  In inverse problems, prior information and a priori-based regularization
techniques play important roles. In this paper, we focus on image restoration
problems, especially on restoring images whose texture mainly follow one
direction. In order to incorporate the directional information, we propose a
new directional total generalized variation (DTGV) functional, which is based
on total generalized variation (TGV) by Bredies \textit{et al}. [SIAM J.
Imaging Sci., 3 (2010)]. After studying the mathematical properties of DTGV, we
utilize it as regularizer and propose the L$^2$-DTGV variational model for
solving image restoration problems. Due to the requirement of the directional
information in DTGV, we give a direction estimation algorithm, and then apply a
primal-dual algorithm to solve the minimization problem. Experimental results
show the effectiveness of the proposed method for restoring the directional
images. In comparison with isotropic regularizers like total variation and TGV,
the improvement of texture preservation and noise removal is significant.
