## Compressive Sampling

- Date: 03/12/2007

Emmanuel Candes (Caltech)

University of British Columbia

One of the central tenets of signal processing is the Shannon/Nyquist

sampling theory: the number of samples needed to reconstruct a signal

without error is dictated by its bandwidth-the length of the shortest

interval which contains the support of the spectrum of the signal under

study. Very recently, an alternative sampling or sensing theory has

emerged which goes against this conventional wisdom. This theory allows

the faithful recovery of signals and images from what appear to be

highly incomplete sets of data, i.e. from far fewer data bits than

traditional methods use. Underlying this methodology is a concrete

protocol for sensing and compressing data simultaneously.

This talk will present the key mathematical ideas underlying this new

sampling or sensing theory, and will survey some of the most important

results. We will argue that this is a robust mathematical theory; not

only is it possible to recover signals accurately from just an

incomplete set of measurements, but it is also possible to do so when

the measurements are unreliable and corrupted by noise. We will see

that the reconstruction algorithms are very concrete, stable (in the

sense that they degrade smoothly as the noise level increases) and

practical; in fact, they only involve solving very simple convex

optimization programs.

An interesting aspect of this theory is that it has bearings on some

fields in the applied sciences and engineering such as statistics,

information theory, coding theory, theoretical computer science, and

others as well. If time allows, we will try to explain these

connections via a few selected examples.

Emmanuel Candes is a Ronald and Maxine Linde Professor of Applied and

Computational Mathematics in the Division of Engineering and Applied

Science at the California Institute of Technology. He is a recent

recipient of the National Science Board's prestigious Alan T. Waterman

Award, the highest honour awarded by the National Science Foundation.

His research interests include compressive sampling, mathematical

signal processing, computational harmonic analysis, approximation

theory, multiscale analysis, and statistical estimation and detection.

IAM - PIMS - MITACS

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