MATLAB小波工具箱2.0用户指南：计算与可视化

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"MATLAB wavlet工具箱使用手册2.0版是专为使用MATLAB进行小波分析而设计的一份详细指南。它涵盖了计算、可视化和编程等方面的内容，旨在帮助用户充分利用该工具箱进行信号处理和数据分析。由Michel Misiti、Yves Misiti、Georges Oppenheim和Jean-Michel Poggi编写，该手册提供了与MATLAB 2.0版本兼容的Wavelet Toolbox的使用方法。" MATLAB wavlet工具箱是MATLAB环境中的一个重要扩展，主要用于小波分析，这是一门结合了时间域和频率域分析的领域。通过小波函数，用户可以对非平稳信号进行高效分析，提取其中的时间局部和频率局部特征。以下是一些关键知识点： 1. **小波基础**：小波分析的基本概念，包括离散小波变换（DWT）、连续小波变换（CWT）以及小波包变换（WPT）。这些变换允许数据在不同尺度和位置上进行分析。 2. **小波函数选择**：工具箱提供了多种类型的小波基函数，如Daubechies、Morlet、Symlets等，每种函数都有其特定的适用场景和特性。 3. **信号去噪**：利用小波系数的阈值处理可以有效去除信号中的噪声，提高信噪比。工具箱提供了软阈值和硬阈值两种去噪方法。 4. **图像处理**：小波分析同样适用于二维图像处理，如图像压缩、边缘检测和图像复原。 5. **计算功能**：包括小波变换、逆变换、多分辨率分析、系数操作（如阈值处理）等功能。 6. **可视化工具**：提供图形用户界面（GUI）或命令行方式，帮助用户直观地查看小波系数和重构信号，以理解分析结果。 7. **编程接口**：MATLAB函数库支持用户自定义算法，可以与其他MATLAB工具箱或自编程序集成。 8. **应用示例**：手册可能包含多个实际应用示例，如信号分类、故障诊断、金融数据分析等，帮助用户理解和应用小波分析。 9. **技术支持**：用户可以通过The MathWorks提供的多种联系方式获取技术帮助，包括网站、新闻组、电子邮件和电话。 10. **版权和许可**：软件遵循特定的许可证协议，用户必须遵守使用规定，未经许可不得非法复制或分发。通过MATLAB wavlet工具箱，用户能够进行深入的小波分析，解决各种科学和工程问题，特别是在信号处理、图像分析、模式识别等领域有着广泛的应用。

Preface

xiv

This chapter contains introductory information.

• “About the Authors” on page xv

• “Notes by Yves Meyer” on page xvi

• “Notes by Ingrid Daubechies” on page xviii

• “Acknowledgments” on page xix

• “What Is the Wavelet Toolbox?” on page xx

• “Using This Guide” on page xxi

• “For More Background” on page xxiii

• “Installing the Wavelet Toolbox” on page xxiv

• “Typographical Conventions” on page xxvi

• “Related Products” on page xxvii

Preface

xvi

Notes by Yves Meyer

The history of wavelets is not very old, at most 10 to 15 years. The field

experienced a fast and impressive start, characterized by a close-knit

international community of researchers who freely circulated scientific

information and were driven by the researchers’ youthful enthusiasm. Even as

the commercial rewards promised to be significant, the ideas were shared, the

trials were pooled together, and the successes were shared by the community.

There are lots of successes for the community to share. Why? Probably because

the time is ripe. Fourier techniques were liberated by the appearance of

windowed Fourier methods that operate locally on a time-frequency approach.

In another direction, Burt-Adelson’s pyramidal algorithms, the quadrature

mirror filters, and filter banks and subband coding are available. The

mathematics underlying those algorithms existed earlier, but new computing

techniques enabled researchers to try out new ideas rapidly. The numerical

image and signal processing areas are blooming.

The wavelets bring their own strong benefits to that environment: a local

outlook, a multiscaled outlook, cooperation between scales, and a time-scale

analysis. They demonstrate that sines and cosines are not the only useful

functions and that other bases made of weird functions serve to look at new

foreign signals, as strange as most fractals or some transient signals.

Recently, wavelets were determined to be the best way to compress a huge

library of fingerprints. This is not only a milestone that highlights the practical

value of wavelets, but it has also proven to be an instructive process for the

researchers involved in the project. Our initial intuition generally was that the

proper way to tackle this problem of interweaving lines and textures was to use

wavelet packets, a flexible technique endowed with quite a subtle sharpness of

analysis and a substantial compression capability. However, it was a

biorthogonal wavelet that emerged victorious and at this time represents the

best method in terms of cost as well as speed. Our intuitions led one way, but

implementing the methods settled the issue by pointing us in the right

direction.

For wavelets, the period of growth and intuition is becoming a time of

consolidation and implementation. In this context, a toolbox is not only

possible, but valuable. It provides a working environment that permits

experimentation and enables implementation.

Notes by Yves Meyer

xvii

Since the field still grows, it has to be vast and open. The MATLAB Wavelet

Toolbox addresses this need, offering an array of tools that can be organized

according to several criteria:

• Synthesis and analysis tools

• Wavelet and wavelet packets approaches

• Signal and image processing

• Discrete and continuous analyses

• Orthogonal and redundant approaches

• Coding, de-noising and compression approaches

What can we anticipate for the future, at least in the short term? It is difficult

to make an accurate forecast. Nonetheless, it is reasonable to think that the

pace of development and experimentation will carry on in many different fields.

Numerical analysis constantly uses new bases of functions to encode its

operators or to simplify its calculations to solve partial differential equations.

The analysis and synthesis of complex transient signals touches musical

instruments by studying the striking up, when the bow meets the cello string.

The analysis and synthesis of multifractal signals, whose regularity (or rather

irregularity) varies with time, localizes information of interest at its

geographic location. Compression is a booming field, and coding and de-noising

are promising.

For each of these areas, the M

ATLAB Wavelet Toolbox provides a way to

introduce, learn, and apply the methods, regardless of the user’s experience. It

includes a command-line mode and a graphical user interface mode, each very

capable and complementing to the other. The user interfaces help the novice to

get started and the expert to implement trials. The command line provides an

open environment for experimentation and addition to the graphical interface.

In the journey to the heart of a signal’s meaning, the toolbox gives the traveler

both guidance and freedom: going from one point to the other, wandering from

a tree structure to a superimposed mode, jumping from low to high scale, and

skipping a breakdown point to spot a quadratic chirp. The time-scale graphs of

continuous analysis are often breathtaking and more often than not

enlightening as to the structure of the signal.

Here are the tools, waiting to be used.

Yves Meyer

Professor at Ecole Normale Supérieure de Cachan and Institut de France

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