Directionally-structured dictionary learning and sparse representation based on subspace projections

This paper presents a new strategy for directionally-structured dictionary learning and component-wise sparse representation. The signal space is divided into directional subspace triplets. Directionally-selective projection operators are designed for this purpose. Each triplet contains two orthogonal subspaces along with a remainder one. For each triplet, a compact dictionary is learned. Sparse representation is done in an analogous manner. The most-fitting dictionary triplet is selected for each signal based on its directional structure. Using the designed projection operators, the signal is decomposed into three subspace components living in the three triplet subspaces. The signal's sparse approximation is obtained as the direct summation of the sparse approximations of these three components, each coded over its subspace dictionary. Experiments conducted over a set of natural images show that the proposed strategy improves the sparse representation coding quality over standard methods, as tested in the problem of image representation.