of Signal Processing the sparse way.pdf" 是一本由Stéphane Mallat撰写，并由Gabriel Peyré贡献的关于小波分析的经典教材。这本书深入探讨了小波理论及其在信号处理中的应用，特别是侧重于稀疏表示的方式。小波分析是一种强大的数学工具，它结合了频域和时域分析的优点，能够对信号进行多尺度分析。在本书中，作者Mallat详细介绍了小波的概念、构造和性质，包括离散小波变换、连续小波变换以及各种小波基函数（如Haar小波、Morlet小波等）的应用。通过小波分析，读者可以了解如何在不同的时间-频率分辨率下分解信号，这对于识别信号中的瞬态特征尤其有用。 "稀疏表示"是本书的另一个核心主题。在信号处理中，稀疏表示是指找到一种方式，使复杂信号可以用尽可能少的基元素来表示。这种理论在图像压缩、噪声去除、信号恢复等领域有广泛的应用。书中可能会涉及压缩感知（Compressed Sensing）理论，这是一种利用信号的稀疏性实现高效数据采集和恢复的方法。此外，Mallat和Peyré可能还讨论了小波分析在实际问题中的应用，例如在音频信号处理、图像处理、医学成像、地震学、金融数据分析等领域的案例研究。他们可能还涵盖了小波包分析、多分辨率分析和小波变换在非平稳信号分析中的作用。本书对于想要深入了解小波理论及其应用的读者来说是一本宝贵的资源。通过阅读，读者不仅能掌握小波分析的基本原理，还能了解到如何将这些理论应用于实际的信号处理问题中，从而实现更高效的信号分析和数据压缩。同时，由于涉及到稀疏表示，这本书也对机器学习和数据科学领域的研究人员具有参考价值。《小波之旅：信号处理的稀疏方式》是一本全面介绍小波分析的教材，不仅适合学术研究，也适用于工程实践，对于提升读者在信号处理和相关领域的专业素养大有裨益。

“Mallat: 04-notation-p374370” — 2009/3/7 — 17:59 — page xix — #1

Notations

f , g Inner product (A.6)

f  Euclidean or Hilbert space norm

f 

or l

norm

f 

⬁

norm

f [n]⫽ O(g[n]) Order of: there exists K such that f [n]⭐ Kg[n]

f [n]⫽ o(g[n]) Small order of: lim

n→⫹⬁

f [n]

g[n]

⫽ 0

f [n]∼g[n] Equivalent to: f [n]⫽ O( g[n]) and g[n]⫽ O( f [n])

A⬍⫹⬁ A is ﬁnite

AB A is much bigger than B

∗

Complex conjugate of z ∈C

x Largest integer n ⭐ x

x Smallest integer n⭓ x

(x)

⫹

max(x, 0)

n mod N Remainder of the integer division of n modulo N

Sets

N Positive integers including 0

Z Integers

R Real numbers

⫹

Positive real numbers

C Complex numbers

|⌳| Number of elements in a set ⌳

Signals

f (t) Continuous time signal

f [n] Discrete signal

␦(t) Dirac distribution (A.30)

␦[n] Discrete Dirac (3.32)

[a,b]

Indicator of a function that is 1 in [a, b] and 0 outside

Spaces

Uniformly continuous functions (7.207)

p times continuously differentiable functions

⬁

Inﬁnitely differentiable functions

(R) Sobolev

times differentiable functions (9.8)

(R) Finite energy functions



|f (t)|

dt ⬍⫹⬁

(R) Functions such that



|f (t)|

dt ⬍⫹⬁



(Z) Finite energy discrete signals



⫹⬁

n⫽⫺⬁

|f [n]|

⬍⫹⬁



(Z) Discrete signals such that



⫹⬁

n⫽⫺⬁

|f [n]|

⬍⫹⬁

Complex signals of size N

U ⊕V Direct sum of two vector spaces

xix

CHAPTER

Sparse Representations

Signals carry overwhelming amounts of data in which relevant information is often

more difﬁcult to ﬁnd than a needle in a haystack. Processing is faster and simpler

in a sparse representation where few coefﬁcients reveal the information we are

looking for. Such representations can be constructed by decomposing signals over

elementary waveforms chosen in a family called a dictionary. But the search for

the Holy Grail of an ideal sparse transform adapted to all signals is a hopeless quest.

The discovery of wavelet orthogonal bases and local time-frequency dictionaries has

opened the door to a huge jungle of new transforms. Adapting sparse representa-

tions to signal properties, and deriving efﬁcient processing operators, is therefore a

necessary survival strategy.

An orthogonal basis is a dictionary of minimum size that can yield a sparse repre-

sentation if designed to concentrate the signal energy over a set of few vectors.This

set gives a geometric signal description. Efﬁcient signal compression and noise-

reduction algorithms are then implemented with diagonal operators computed

with fast algorithms. But this is not always optimal.

In natural languages,a richer dictionary helps to build shorter and more precise

sentences. Similarly, dictionaries of vectors that are larger than bases are needed

to build sparse representations of complex signals. But choosing is difﬁcult and

requires more complex algorithms. Sparse representations in redundant dictionaries

can improve pattern recognition,compression,and noise reduction,but also the res-

olution of new inverse problems. This includes superresolution, source separation,

and compressive sensing.

This ﬁrst chapter is a sparse book representation, providing the story line and

the main ideas. It gives a sense of orientation for choosing a path to travel.

1.1 COMPUTATIONAL HARMONIC ANALYSIS

Fourier and wavelet bases are the journey’s starting point. They decompose sig-

nals over oscillatory waveforms that reveal many signal properties and provide

a path to sparse representations. Discretized signals often have a very large

size N ⭓ 10

, and thus can only be processed by fast algorithms, typically imple-

mented with O(N log N ) operations and memories. Fourier and wavelet transforms

2 CHAPTER 1 Sparse Representations

illustrate the strong connection between well-structured mathematical tools and

fast algorithms.

1.1.1 The Fourier Kingdom

The Fourier transform is everywhere in physics and mathematics because it diago-

nalizes time-invariant convolution operators. It rules over linear time-invariant signal

processing, the building blocks of which are frequency ﬁltering operators.

Fourier analysis represents any ﬁnite energy function f (t) as a sum of sinusoidal

waves e

i␻t

f (t) ⫽

2␲



⫹⬁

⫺⬁

f (␻) e

i␻t

d␻. (1.1)

The amplitude

f (␻) of each sinusoidal wave e

i␻t

is equal to its correlation with f ,

also called Fourier transform:

f (␻) ⫽



⫹⬁

⫺⬁

f (t) e

⫺i␻t

dt. (1.2)

The more regular f (t), the faster the decay of the sinusoidal wave amplitude |

f (␻)|

when frequency ␻ increases.

When f (t) is deﬁned only on an interval, say [0, 1], then the Fourier transform

becomes a decomposition in a Fourier orthonormal basis {e

i2␲mt

}

m∈Z

of L

[0, 1].

If f (t) is uniformly regular, then its Fourier transform coefﬁcients also have a fast

decay when the frequency 2␲m increases, so it can be easily approximated with

few low-frequency Fourier coefﬁcients. The Fourier transform therefore deﬁnes a

sparse representation of uniformly regular functions.

Over discrete signals, the Fourier transform is a decomposition in a discrete

orthogonal Fourier basis {e

i2␲kn/N

}

0⭐k⬍N

of C

, which has properties similar to a

Fourier transform on functions. Its embedded structure leads to fast Fourier trans-

form (FFT) algorithms,which compute discrete Fourier coefﬁcients with O(N log N )

instead of N

.This FFT algorithm is a cornerstone of discrete signal processing.

As long as we are satisﬁed with linear time-invariant operators or uniformly

regular signals, the Fourier transform provides simple answers to most questions.

Its richness makes it suitable for a wide range of applications such as signal

transmissions or stationary signal processing. However, to represent a transient

phenomenon—a word pronounced at a particular time, an apple located in the

left corner of an image—the Fourier transform becomes a cumbersome tool that

requires many coefﬁcients to represent a localized event. Indeed, the support of

i␻t

covers the whole real line, so

f (␻) depends on the values f (t) for all times

t ∈R. This global “mix”of information makes it difﬁcult to analyze or represent any

local property of f (t) from

f (␻).

1.1.2 Wavelet Bases

Wavelet bases, like Fourier bases, reveal the signal regularity through the ampli-

tude of coefﬁcients, and their structure leads to a fast computational algorithm.

1.1 Computational Harmonic Analysis 3

However, wavelets are well localized and few coefﬁcients are needed to represent

local transient structures. As opposed to a Fourier basis, a wavelet basis deﬁnes a

sparse representation of piecewise regular signals,which may include transients and

singularities. In images, large wavelet coefﬁcients are located in the neighborhood

of edges and irregular textures.

The story began in 1910, when Haar [291] constructed a piecewise constant

function

␺(t) ⫽

⎧

⎨

⎩

1if0⭐ t ⬍ 1/2

⫺1if1/2⭐ t ⬍ 1

0 otherwise

the dilations and translations of which generate an orthonormal basis



␺

j,n

(t) ⫽

√

␺



t ⫺ 2



( j,n)∈Z

of the space L

(R) of signals having a ﬁnite energy

f 

⫽



⫹⬁

⫺⬁

|f (t)|

dt ⬍⫹⬁.

Let us write f, g⫽



⫹⬁

⫺⬁

f (t) g

∗

(t) dt—the inner product in L

(R). Any ﬁnite energy

signal f can thus be represented by its wavelet inner-product coefﬁcients

f , ␺

j,n

⫽



⫹⬁

⫺⬁

f (t) ␺

j,n

(t) dt

and recovered by summing them in this wavelet orthonormal basis:

f ⫽

⫹⬁



j⫽⫺⬁

⫹⬁



n⫽⫺⬁

f , ␺

j,n

␺

j,n

. (1.3)

Each Haar wavelet ␺

j,n

(t) has a zero average over its support [2

n, 2

(n⫹ 1)].If f

is locally regular and 2

is small, then it is nearly constant over this interval and the

wavelet coefﬁcient f , ␺

j,n

is nearly zero.This means that large wavelet coefﬁcients

are located at sharp signal transitions only.

With a jump in time, the story continues in 1980, when Strömberg [449] found

a piecewise linear function ␺ that also generates an orthonormal basis and gives

better approximations of smooth functions. Meyer was not aware of this result,

and motivated by the work of Morlet and Grossmann over continuous wavelet

transform, he tried to prove that there exists no regular wavelet ␺ that generates

an orthonormal basis. This attempt was a failure since he ended up constructing

a whole family of orthonormal wavelet bases, with functions ␺ that are inﬁnitely

continuously differentiable [375]. This was the fundamental impulse that led to a

widespread search for new orthonormal wavelet bases, which culminated in the

celebrated Daubechies wavelets of compact support [194].

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