Numerical Methods for Parameter Estimation in Poisson Data Inversion

In a regularized approach to Poisson data inversion, the problem is reduced to the minimization of an objective function which consists of two terms: a data-fidelity function, related to a generalized Kullback---Leibler divergence, and a regularization function expressing a priori information on the unknown image. This second function is multiplied by a parameter $$\beta $$β, sometimes called regularization parameter, which must be suitably estimated for obtaining a sensible solution. In order to estimate this parameter, a discrepancy principle has been recently proposed, that implies the minimization of the objective function for several values of $$\beta $$β. Since this approach can be computationally expensive, it has also been proposed to replace it with a constrained minimization, the constraint being derived from the discrepancy principle. In this paper we intend to compare the two approaches from the computational point of view. In particular, we propose a secant-based method for solving the discrepancy equation arising in the first approach; when this root-finding algorithm can be combined with an efficient solver of the inner minimization problems, the first approach can be competitive and sometimes faster than the second one.