Wavelets. Tools for science and technology.

*(English)*Zbl 0970.42020
Philadelphia, PA: SIAM. xiii, 256 p. (2001).

This book is an updated and extended version of [Y. Meyer, “Wavelets: algorithms and applications” (1995; Zbl 0821.42018)]. The new book contains extra chapters on wavelets and turbulence, wavelets and fractals, data compression, and wavelets in astronomy. Rather than being a mathematical introduction to wavelets, the book exhibits the connection between wavelets and several topics in applied science. It is a very good source of information for as well applied scientists as mathematicians who want to know about applications of wavelet theory. Furthermore, the books gives a nice historical presentation of ideas from the last century which eventually lead to wavelets.

Chapter 1 is an introduction to the subjects that are developed in the later chapters. The basic concepts (signals, wavelets, representation of signals, optimal representation) are introduced in an easy understandable way, and analogies are drawn whenever possible. By example, it is made clear that “optimal representation” depends on the information one is interested in. Chapter 2 is devoted to the historical perspective. Most sources only mention Fourier and Haar as the ancestors of wavelets, but here the developement is connected with, e.g., Brownian motions, Littlewood-Paley theory, and the work by Lusin and Franklin. Chapter 3 is about quadrature mirror filters and its close relationship to Daubechies’ pioneering work. Chapter 4 discusses pyramid algorithms and their applications to image processing. A short introduction to multiresolution analysis and biorthogonal wavelets is included here. Chapter 5 deals with time-frequency analysis. An example explains why the Fourier transform is not the optimal tool for analyzing music (the time-localization of the notes is lost). Time-frequency atoms (corresponding to dilations, modulations, and translates of a single function \(g\)) of the type \(W_{h,w,t_0}(t)=h^{-1/2}e^{-iwt}g(\frac{t-t_0}{h})\) are introduced. The special case \(h=1\) corresponds to a Gabor system, while the case \(w=0\) gives a wavelet system. The Wigner-Ville transform is defined and related to Mallat’s matching pursuit algorithm and pseudodifferential calculus. Chapter 6 concerns Malvar-Wilson bases consisting of translated versions of a window multiplied with certain sine and cosine functions. Bases of this type are for instance used to overcome the problems caused by Balian-Low’s theorem. An algorithm for finding the optimal Malvar-Wilson basis is described. This is strongly related to Chapter 7 about wavelet packets. Chapter 8 describes work by Marr, in particular his conjecture about zero-crossings (a mathematical description of an image with sudden intensity changes) and the more precise version by Mallat. Counterexamples are given in Chapter 8 and Appendix C. Chapter 9 is about turbulence and the need for simultaneous space and scale representations, making wavelets an obvious tool. Chapter 10 (about wavelets and multifractal functions) gives an interesting proof of the non-differentiability of the Weierstrass function which highlights the importance of selecting the right wavelet, and analyses Riemann’s function. It is also shown how to find the Hölder exponents of a multifractal function using wavelet analysis. Chapter 11 defines sparse wavelet expansions and discusses its role in data compression and signal transmission. The relationship between wavelets, Besov spaces, and nonlinear approximation is examined. In Chapter 12 a problem from astronomy (recovering of an object from data) is formulated as an inverse convolution problem, and wavelets are used to attack it. It is also explained that astronomy leads to very large data sets, and therefore compression is an important issue here, too.

Chapter 1 is an introduction to the subjects that are developed in the later chapters. The basic concepts (signals, wavelets, representation of signals, optimal representation) are introduced in an easy understandable way, and analogies are drawn whenever possible. By example, it is made clear that “optimal representation” depends on the information one is interested in. Chapter 2 is devoted to the historical perspective. Most sources only mention Fourier and Haar as the ancestors of wavelets, but here the developement is connected with, e.g., Brownian motions, Littlewood-Paley theory, and the work by Lusin and Franklin. Chapter 3 is about quadrature mirror filters and its close relationship to Daubechies’ pioneering work. Chapter 4 discusses pyramid algorithms and their applications to image processing. A short introduction to multiresolution analysis and biorthogonal wavelets is included here. Chapter 5 deals with time-frequency analysis. An example explains why the Fourier transform is not the optimal tool for analyzing music (the time-localization of the notes is lost). Time-frequency atoms (corresponding to dilations, modulations, and translates of a single function \(g\)) of the type \(W_{h,w,t_0}(t)=h^{-1/2}e^{-iwt}g(\frac{t-t_0}{h})\) are introduced. The special case \(h=1\) corresponds to a Gabor system, while the case \(w=0\) gives a wavelet system. The Wigner-Ville transform is defined and related to Mallat’s matching pursuit algorithm and pseudodifferential calculus. Chapter 6 concerns Malvar-Wilson bases consisting of translated versions of a window multiplied with certain sine and cosine functions. Bases of this type are for instance used to overcome the problems caused by Balian-Low’s theorem. An algorithm for finding the optimal Malvar-Wilson basis is described. This is strongly related to Chapter 7 about wavelet packets. Chapter 8 describes work by Marr, in particular his conjecture about zero-crossings (a mathematical description of an image with sudden intensity changes) and the more precise version by Mallat. Counterexamples are given in Chapter 8 and Appendix C. Chapter 9 is about turbulence and the need for simultaneous space and scale representations, making wavelets an obvious tool. Chapter 10 (about wavelets and multifractal functions) gives an interesting proof of the non-differentiability of the Weierstrass function which highlights the importance of selecting the right wavelet, and analyses Riemann’s function. It is also shown how to find the Hölder exponents of a multifractal function using wavelet analysis. Chapter 11 defines sparse wavelet expansions and discusses its role in data compression and signal transmission. The relationship between wavelets, Besov spaces, and nonlinear approximation is examined. In Chapter 12 a problem from astronomy (recovering of an object from data) is formulated as an inverse convolution problem, and wavelets are used to attack it. It is also explained that astronomy leads to very large data sets, and therefore compression is an important issue here, too.

Reviewer: Ole Christensen (Lyngby)

##### MSC:

42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |

42-02 | Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces |

94A12 | Signal theory (characterization, reconstruction, filtering, etc.) |