  The recovery of images from the observations that are degraded by a linear
operator and further corrupted by Poisson noise is an important task in modern
imaging applications such as astronomical and biomedical ones. Gradient-based
regularizers involve the popular total variation semi-norm have become standard
techniques for Poisson image restoration due to its edge-preserving ability.
Various efficient algorithms have been developed for solving the corresponding
minimization problem with non-smooth regularization terms. In this paper,
motivated by the idea of the alternating direction minimization algorithm and
the Newton's method with upper convergent rate, we further propose inexact
alternating direction methods utilizing the proximal Hessian matrix information
of the objective function, in a way reminiscent of Newton descent methods.
Besides, we also investigate the global convergence of the proposed algorithms
under certain conditions. Finally, we illustrate that the proposed algorithms
outperform the current state-of-the-art algorithms through numerical
experiments on Poisson image deblurring.
