Elements of Wavelets for Engineers and Scientists

Elements of Wavelets for Engineers and Scientists , Dwight F. Mix Kraig J. Olejniczak Wiley, Hoboken, NJ, 2003. $59.95 (236 pp.). ISBN 0-471-46617-4

Nowadays, wavelets are ubiquitous in science, whether in physics, applied mathematics, or engineering. New textbooks on the subject keep appearing on the market, but each differs widely in style, aim, and target audience. Some books are clearly mathematically minded in content, such as Ingrid Daubechies’s Ten Lectures on Wavelets (SIAM, 1992) or Yves Meyer’s Wavelets: Algorithms & Applications (SIAM, 1993). Others have a distinct signal-processing flavor, such as Stéphane Mallat’s A Wavelet Tour of Signal Processing (2nd ed., Academic Press, 1999), and some are rather elementary, such as Wavelets: Tools for Science & Technology (SIAM, 2001) by Stéphane Jaffard, Yves Meyer, and Robert D. Ryan and A Friendly Guide to Wavelets (Birkhäuser, 1994) by Gerald Kaiser.

In Elements of Wavelets for Engineers and Scientists, Dwight Mix and Kraig Olejniczak claim “there is a need for an introductory book on wavelets for scientists and engineers that acts as a prerequisite for the other 150 books mentioned above…. This book is for the rest of us, the non-mathematicians who want to understand wavelets.” Thus the authors have written a text at a very elementary level, full of worked-out examples and drill problems. They claim the mathematical background of the reader is that achieved by a typical holder of a bachelor of science degree, but I disagree. The first three or four chapters of the book should normally be known to undergraduate fresh-men—if not high-school seniors—studying science curricula: vector spaces, bases, matrices, and so on. On the other hand, the authors assume the reader knows a lot about signal processing—a topic mostly foreign to physics undergraduates until their junior or senior year, at least. We are probably witnessing, in Mix and Olejniczak’s book, a cultural split between US and European science curricula.

The authors’ general vantage point is that of signal processing: They define a signal as being a finite string of numbers. Of course, that definition is correct but very reductive. The chapter titles are quite revealing of the general attitude of the authors. Indeed, after covering the basic prerequisites, they introduce readers to “Sampling Theory,” “Multirate Processing,” “Fast Fourier Transform,” “Wavelet Transform” (which, at last, is covered in chapter 8), “Quadratic Mirror Filters,” and “Practical Wavelets and Filters.” Such an array of topics can provide readers a very down-to-earth approach to the subject that is perfectly suitable for self-study. But it has the drawback of ignoring a large part of the field. For instance, the Haar wavelets are the only wavelets mentioned with some detail, and the continuous wavelet transform is barely mentioned. Thus the reader fails to understand the basic motivation for using wavelets and, more generally, time-frequency analysis—namely, to overcome the inadequacy of Fourier transforms for treating nonstationary signals. (Yet, strictly speaking, stationary signals exist only in textbooks.) In addition, the book has no bibliography, which is strange for a work whose explicit goal is “to conceptually prepare you for the more mathematically rigorous wavelet texts that lie ahead.” One would have expected a list of recommended texts, with comments about their aim and difficulty.

In conclusion, my feelings about Elements of Wavelets for Engineers and Scientists are ambivalent. No other textbook on wavelets offers such an elementary treatment of the subject—so elementary, in fact, that it is often irritating. But the authors’ treatment is pedagogically careful, with each chapter preceded by chapter goals and worked-out examples and drill problems. (By the way, wavelet theory is notorious for using rather elementary mathematical techniques: No sign of the “esoteric math” mentioned by the authors is to be found in the theory!)

Readers should take the book as what the authors explicitly state it to be in the preface: a prerequisite to tackling a genuine textbook on wavelets. Otherwise, they will quit with a seriously flawed, incomplete understanding of this remarkably efficient tool in science and engineering.