##
**A wavelet tour of signal processing. The sparse way.
3rd ed.**
*(English)*
Zbl 1170.94003

Amsterdam: Elsevier/Academic Press (ISBN 978-0-12-374370-1/hbk). xx, 805 p. (2009).

The monograph under review is the third edition of the classic book that appeared in 1998 and is reviewed in Zbl 0937.94001. For the second edition from 1999 see Zbl 0998.94510. This graduate-level textbook presents an excellently written, comprehensive survey of all major concepts, techniques, and applications of sparse representations which play a key role in signal processing. The book clearly presents the standard representations with Fourier, wavelets, and time-frequency transforms, and the construction of orthogonal bases with fast algorithms. But wavelets are no longer the central topic. Sparse representation and processing are now at the core, since sparsity decreases computations, memory, and data communications. The central concept of sparsity is applied to signal compression, noise reduction, and inverse problems.

This book consists of 13 chapters, an appendix (on mathematical complements), an updated bibliography (with 501 references), and a subject index. Putting sparsity at the center of the book implied rewriting many parts and adding sections. Chapter 1 provides the main ideas and gives an orientation of this book. The titles of Chapters 2–9 are not changed. Chapter 10 introduces the signal compression together with the image compression standards JPEG and JPEG-2000. Chapter 11 presents methods for signal denoising. Chapters 12 and 13 are completely new. Chapter 12 introduces sparse signal approximations in redundant dictionaries with pursuit algorithms. Chapter 13 deals with the resolution of inverse problems, super-resolution, compressive sensing, and source separation. Further new topics in this book are computer tomography, lifting for wavelets on surfaces, geometric image processing with curvelets and bandlets, time-frequency audio processing and denoising.

More exercises (ordered by level of difficulty) have been added at the end of each chapter. A new website http://wavelet-tour.com provides exercise solutions, figures, MATLAB toolboxes, C/C++ codes and teaching material (slides with figures). The author proposes 6 course programs which can be covered by selected chapters. This survey of the basic features of most topics of sparse signal processing is an invaluable resource for researchers and graduate students in applied mathematics and signal processing. Without doubt, this work will stimulate the further research.

This book consists of 13 chapters, an appendix (on mathematical complements), an updated bibliography (with 501 references), and a subject index. Putting sparsity at the center of the book implied rewriting many parts and adding sections. Chapter 1 provides the main ideas and gives an orientation of this book. The titles of Chapters 2–9 are not changed. Chapter 10 introduces the signal compression together with the image compression standards JPEG and JPEG-2000. Chapter 11 presents methods for signal denoising. Chapters 12 and 13 are completely new. Chapter 12 introduces sparse signal approximations in redundant dictionaries with pursuit algorithms. Chapter 13 deals with the resolution of inverse problems, super-resolution, compressive sensing, and source separation. Further new topics in this book are computer tomography, lifting for wavelets on surfaces, geometric image processing with curvelets and bandlets, time-frequency audio processing and denoising.

More exercises (ordered by level of difficulty) have been added at the end of each chapter. A new website http://wavelet-tour.com provides exercise solutions, figures, MATLAB toolboxes, C/C++ codes and teaching material (slides with figures). The author proposes 6 course programs which can be covered by selected chapters. This survey of the basic features of most topics of sparse signal processing is an invaluable resource for researchers and graduate students in applied mathematics and signal processing. Without doubt, this work will stimulate the further research.

Reviewer: Manfred Tasche (Rostock)

### MSC:

94-02 | Research exposition (monographs, survey articles) pertaining to information and communication theory |

94-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to information and communication theory |

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

94A08 | Image processing (compression, reconstruction, etc.) in information and communication theory |

65T60 | Numerical methods for wavelets |

42C15 | General harmonic expansions, frames |

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