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Communications-Inspired Projection Design with Application to Compressive Sensing
We consider the recovery of an underlying signal x \in C^m based on
projection measurements of the form y=Mx+w, where y \in C^l 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,S_w), for known S_W. The objective is
to design a projection matrix M \in 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 h_a(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.Comment: 25 pages, 7 figures, parts of material published in IEEE ICASSP 2012,
submitted to SIIM
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