I can not help but find striking resemblances between scientific communities and schools of fish. We interact in conferences and through articles, we move together while a global trajectory emerges from individual contributions. Some of us like to be at the center of the school, others prefer to wander around, and few swim in multiple directions in front. To avoid dying by starvation in a progressively narrower and specialized domain, a scientific community needs to move on. Computational harmonic analysis is still well alive because it went beyond wavelets. Writing such a book is about decoding the trajectory of the school, and gathering the pearls that have been uncovered on the way. Wavelets are not any more the central topic, despite the original title. It is just an important tool, as the Fourier transform is. Sparse representation and processing are now at the core.
In the 80's, many researchers were focused on building time-frequency decompositions, trying to avoid the uncertainty barrier, and hoping to discover the ultimate representation. Along the way came the construction of wavelet orthogonal bases, which opened new perspectives through collaborations with physicists and mathematicians. Designing orthogonal bases with Xlets became a popular sport, with compression and noise reduction applications. Connections with approximations and sparsity also became more apparent. The search for sparsity has taken over, leading to new grounds, where orthonormal bases are replaced by redundant dictionaries of waveforms. Geometry is now also becoming more apparent through sparse approximation supports in dictionaries.
During these last 7 years, I also encountered the industrial world. With a lot of naiveness, some bandlets and more mathematics, we created a start-up with Christophe Bernard, Jérome Kalifa and Erwan Le Pennec. It took us some time to learn that in 3 months good engineering should produce robust algorithms that operate in real time, as opposed to the 3 years we were used to have for writing new ideas with promissing perspectives. Yet, we survived because mathematics is a major source of industrial innovations for signal processing. Semi-conductor technology offers amazing computational power and flexibility. However, ad-hoc algorithms often do not scale easily and mathematics accelerates the trial and error development process. Sparsity decreases computations, memory and data communications. Although it brings beauty, mathematical understanding is not a luxury. It is required by increasingly sophisticated information processing devices.
New Additions
Putting sparsity at the center of the book implied rewriting many parts and adding sections. Chapter 12 and Chapter 13 are new. They introduce sparse representations in redundant dictionaries, and inverse problems, super-resolution and compressive sensing. Here is a small catalogue of new elements in this third edition.
Teaching
This book is intended as a graduate textbook. Its evolution is also the result of teaching courses in electrical engineering and applied mathematics. A new web site provides softwares for reproducible experimentations, exercise solutions, together with teaching material such as slides with figures, and Matlab softwares for numerical classes: http://wavelet-tour.com.
More exercises have been added at the end of each chapter, ordered by level of difficulty. Level1 exercises are direct applications of the course. Level2 requires more thinking. Level3 includes some technical derivations. Level4 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 important mathematical and algorithmic tools needed afterwards. Many trajectories are then possible to explore and teach sparse signal processing. The following list gives several topics that can orient the course structure, with elements that can be covered along the way.
Sparse representations with bases and applications
Sparse time-frequency representations
Sparse signal estimation
Sparse compression and information theory
Dictionary representations and inverse problems
Geometric sparse processing
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 the ones 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.
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