Communications-Inspired Projection Design with Application to Compressive Sensing
Abstract
We consider the recovery of an underlying signal x ∈ Cm based on projection measurements of the form y=Mx+w, where y ∈ Cl and w is measurement noise; we are interested in the case l < m. It is assumed that the signal model p(x) is known, and w CN(w;0,Sw), for known SW. The objective is to design a projection matrix M ∈ C(l x m) to maximize key information-theoretic quantities with operational significance, including the mutual information between the signal and the projections I(x;y) or the Renyi entropy of the projections ha(y) (Shannon entropy is a special case). By capitalizing on explicit characterizations of the gradients of the information measures with respect to the projections matrix, where we also partially extend the well-known results of Palomar and Verdu from the mutual information to the Renyi entropy domain, we unveil the key operations carried out by the optimal projections designs: mode exposure and mode alignment. Experiments are considered for the case of compressive sensing (CS) applied to imagery. In this context, we provide a demonstration of the performance improvement possible through the application of the novel projection designs in relation to conventional ones, as well as justification for a fast online projections design method with which state-of-the-art adaptive CS signal recovery is achieved.