  We develop a theory of discrete directional Gabor frames for functions
defined on the $d$-dimensional Euclidean space. Our construction incorporates
the concept of ridge functions into the theory of isotropic Gabor systems, in
order to develop an anisotropic Gabor system with strong directional
sensitivity. We present sufficient conditions on a window function $g$ and a
sampling set $\Lambda$ for the corresponding directional Gabor system
$\{g_{m,t,u}\}_{(m,t,u)\in\Lambda}$ to form a discrete frame. Explicit
estimates on the frame bounds are developed. A numerical implementation of our
scheme is also presented, and is shown to perform competitively in compression
and denoising against state-of-the-art anisotropic methods.
