Learning Dictionary-Based Unions of Subspaces for Image Denoising.

Many signals of interest are well-approximated by sparse linear combinations of atomic signals from a dictionary. Equivalently, they are well-approximated by low-dimensional subspaces in a union of subspaces generated by the dictionary. A given sparsity level has an associated union of subspaces (UoS) generated by sparse combinations of correspondingly many atoms. When considering a sequence of sparsity levels, we have a sequence of unions of subspaces (SUoS) of increasing dimension. This paper considers the problem of learning such an SUoS from data. While each UoS is combinatorially large with respect to sparsity level, our learning approach exploits the fact that sparsity is structured for many signals of interest, i.e., that certain collections of atoms are more frequently used together than others. This is known as group sparsity structure and has been studied extensively when the structure is known a priori. We consider the setting where the structure is unknown, and we seek to learn it from training data. We also adapt the subspaces we obtain to improve representation and parsimony, similar to the goal of adapting atoms in dictionary learning. We illustrate the benefits of the learned dictionary-based SUoS for the problem of denoising; using a more parsimonious and representative SUoS results in improved recovery of complicated structures and edges.