小波分析深入探索：A Wavelet Tour of Signal Processing(英文第3版)

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"A Wavelet Tour of Signal Processing" 是一本由 Stéphane Mallat 编写的关于小波分析在信号处理中的应用的书籍，第三版加入了Gabriel Peyré的贡献。这本书是信号处理领域的经典著作，旨在引导读者深入理解小波分析的原理和实践。在信号处理领域，小波分析是一种强大的工具，它结合了频域和时域分析的优点，能对信号进行多尺度、多分辨率的分析。Mallat的这本书详细介绍了这一理论和技术，涵盖了从小波的基础概念到高级应用的各个方面。书中可能包含的内容有： 1. **小波基础**：讲解小波函数的定义、性质，包括连续小波和离散小波的区别，以及常用的Haar、Morlet、Daubechies等小波基函数。 2. **小波变换**：详述小波变换的理论，如连续小波变换（CWT）和离散小波变换（DWT），以及它们如何提供信号的时间-频率局部化特性。 3. **多分辨率分析**：解释多分辨率分析的概念，它是小波理论的核心，能够展示信号在不同尺度上的细节。 4. **信号压缩与去噪**：阐述小波在信号压缩中的应用，以及如何利用小波系数进行信号去噪，比如硬阈值和软阈值方法。 5. **图像处理**：讨论小波在图像压缩、图像增强和图像重建中的应用，以及与傅立叶变换相比的优势。 6. **时间频率分析**：介绍小波在非平稳信号分析中的作用，如瞬时频率估计和瞬时振幅分析。 7. **稀疏表示**：鉴于标题中的"The Sparse Way"，本书可能会深入探讨信号的稀疏表示理论，这是现代压缩感知理论的基础。 8. **应用案例**：书中可能包含各种实际应用案例，如音频处理、地震数据解析、医学成像分析等。 9. **算法实现**：可能提供了小波变换相关算法的实现细节，帮助读者理解和编程实践。 10. **数学工具**：为了帮助读者掌握小波分析，书里可能会涉及一些必要的数学背景知识，如泛函分析、傅立叶分析和概率统计。这本书是为那些希望深入了解小波分析和其在信号处理中应用的读者准备的。通过阅读，读者不仅可以获得理论知识，还能学习如何将这些理论应用于实际问题中。由于是英文版，对于提高专业英语阅读能力也有一定帮助。

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Preface to the Sparse Edition

I cannot help but ﬁnd striking resemblances between scientiﬁc communities and

schools of ﬁsh. We interact in conferences and through articles, and we move

together while a global trajectory emerges from individual contributions. Some of

us like to be at the center of the school, others prefer to wander around, and a few

swim in multiple directions in front. To avoid dying by starvation in a progressively

narrower and specialized domain, a scientiﬁc community needs also to move on.

Computational harmonic analysis is still very much alive because it went beyond

wavelets. Wr iting such a book is about decoding the trajectory of the school and

gathering the pearls that have been uncovered on the way. Wavelets are no longer

the central topic, despite the previous edition’s original title. It is just an important

tool, as the Fourier transform is. Sparse representation and processing are now at

the core.

In the 1980s,many researchers were focused on building time-frequency decom-

positions,trying to avoid the uncertainty bar rier,and hoping to discover the ultimate

representation. Along the way came the construction of wavelet orthogonal bases,

which opened new perspectives through collaborations with physicists and math-

ematicians. Designing orthogonal bases with Xlets became a popular sport with

compression and noise-reduction applications. Connections with approximations

and sparsity also became more apparent. The search for sparsity has taken over,

leading to new grounds where orthonormal bases are replaced by redundant dictio-

naries of waveforms.

During these last seven years, I also encountered the industrial world. With

a lot of naiveness, some bandlets, and more mathematics, I cofounded a start-up

with Christophe Bernard, Jérome Kalifa, and Erwan Le Pennec. It took us some

time to learn that in three months good engineering should produce robust algo-

rithms that operate in real time, as opposed to the three years we were used

to having for writing new ideas with promising perspectives. Yet, we survived

because mathematics is a major source of industrial innovations for signal process-

ing. Semiconductor technology offers amazing computational power and ﬂexibility.

However, ad hoc algorithms often do not scale easily and mathematics accelerates

the trial-and-error development process. Sparsity decreases computations, memory,

and data communications. Although it brings beauty, mathematical understanding

is not a luxury. It is required by increasingly sophisticated information-processing

devices.

New Additions

Putting sparsity at the center of the book implied rewriting many parts and

adding sections. Chapters 12 and 13 are new. They introduce sparse represen-

tations in redundant dictionaries, and inverse problems, super-resolution, and

“Mallat: 03-preface-p374370” — 2009/3/7 — 17:58 — page xvi — #2

xvi Preface to the Sparse Edition

compressive sensing. Here is a small catalog of new elements in this third

edition:

■ Radon transform and tomography

■ Lifting for wavelets on surfaces, bounded domains, and fast computations

■ JPEG-2000 image compression

■

Block thresholding for denoising

■ Geometric representations with adaptive triangulations, curvelets, and

bandlets

■ Sparse approximations in redundant dictionaries with pursuit algorithms

■ Noise reduction with model selection in redundant dictionaries

■

Exact recovery of sparse approximation supports in dictionaries

■ Multichannel signal representations and processing

■

Dictionary learning

■ Inverse problems and super-resolution

■

Compressive sensing

■ Source separation

Teaching

This book is intended as a graduate-level textbook. Its evolution is also the result

of teaching courses in electrical engineering and applied mathematics. A new

website provides software for reproducible experimentations, exercise solutions,

together with teaching material such as slides with ﬁgures and MATLAB software

for numerical classes of http://wavelet-tour.com.

More exercises have been added at the end of each chapter, ordered by level

of difﬁculty. Level

exercises are direct applications of the course. Level

exercises

requires more thinking. Level

includes some technical derivation exercises. Level

are projects at the interface of research that are possible topics for a ﬁnal course

project or independent study. More exercises and projects can be found in the

website.

Sparse Course Programs

The Fourier transform and analog-to-digital conversion through linear sampling

approximations provide a common ground for all courses (Chapters 2 and 3).

It introduces basic signal representations and reviews important mathematical

and algorithmic tools needed afterward. Many trajectories are then possible to

explore and teach sparse signal processing. The following list notes several top-

ics that can orient a course’s structure with elements that can be covered along

the way.

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Preface to the Sparse Edition xvii

Sparse representations with bases and applications:

■ Principles of linear and nonlinear approximations in bases (Chapter 9)

■ Lipschitz regularity and wavelet coefﬁcients decay (Chapter 6)

■ Wavelet bases (Chapter 7)

■

Properties of linear and nonlinear wavelet basis approximations (Chapter 9)

■ Image wavelet compression (Chapter 10)

■ Linear and nonlinear diagonal denoising (Chapter 11)

Sparse time-frequency representations:

■ Time-frequency wavelet and windowed Fourier ridges for audio processing

(Chapter 4)

■ Local cosine bases (Chapter 8)

■ Linear and nonlinear approximations in bases (Chapter 9)

■ Audio compression (Chapter 10)

■

Audio denoising and block thresholding (Chapter 11)

■

Compression and denoising in redundant time-frequency dictionaries with

best bases or pursuit algorithms (Chapter 12)

Sparse signal estimation:

■

Bayes versus minimax and linear versus nonlinear estimations (Chapter 11)

■

Wavelet bases (Chapter 7)

■ Linear and nonlinear approximations in bases (Chapter 9)

■ Thresholding estimation (Chapter 11)

■ Minimax optimality (Chapter 11)

■ Model selection for denoising in redundant dictionaries (Chapter 12)

■ Compressive sensing (Chapter 13)

Sparse compression and information theory:

■ Wavelet orthonormal bases (Chapter 7)

■ Linear and nonlinear approximations in bases (Chapter 9)

■ Compression and sparse transform codes in bases (Chapter 10)

■

Compression in redundant dictionaries (Chapter 12)

■ Compressive sensing (Chapter 13)

■ Source separation (Chapter 13)

Dictionary representations and inverse problems:

■ Frames and Riesz bases (Chapter 5)

■ Linear and nonlinear approximations in bases (Chapter 9)

■ Ideal redundant dictionary approximations (Chapter 12)

■ Pursuit algorithms and dictionary incoherence (Chapter 12)

■ Linear and thresholding inverse estimators (Chapter 13)

■ Super-resolution and source separation (Chapter 13)

■ Compressive sensing (Chapter 13)

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