Matrix optimization for poisson compressed sensing

For compressed sensing of Poissonian measurements, there is a need for nonnegative measurement matrices. We seek an optimal measurement matrix that conserves energy. Moreover, the signals pass a known but uncontrolled mixing matrix, before being multiplexed and measured. This situation is relevant to various optical applications. We optimize the measurement matrix by mutual coherence minimization, under nonnegativity and energy conservation constraints. Nonnegativity excludes the known approach of seeking an equiangular tight frame as the optimal matrix. We thus seek a quasi-equiangular frame, which is approximated by a tight frame. Simulation results demonstrate superior reconstruction using our optimized matrices, compared to random nonnegative matrices.