Preface to "a wavelet tour of signal processing"

Preface to the sparse edition

I can not help but find striking resemblances between scientific communities and schoolsof fish. We interact in conferences and through articles, we move together while a globaltrajectory 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 alivebecause it went beyond wavelets. Writing such a book is about decoding the trajectory of the school, and gathering the pearls that havebeen 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 avoidthe 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 physicistsand mathematicians. Designing orthogonal bases with Xlets became a popular sport, withcompression 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 dictionariesof 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 moremathematics, 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 producerobust algorithms that operate in real time, as opposed to the 3 years we wereused to have for writing new ideas with promissing perspectives. Yet, we survived because mathematics isa major source of industrial innovations for signal processing. Semi-conductor technology offers amazingcomputational power and flexibility. However, ad-hoc algorithms often do not scale easily andmathematics accelerates the trial and error development process. Sparsitydecreases computations, memory and data communications. Although it brings beauty,mathematical understanding is not a luxury. It is required by increasinglysophisticated information processing devices.

New additions

Putting sparsity at the center of the book implied rewriting many parts and adding sections. Chapter 12 andChapter 13 are new. They introduce sparse representations in redundant dictionaries, and inverseproblems, super-resolution and compressive sensing. Here is a small catalogue of new elements in this third edition.