A convergent blind deconvolution method for post-adaptive-optics astronomical imaging

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 (KL) 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. Therefore the method is similar to other proposed methods based on the Richardson-Lucy (RL) algorithm, with SGP replacing RL. 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 (SR), which is the ratio between the peak value of an aberrated versus a perfect wavefront. Therefore a typical application, but not the unique one, is to 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 in detail 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. However the proposed method, based on SGP, allows the generalization to the case of differentiable regularization terms added to the KL divergence, even if this generalization is outside the scope of this paper.