Sparsity-regularized photon-limited imaging

In many medical imaging applications (e.g., SPECT, PET), the data are a count of the number of photons incident on a detector array. When the number of photons is small, the measurement process is best modeled with a Poisson distribution. The problem addressed in this paper is the estimation of an underlying intensity from photon-limited projections where the intensity admits a sparse or low-complexity representation. This approach is based on recent inroads in sparse reconstruction methods inspired by compressed sensing. However, unlike most recent advances in this area, the optimization formulation we explore uses a penalized negative Poisson log-likelihood objective function with nonnegativity constraints (since Poisson intensities are naturally nonnegative). This paper describes computational methods for solving the nonnegatively constrained sparse Poisson inverse problem. In particular, the proposed approach incorporates sequential separable quadratic approximations to the log-likelihood and computationally efficient partition-based multiscale estimation methods.