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Travel Guide

Module by: Stephane Mallat. E-mail the author

Summary: This collection comprises Chapter 1 of the book A Wavelet Tour of Signal Processing, The Sparse Way (third edition, 2009) by Stéphane Mallat. The book's website at Academic Press is http://www.elsevier.com/wps/find/bookdescription.cws_home/714561/description#description The book's complementary materials are available at http://wavelet-tour.com

Travel Guide

Reproducible Computational Science

This book covers the whole spectrum from theorems on functions of continuous variables to fast discrete algorithms and their applications. (Reference) argues that models based on continuous time functions give useful asymptotic results for understanding the behavior of discrete algorithms. Still, a mathematical analysis alone is often unable to fully predict the behavior and suitability of algorithms for specific signals. Experiments are necessary and such experiments should be reproducible, just like experiments in other fields of science (DonohoB:95).

The reproducibility of experiments requires having complete software and full source code for inspection, modification, and application under varied parameter settings. Following this perspective, computational algorithms presented in this book are available as MATLAB subroutines or in other software packages. Figures can be reproduced and the source code is available. Software demonstrations and selected exercise solutions are available at http://wavelet-tour.com. For the instructor, solutions are available at www.elsevierdirect.com/9780123743701.

Book Road Map

Some redundancy is introduced between sections to avoid imposing a linear progression through the book. The preface describes several possible programs for a sparse signal-processing course.

All theorems are explained in the text and reading the proofs is not necessary to understand the results. Most of the book's theorems are proved in detail, and important techniques are included. Exercises at the end of each chapter give examples of mathematical, algorithmic, and numeric applications, ordered by level of difficulty from 1 to 4, and selected solutions can be found at http://wavelet-tour.com.

The book begins with Chapters 2 and 3, which review the Fourier transform and linear discrete signal processing. They provide the necessary background for readers with no signal-processing background. Important properties of linear operators, projectors, and vector spaces can be found in the Appendix. Local time-frequency transforms and dictionaries are presented in Chapter 4;  the wavelet and windowed Fourier transforms are introduced and compared. The measurement of instantaneous frequencies illustrates the limitations of time-frequency resolution. Dictionary stability and redundancy are introduced in Chapter 5 through the frame theory, with examples of windowed Fourier, wavelet, and curvelet frames. Chapter 6 explains the relationship between wavelet coefficient amplitude and local signal regularity. It is applied to the detection of singularities and edges and to the analysis of multifractals.

Wavelet bases and fast filter bank algorithms are important tools presented in Chapter 7. An overdose of orthonormal bases can strike the reader while studying the construction and properties of wavelet packets and local cosine bases in Chapter 8. It is thus important to read Chapter 9, which describes sparse approximations in bases. Signal-compression and denoising applications described in Chapters 10 and 11 give life to most theoretical and algorithmic results in the book. These chapters offer a practical perspective on the relevance of linear and nonlinear signal-processing algorithms. Chapter 12 introduces sparse decompositions in redundant dictionaries and their applications. The resolution of inverse problems is studied in Chapter 13, with super-resolution, compressive sensing, and source separation.

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