Compressive Sampling

Dr. Brian Robinson

Center for Applied Optics
UA Huntsville

February 27, 2009
219 Shelby Center
3:00 (Refreshements at 2:30)

Abstract

Conventional sampling is ruled by the Whittaker-Shannon Theorem, which states that, in order to faithfully recover a signal from a set of discrete samples, the samples must be taken with a frequency (the Nyquist frequency) which is twice the highest frequency present in the original signal. Under this regime, the sampling function is the classical comb function, consisting of a train of delta functions equally spaced with a period that is the reciprocal of twice the highest frequency in the signal. This principle underlies nearly all signal sampling protocols used in contemporary electronic devices. As it turns out, however, this method of sampling is not optimal. In other words, there are other methods that require far fewer samples than sampling with the canonical "spike" basis of W-S theory. Compressed Sensing (CS) is a novel sensing/sampling paradigm that goes against the common wisdom in data acquisition in that it demonstrates the feasibility of reconstructing data sets to a high degree of fidelity (and even exactly), with far fewer samples than is required by the W-S sampling paradigm. It is important to note that the methods are also robust to noise. CS relies on two related phenomena to make this possible: sparseness and incoherence. The properties of CS were largely discovered by applied mathematicians working in the field of probability theory and it has profound implications for practical systems such as optical imaging systems. In this talk we will illuminate the ideas of CS theory, discuss the roles of sparseness and incoherence, talk about the relationship that randomness plays in constructing a suitable set of sensing waveforms, and talk about the practical applications of this theory to modern sensing systems.