  We are interested in the restoration of noisy and blurry images where the
texture mainly follows a single direction (i.e., directional images). Problems
of this type arise, for example, in microscopy or computed tomography for
carbon or glass fibres. In order to deal with these problems, the Directional
Total Generalized Variation (DTGV) was developed by Kongskov et al. in 2017 and
2019, in the case of impulse and Gaussian noise. In this article we focus on
images corrupted by Poisson noise, extending the DTGV regularization to image
restoration models where the data fitting term is the generalized
Kullback-Leibler divergence. We also propose a technique for the identification
of the main texture direction, which improves upon the techniques used in the
aforementioned work about DTGV. We solve the problem by an ADMM algorithm with
proven convergence and subproblems that can be solved exactly at a low
computational cost. Numerical results on both phantom and real images
demonstrate the effectiveness of our approach.
