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 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 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, which in principle are equivalent, 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.