  In this paper, we propose two algorithms for solving linear inverse problems
when the observations are corrupted by Poisson noise. A proper data fidelity
term (log-likelihood) is introduced to reflect the Poisson statistics of the
noise. On the other hand, as a prior, the images to restore are assumed to be
positive and sparsely represented in a dictionary of waveforms. Piecing
together the data fidelity and the prior terms, the solution to the inverse
problem is cast as the minimization of a non-smooth convex functional. We
establish the well-posedness of the optimization problem, characterize the
corresponding minimizers, and solve it by means of primal and primal-dual
proximal splitting algorithms originating from the field of non-smooth convex
optimization theory. Experimental results on deconvolution and comparison to
prior methods are also reported.
