Communications-inspired projection design with application to compressive sensing

Published

Journal Article

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 that w ~ CN(w; 0,Σw) for known Σ w. The objective is to design a projection matrix M ∈ ℂℓ×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 Rényi entropy of the projections hα (y) (Shannon entropy is a special case). By capitalizing on explicit characterizations of the gradients of the information measures with respect to the projection matrix, where we also partially extend the well-known results of Palomar and Verdu ́ from the mutual information to the Rényi entropy domain, we reveal the key operations carried out by the optimal projection 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 projection design method with which state-of-the-art adaptive CS signal recovery is achieved. © 2012 Society for Industrial and Applied Mathematics.

Full Text

Duke Authors

Cited Authors

  • Carson, WR; Chen, M; Rodrigues, MRD; Calderbank, R; Carin, L

Published Date

  • January 1, 2012

Published In

Volume / Issue

  • 5 / 4

Start / End Page

  • 1182 - 1212

Electronic International Standard Serial Number (EISSN)

  • 1936-4954

Digital Object Identifier (DOI)

  • 10.1137/120878380

Citation Source

  • Scopus