Restoration of Poissonian images is a class of inverse problem arising in fields such medical and astronomical imaging. Regularization criteria that combine the Poisson log-likelihood with a non-smooth convex regularizer lead to optimization
problems with several difficulties: the log-likelihood does not
have a Lipschitzian gradient; the regularizer is non-smooth; there is a non-negativity constraint. Using convex analysis tools, we give sufficient conditions for existence and uniqueness of solutions of these optimization problems for (frame-based) analysis and synthesis formulations. Then, we attack these problems with an adapted version of the alternating direction method of multipliers and show that sufficient conditions for convergence are met. The algorithm is shown to be competitive, often outperform, state-of-the-art methods.