Discrete Directional Gabor Frames

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.