Automatic parameter selection for the TGV regularizer in image restoration under Poisson noise

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.