Wavelets are recognized today as an essential tool in information processing and transmission. The authors have realized a valuable work, which explains what wavelets are, and provides a large collection of algorithmic constructions based on them.

Structured in twelve chapters and four appendices, this book covers the historical origins of wavelets, the algorithms of wavelets, and the applications of wavelets to signal and image processing, fractals, turbulence and astronomy. The first chapter describes the themes under study. The path from Fourier analysis to wavelet analysis, and to actual applied mathematics, is presented in the second chapter.

The next five chapters deal with time-scale algorithms and time-frequency algorithms. In the third chapter, quadrature mirror filters are studied in the context of digital telephony. Such filters are used by pyramid algorithms (described in chapter 4) for numerical image processing. In chapters 5 through 7, time-frequency algorithms are studied: the Wigner-Ville transform (chapter 5), the Malvar-Wilson wavelets (chapter 6), and Wavelet packets (chapter 7).

The last five chapters of the book treat applications of wavelets and time-scale methods: image coding by zero-crossing of the image’s wavelet transform (chapter 8), multifractal modeling and turbulence (chapter 9), multifractal functions (chapter 10), data compression and restoration of noisy images (chapter 11), and some applications in astronomy (chapter 12). The appendices provide some background to the information used in the textbook.

Readers will find this an excellent mathematical monograph, with a good bibliography (including papers and books of general and historical interest), a complete author index, and a well-designed subject index. The book’s appearance and layout are both adequate. If someone is seeking algorithms described in pseudo-code or translated into a programming language, this is not the best place to look for them. The subject is presented from mathematical point of view.

The mathematical background is presented in a clear manner, but given the level of knowledge necessary to completely understand the subject, this book is not recommended for undergraduate students. In my opinion, this book provides a valuable reference for scientists and engineers on the recent developments in this branch of applied mathematics. It is appropriate for people interested in both research and applications.