  In this paper we propose a blind deconvolution method which applies to data
perturbed by Poisson noise. The objective function is a generalized
Kullback-Leibler divergence, depending on both the unknown object and unknown
point spread function (PSF), without the addition of regularization terms;
constrained minimization, with suitable convex constraints on both unknowns, is
considered. The problem is nonconvex and we propose to solve it by means of an
inexact alternating minimization method, whose global convergence to stationary
points of the objective function has been recently proved in a general setting.
The method is iterative and each iteration, also called outer iteration,
consists of alternating an update of the object and the PSF by means of fixed
numbers of iterations, also called inner iterations, of the scaled gradient
projection (SGP) method. The use of SGP has two advantages: first, it allows to
prove global convergence of the blind method; secondly, it allows the
introduction of different constraints on the object and the PSF. The specific
constraint on the PSF, besides non-negativity and normalization, is an upper
bound derived from the so-called Strehl ratio, which is the ratio between the
peak value of an aberrated versus a perfect wavefront. Therefore a typical
application is the imaging of modern telescopes equipped with adaptive optics
systems for partial correction of the aberrations due to atmospheric
turbulence. In the paper we describe the algorithm and we recall the results
leading to its convergence. Moreover we illustrate its effectiveness by means
of numerical experiments whose results indicate that the method, pushed to
convergence, is very promising in the reconstruction of non-dense stellar
clusters. The case of more complex astronomical targets is also considered, but
in this case regularization by early stopping of the outer iterations is
required.
