You are here

A Primer on Wavelets and Their Scientific Applications

James S. Walker
Chapman & Hall/CRC
Publication Date: 
Number of Pages: 
[Reviewed by
David A. Huckaby
, on

This book is aimed at those who want to progress as quickly as possible from knowing nothing about wavelets to knowledgeably engaging in wavelet-based audio and image processing. Flipping through its pages reveals figure after figure, most of them graphs of audio signals or grayscale images being analyzed and processed in various ways. A companion web site is also packed with sound and image files ready for download along with free wavelet software with which to manipulate them. The author clearly envisions readers not as observers, but as budding practitioners.

That he enables wavelet beginners to achieve this level of proficiency in so few pages is quite a feat. He manages it by demanding scant mathematical background (mostly elementary algebra), by focusing on fundamental ideas rather than formalism, and by driving the exposition with current, exciting applications.

The modest prerequisite mathematics eliminates the need for a preliminary chapter or two of advanced material. It helps that the book focuses on the discrete theory rather than on the more difficult continuous theory. Sections containing advanced theory (for example, the z-transform) are marked with an asterisk; this material is generally not needed later, except in other asterisked sections. These sections fill about 14 pages of the book altogether.

Rather than insisting on mathematical rigor, the author highlights basic ideas. In lieu of precisely stated theorems, important wavelet results are written in informal English and sometimes called a “principle” or a “feature.” One example is the “Compaction of Energy” principle, stated thus: “The energy of the trend subsignal a1 accounts for a large percentage of the energy of the transformed signal ( a1 | d1 )”. This statement is the fruit of a discussion driven by a simple eight-value example signal to which has been applied a Haar transform. That the four values in a1 are large and the four values in d1 small is obvious by inspection. After the statement of the principle, readers are promised an application to signal compression in a few pages. Most will already guess that the trick is to store most of a signal’s energy in a few values and discard the rest.

In this text theory is an on-ramp to applications, most of which involve audio or image processing. Compression and denoising are discussed many times using various algorithms. Image recognition and edge detection and enhancement are touched on. There are sections on speech analysis and music analysis. Room is given to describe the JPEG standard, to detect features in an electrocardiogram, and to analyze bird calls. Most of the copious figures that illustrate all of this are composed of multiple frames. For example, frame a might display an original image, frame b a noisy version, and frames c and d the results from two denoising algorithms. Or frame a might display a signal, b its Haar transform, c the energy map, and d the compressed signal. Moreover, the reader is often invited to make comparisons not only between frames within a figure but also between figures. Performing the many possible comparisons leads to a wealth of insight. The figures almost constitute a book on their own.

Indeed, the entire supporting apparatus is comprehensive and useful. Each chapter ends with a list of references ordered by appearance. These lists are all combined into one alphabetical bibliography in an appendix. Also capping each chapter (except the first) are worked examples and exercises, many with solutions provided in an appendix. Some require routine calculations (e.g., computing a Daub4 transform) while others introduce new material (e.g., Huffman coding). The majority involve reproducing figures or data tables given in the chapter or performing analysis on files contained at the companion web site using the wavelet software freely available there. For instance, a typical example begins, “To produce Figure 4.10(a) you load the image boat.pgm, select Transform/Wavelet, and then plot a 5-level Daub 9/7 wavelet transform of the image.” A typical exercise reads, “Produce images like in Figure 4.14, but this time use the dog_head.bmp image and select a region of interest that contains just the jaw region of the dog. How much savings in file size do you get over sending a lossless compression of the entire image?” Altogether these examples and exercises fill about 56 pages (with an additional 23 pages of selected exercise solutions in an appendix)!

The companion web site supplements the book and provides updates. From here is downloadable the free wavelet software FAWAV and many image and sounds files to be used with it. The site also has links to papers and other sites along with a list of project ideas. There is also an Updates page, which at the time of this review included some new references and additional analysis of music using Gabor transforms and scalograms.

This book could serve as the text in an applications-based undergraduate course on wavelets. While the mathematical theory serves mainly to facilitate the applications, this makes the theory that much more meaningful. Denoising images from a spy plane or analyzing a Chinese folk song compels students to look backwards and forwards at their mathematics education and also sideways at other subjects. The basic theory required to do the applications is presented simply and clearly and instills in students a deeper respect for the mathematics they have already learned. More advanced theory is optional, but evidently valuable, and beckons students onward. Linear algebra seems a lot more important when it is being used to compress an image! Students will also enjoy connections to related subject areas. The idea of entropy in information theory can be compared and contrasted to what is being learned in physics class. Concepts like conservation of energy will be seen in a new context. There is even a one-paragraph nod to Heisenberg’s Uncertainty Principle. Students will be enchanted by the ideas along with the applications.

A course based on the book could be offered at an introductory level, since elementary algebra is the main prerequisite, or at a more advanced level, due to the starred sections and references, links to which are on the web site. The book’s relative lack of formalism results in a low symbol-to-word ratio; here is a math text that can be read quickly yet with comprehension. This is in keeping with the goal of engaging the applications as quickly and painlessly as possible. With such accessibility and a first-rate supporting apparatus, this flexible book can teach various audiences the basics of wavelets, involve them in applications, and inspire them to learn more.

David A. Huckaby is an assistant professor of mathematics at Angelo State University.

Haar Wavelets. Daubechies Wavelets. Two-Dimensional Wavelets. Frequency Analysis. Beyond Wavelets. Software for Wavelet Analysis.