Exact distribution of edge-preserving MAP estimators for linear signal models with Gaussian measurement noise

We derive the exact statistical distribution of maximum a posteriori (MAP) estimators having edge-preserving non-Gaussian priors, Such estimators have been widely advocated for image restoration and reconstruction problems, P revious investigations of these image recovery methods have been primarily empirical; the distribution we derive enables theoretical analysis, The sig nal model is linear with Gaussian measurement noise. We assume that the ene rgy function of the prior distribution is chosen to ensure a unimodal poste rior distribution (for which convexity of the energy function is sufficient ), and that the energy function satisfies a uniform Lipschitz regularity co ndition. The regularity conditions are sufficiently general to encompass po pular priors such as the generalized Gaussian Markov random field prior and the Huber prior, even though those priors are not everywhere twice continu ously differentiable.