  We address the image restoration problem under Poisson noise corruption. The
Kullback-Leibler divergence, which is typically adopted in the variational
framework as data fidelity term in this case, is coupled with the second-order
Total Generalized Variation (TGV$^2$). The TGV$^2$ regularizer is known to be
capable of preserving both smooth and piece-wise constant features in the
image, however its behavior is subject to a suitable setting of the parameters
arising in its expression. We propose a hierarchical Bayesian formulation of
the original problem coupled with a Maximum A Posteriori estimation approach,
according to which the unknown image and parameters can be jointly and
automatically estimated by minimizing a given cost functional. The minimization
problem is tackled via a scheme based on the Alternating Direction Method of
Multipliers, which also incorporates a procedure for the automatic selection of
the regularization parameter by means of a popular discrepancy principle.
Computational results show the effectiveness of our proposal.
