Emmanuel Candès: Wavelets, sparsity and its consequences

Abstract:
Soon after they were introduced, it was realized that wavelets offered representations of signals and images of interest that are far more sparse than those offered by more classical representations; for instance, Fourier series. Owing to their increased spatial localization at finer scales, wavelets prove to be better adapted to represent signals with discontinuities or transient phenomena because only a few wavelets actually interact with those discontinuities. It turns out that sparsity has extremely important consequences and this lecture will briefly discuss three vignettes. First, enhanced sparsity yields the same quality of approximation with fewer terms, a feat which has implications for lossy image compression since it roughly says that fewer bits are needed to achieve the same distortion. Second, enhanced sparsity yields superior statistical accuracy since there are fewer degrees of freedom or parameters to estimate. This gives scientists better methods to tease apart the signal from the noise. Third, enhanced sparsity has important consequences for data acquisition itself: a new technique known as compressed sensing is turning a few fields a bit upside down for it effectively says that to make a high-resolution image we need to collect far fewer samples than were thought necessary.

This lecture was held at The University of Oslo, May 24, 2017 and was part of the Abel Prize Lectures in connection with the Abel Prize Week celebrations.

Program for the Abel Lectures 2017:
1. Detection of gravitational waves and time-frequency wavelets, by Abel Laureate Yves Meyer, École Normale Supérieure Paris-Saclay
2. A Wavelet Zoom to Analyze a Multiscale World, by professor Stéphane Mallat, École Normale Supérieure
3. Wavelet bases: roots, surprises and applications, by professor Ingrid Daubechies, Duke University
4. Wavelets, sparsity and its consequences, professor Emmanuel Jean Candès, Stanford University

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