Recently, Poisson noise has become of great interest in many imaging applications. When regularization strategies are used in the so-called Bayesian approach, a relevant issue is to find rules for selecting a proper value of the regularization parameter. In this work we compare three different approaches which deal with this topic. The first model aims to find the root of a discrepancy equation, while the second one estimates such parameter by adopting a constrained, approach. These two models do not always provide reliable results in presence of low counts images. The third approach presented is the inexact Bregman procedure, which allows to use an overestimation of the regularization parameter and moreover enables to obtain very promising results in case of low counts images and High Dynamic Range astronomical images.