Abstract

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 metholdology 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.