Analysis of an approximate model for Poisson data reconstruction and a related discrepancy principle

In this paper, we investigate an approximate model for Poisson data reconstruction inspired by a discrepancy principle for the selection of the regularization parameter, recently proposed by Bardsley and Goldes. The model can be obtained by approximating the generalized Kullback-Leibler (KL) divergence in terms of a weighted least-squares function, with weights depending on the object to be reconstructed. We show that it is possible to develop a complete theory, based on this approximation, including results of existence and uniqueness of regularized solutions and simple gradient-based reconstruction algorithms for their computation. Moreover, in this context, the criterion of Bardsley and Goldes is a natural one and it is possible to prove that, in several important cases, it provides a unique value of the regularization parameter. We describe a few numerical tests for comparing the approximate approach with the exact one based on the generalized KL divergence. In the case of a moderate or large number of photons, they provide essentially the same results and therefore the approximate model can be considered as a possible alternative to the exact one.