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Design of projection matrix for compressive sensing by nonsmooth optimization

Sparsity and incoherence are the two key ingredients in compressive sensing (CS). Given a sparsifying dictionary D, the projection matrix P must be as incoherent with D as possible for the CS system to be efficient. Thus the design of projection matrix is naturally a problem of minimizing the coherence between P andD. Unfortunately, this turns out to be a nonconvex, nonsmooth, large-scale problem even for a CS system of moderate size. In this paper, the above-mentioned problem is investigated in a formulation where the problem is converted into a sequence of nonsmooth but convex subproblems. A subgradient projection algorithm is proposed to solve the nonsmooth subproblems that converges to a projection matrix with improved performance. The performance of the proposed algorithm is evaluated by simulations and comparisons with several existing techniques.