This book develops the ideas behind and properties of wavelets and shows how they can be used as analytical tools for signal processing, numerical analysis, and mathematical modeling. We try to present this in a way that is accessible to the engineer, scientist, and applied mathematician both as a theoretical approach and as a potentially practical method to solve problems. Although the roots of this subject go back some time, the modern interest and development have a history of only a few years.

The early work was in the 1980's by Morlet, Grossmann, Meyer, Mallat, and others, but it was the paper by Ingrid Daubechies 4 in 1988 that caught the attention of the larger applied mathematics communities in signal processing, statistics, and numerical analysis. Much of the early work took place in France 3, 13 and the USA 4, 14, 5, 15. As in many new disciplines, the first work was closely tied to a particular application or traditional theoretical framework. Now we are seeing the theory abstracted from application and developed on its own and seeing it related to other parallel ideas. Our own background and interests in signal processing certainly influence the presentation of this book.

The goal of most modern wavelet research is to create a set of basis functions (or general expansion functions) and transforms that will give an informative, efficient, and useful description of a function or signal. If the signal is represented as a function of time, wavelets provide efficient localization in both time and frequency or scale. Another central idea is that of multiresolution where the decomposition of a signal is in terms of the resolution of detail.

For the Fourier series, sinusoids are chosen as basis functions, then the properties of the resulting expansion are examined. For wavelet analysis, one poses the desired properties and then derives the resulting basis functions. An important property of the wavelet basis is providing a multiresolution analysis. For several reasons, it is often desired to have the basis functions orthonormal. Given these goals, you will see aspects of correlation techniques, Fourier transforms, short-time Fourier transforms, discrete Fourier transforms, Wigner distributions, filter banks, subband coding, and other signal expansion and processing methods in the results.

Wavelet-based analysis is an exciting new problem-solving tool for the mathematician, scientist, and engineer. It fits naturally with the digital computer with its basis functions defined by summations not integrals or derivatives. Unlike most traditional expansion systems, the basis functions of the wavelet analysis are not solutions of differential equations. In some areas, it is the first truly new tool we have had in many years. Indeed, use of wavelets and wavelet transforms requires a new point of view and a new method of interpreting representations that we are still learning how to exploit.

More recently, work by Donoho, Johnstone, Coifman, and others have added theoretical reasons for why wavelet analysis is so versatile and powerful, and have given generalizations that are still being worked on. They have shown that wavelet systems have some inherent generic advantages and are near optimal for a wide class of problems 8. They also show that adaptive means can create special wavelet systems for particular signals and classes of signals.

The multiresolution decomposition seems to separate components of a signal in a way that is superior to most other methods for analysis, processing, or compression. Because of the ability of the discrete wavelet transform to decompose a signal at different independent scales and to do it in a very flexible way, Burke calls wavelets “The Mathematical Microscope" 2, 11. Because of this powerful and flexible decomposition, linear and nonlinear processing of signals in the wavelet transform domain offers new methods for signal detection, filtering, and compression 8, 9, 7, 17, 19, 10. It also can be used as the basis for robust numerical algorithms.

You will also see an interesting connection and equivalence to filter bank theory from digital signal processing 18, 1. Indeed, some of the results obtained with filter banks are the same as with discrete-time wavelets, and this has been developed in the signal processing community by Vetterli, Vaidyanathan, Smith and Barnwell, and others. Filter banks, as well as most algorithms for calculating wavelet transforms, are part of a still more general area of multirate and time-varying systems.

The presentation here will be as a tutorial or primer for people who know little or nothing about wavelets but do have a technical background. It assumes a knowledge of Fourier series and transforms and of linear algebra and matrix theory. It also assumes a background equivalent to a B.S. degree in engineering, science, or applied mathematics. Some knowledge of signal processing is helpful but not essential. We develop the ideas in terms of one-dimensional signals 15 modeled as real or perhaps complex functions of time, but the ideas and methods have also proven effective in image representation and processing 16, 12 dealing with two, three, or even four dimensions. Vector spaces have proved to be a natural setting for developing both the theory and applications of wavelets. Some background in that area is helpful but can be picked up as needed. The study and understanding of wavelets is greatly assisted by using some sort of wavelet software system to work out examples and run experiments. Matlab^TM programs are included at the end of this book and on our web site (noted at the end of the preface). Several other systems are mentioned in Chapter: Wavelet-Based Signal Processing and Applications .

There are several different approaches that one could take in presenting wavelet theory. We have chosen to start with the representation of a signal or function of continuous time in a series expansion, much as a Fourier series is used in a Fourier analysis. From this series representation, we can move to the expansion of a function of a discrete variable (e.g., samples of a signal) and the theory of filter banks to efficiently calculate and interpret the expansion coefficients. This would be analogous to the discrete Fourier transform (DFT) and its efficient implementation, the fast Fourier transform (FFT). We can also go from the series expansion to an integral transform called the continuous wavelet transform, which is analogous to the Fourier transform or Fourier integral. We feel starting with the series expansion gives the greatest insight and provides ease in seeing both the similarities and differences with Fourier analysis.

This book is organized into sections and chapters, each somewhat self-contained. The earlier chapters give a fairly complete development of the discrete wavelet transform (DWT) as a series expansion of signals in terms of wavelets and scaling functions. The later chapters are short descriptions of generalizations of the DWT and of applications. They give references to other works, and serve as a sort of annotated bibliography. Because we intend this book as an introduction to wavelets which already have an extensive literature, we have included a rather long bibliography. However, it will soon be incomplete because of the large number of papers that are currently being published. Nevertheless, a guide to the other literature is essential to our goal of an introduction.

A good sketch of the philosophy of wavelet analysis and the history of its development can be found in a recent book published by the National Academy of Science in the chapter by Barbara Burke 2. She has written an excellent expanded version in 11, which should be read by anyone interested in wavelets. Daubechies gives a brief history of the early research in 6.

Many of the results and relationships presented in this book are in the form of theorems and proofs or derivations. A real effort has been made to ensure the correctness of the statements of theorems but the proofs are often only outlines of derivations intended to give insight into the result rather than to be a formal proof. Indeed, many of the derivations are put in the Appendix in order not to clutter the presentation. We hope this style will help the reader gain insight into this very interesting but sometimes obscure new mathematical signal processing tool.

We use a notation that is a mixture of that used in the signal processing literature and that in the mathematical literature. We hope this will make the ideas and results more accessible, but some uniformity and cleanness is lost.

The authors acknowledge AFOSR, ARPA, NSF, Nortel, Inc., Texas Instruments, Inc. and Aware, Inc. for their support of this work. We specifically thank H. L. Resnikoff, who first introduced us to wavelets and who proved remarkably accurate in predicting their power and success. We also thank W. M. Lawton, R. O. Wells, Jr., R. G. Baraniuk, J. E. Odegard, I. W. Selesnick, M. Lang, J. Tian, and members of the Rice Computational Mathematics Laboratory for many of the ideas and results presented in this book. The first named author would like to thank the Maxfield and Oshman families for their generous support. The students in EE-531 and EE-696 at Rice University provided valuable feedback as did Bruce Francis, Strela Vasily, Hans Schüssler, Peter Steffen, Gary Sitton, Jim Lewis, Yves Angel, Curt Michel, J. H. Husoy, Kjersti Engan, Ken Castleman, Jeff Trinkle, Katherine Jones, and other colleagues at Rice and elsewhere.

We also particularly want to thank Tom Robbins and his colleagues at Prentice Hall for their support and help. Their reviewers added significantly to the book.

We would appreciate learning of any errors or misleading statements that any readers discover. Indeed, any suggestions for improvement of the book would be most welcome. Send suggestions or comments via email to csb@rice.edu. Software, articles, errata for this book, and other information on the wavelet research at Rice can be found on the world-wide-web URL: http:www-dsp.rice.edu/ with links to other sites where wavelet research is being done.

C. Sidney Burrus, Ramesh A. Gopinath, and Haitao Guo

Houston, Texas, Yorktown Heights, New York, and Sunnyvale, California