On the Uniqueness of the Solution of Image Reconstruction Problems with Poisson Data

The paper is concerned with the uniqueness of the Maximum a Posteriori estimate for restoration problems of data corrupted by Poisson noise, when we have to minimize a combination of the generalized Kullback-Leibler divergence and a regularization penalty function. The aim of this paper is to prove the uniqueness result for 2D and 3D problems for several penalty functions, such as an edge preserving functional, a simple case of the class of Markov Random Field (MRF) regularization functionals and the classical Tikhonov regularization.