Preface to "a wavelet tour of signal processing"

$•$ Radon transform and tomography.

$•$ Lifting for wavelets on surfaces, bounded domains and fast computations.

$•$ JPEG-2000 image compression.

$•$ Block thresholding for denoising.

$•$ Geometric representations with adaptive triangulations, curvelets and bandlets.

$•$ Sparse approximations in redundant dictionaries with pursuits algorithms.

$•$ Noise reduction with model selection, in redundant dictionaries.

$•$ Exact recovery of sparse approximation supports in dictionaries.

$•$ Multichannel signal representations and processing.

$•$ Dictionary learning.

$•$ Inverse problems and super-resolution.

$•$ Compressive sensing.

$•$ Source separation.

Teaching

This book is intended as a graduate textbook. Its evolution is also the result ofteaching courses in electrical engineering and applied mathematics. A new web siteprovides softwares for reproducible experimentations, exercise solutions, together with teaching material such as slides with figures, andMatlab softwares for numerical classes: http://wavelet-tour.com .

More exercises have been added at the end of each chapter, ordered by level of difficulty.Level ¹ exercises are direct applications of the course. Level ² requires more thinking. Level ³ includes some technical derivations. Level ⁴ are projects at the interface of research, that are possible topics for a final course project or an independent study.More exercises and projects can be found in the web site.

Sparse course programs

The Fourier transform and analog to digital conversion through linear sampling approximations provide a common ground for all courses(Chapters 2 and 3). It introduces basic signal representations, and reviews importantmathematical and algorithmic tools needed afterwards. Many trajectories are then possible to explore and teach sparsesignal processing. The following list gives several topics that can orient the coursestructure, with elements that can be covered along the way.

Sparse representations with bases and applications

$•$ Principles of linear and non-linear approximations in bases (Chapter 9).

$•$ Lipschitz regularity and wavelet coefficients decay (Chapter 6).

$•$ Wavelet bases (Chapter 7).

$•$ Properties of linear and non-linear wavelet basis approximations (Chapter 9).

$•$ Image wavelet compression (Chapter 10).

$•$ Linear and non-linear diagonal denoising (Chapter 11).

Sparse time-frequency representations

$•$ Time-frequency wavelet and windowed Fourier ridges for audio processing (Chapter 4).

$•$ Local cosine bases (Chapter 8).

$•$ Linear and non-linear approximations in bases (Chapter 9).

$•$ Audio compression (Chapter 10).

$•$ Audio denoising and block thresholding (Chapter 11).

$•$ Compression and denoising in redundant time-frequency dictionaries, with best bases or pursuit algorithms(Chapter 12).

Sparse signal estimation

$•$ Bayes versus minimax, and linear versus non-linear estimations (Chapter 11).

$•$ Wavelet bases (Chapter 7).

$•$ Linear and non-linear approximations in bases (Chapter 9).

$•$ Thresholding estimation (Chapter 11).

$•$ Minimax optimality (Chapter 11).

$•$ Model selection for denoising in redundant dictionaries (Chapter 12).

$•$ Compressive sensing (Chapter 13).

Sparse compression and information theory

$•$ Wavelet orthonormal bases (Chapter 7).

$•$ Linear and non-linear approximations in bases (Chapter 9).

$•$ Compression and sparse transform codes in bases (Chapter 10).

$•$ Compression in redundant dictionaries (Chapter 12).

$•$ Compressive sensing (Chapter 13).

$•$ Source separation (Chapter 13).

Dictionary representations and inverse problems

$•$ Frames and Riesz bases (Chapter 5).

$•$ Linear and non-linear approximations in bases (Chapter 9).

$•$ Ideal redundant dictionary approximations (Chapter 12).

$•$ Pursuit algorithms and dictionary incoherence (Chapter 12).

$•$ Linear and thresholding inverse estimators (Chapter 13).

$•$ Super-resolution and source separation (Chapter 13).

$•$ Compressive sensing (Chapter 13).

Geometric sparse processing

$•$ Time-frequency spectral lines and ridges (Chapter 4).

$•$ Frames and Riesz bases (Chapter 5).

$•$ Multiscale edge representations with wavelet maxima (Chapter 6).

$•$ Sparse approximation supports in bases (Chapter 9).

$•$ Approximations with geometric regularity, curvelets and bandlets (Chapters 9 and 12).

$•$ Sparse signal compression and geometric bit budget(Chapters 10 and 12).

$•$ Exact recovery of sparse approximation supports (Chapter 12).

$•$ Super-resolution (Chapter 13).

Acknowledgments

Some things do not change with new editions, in particular the traces left by the ones that were, and remain important references for me.As always, I am deeply grateful to Ruzena Bajcsy and Yves Meyer.

I spent the last few years, with three brilliant and kind colleagues, Christophe Bernard, Jérome Kalifa,and Erwan Le Pennec, in a pressure cooker called a start-up. Pressure means stress, despite very good moments.The resulting sauce was a blend of what all of us could provide, and which brought new flavors to our personalities. I am thankful to them for theones I got, some of which I am still discovering.

This new edition is the result of a collaboration with Gabriel Peyré, who made these changes not only possible, but also very interesting to do.I thank him for his remarkable work and help.