A Wavelet Tour of Signal Processing (review)

A Wavelet Tour of Signal Processing (AWT) is a complete guide to not only the important results and applications of wavelets—also known as multiresolution analysis—but also many of their lesser-known aspects. It contains eleven well-written chapters, two appendixes, and 361 references. Each chapter, except for the first, contains several theorems—most accompanied by thorough proofs—and concludes with a set of problems for students and further exploration. Stéphane Mallat, having made seminal contributions to the theory behind and application of wavelets to signal processing, is certainly qualified to be an official tour guide to the world of wavelets.

The history of multiresolution analysis is quite diverse, extending from contributions made by mathematicians, physicists, statisticians, and engineers. As such, it has found application in a broad range of disciplines, of which signal processing is only one. In fact, it is especially applicable to signal processing, whether working with sounds, images, or higher-dimensional data. Mr. Mallat makes this clear in his first chapter, where he motivates the utility of wavelets by discussing problems difficult to approach using Fourier methods, but easier using wavelets. He writes, "If we are interested in transient phenomena—a word pronounced at a particular time, an apple located in the left corner of an image—the Fourier transform becomes a cumbersome tool" (p. 2). Wavelets provide an intuitive means of analyzing signals at various scales for, among other things, segmenting and representing these transient structures.

The formal definition of the wavelet transform (WT) is no different from the Fourier transform, save for the fact that the signal is correlated with a basis created by scaling and translating a single zero-mean time-localized function. In contrast, the Fourier transform basis consists of a set of infinite duration sinusoids, and the short-time Fourier transform (STFT) uses a basis of windowed sinusoids at a fixed time resolution. Therein lies a world of difference. A wavelet basis allows one to resolve a signal at multiple resolutions in a very efficient manner, which is not possible with the Fourier transform. Like the fast Fourier transform, fast methods exist for computing wavelet transforms. Furthermore, one is free in designing wavelet functions that are "intrinsically well adapted to represent a class of signals" (p. 11). Indeed, the largest chapter in the book is devoted to this subject.

AWT functions as an indispensable guide to any serious "tourists"—be they students, teachers, or professionals—intrigued by the technical aspects of wavelets and how they are and can be applied to signal processing. It is a beautifully produced book with nice paper, crisp text, and plenty of well-produced and thought-out graphics complementing the contents. One cannot help but want to visit these places after viewing the pictures alone. Like a good tour guide, AWT provides many paths of interest suited to particular levels of complexity and depth. Material is graded throughout the text by three levels of "difficulty." Essentials and easy proofs are marked with a "1." Important results for particular applications are marked "2." And items "at the frontier of research" (p. 18) are marked "3." This makes it easy to know where time must be necessarily spent, and which content can be skimmed. AWT is in no sense an introductory text; it is rigorous and complete in its presentation. The first four chapters, however, provide an excellent and thorough introduction to time–frequency (TF) representations, and motivate the utility and applicability of wavelets.

AWT begins with an introduction to the TF domain and transforms. A well-known example is the STFT, alluded to as early as 1946 by Nobel Prize physicist Dennis Gabor (inventor of the hologram). The discrete STFT partitions the TF plane uniformly, but with a trade...