Double Sparsity: Learning Sparse Dictionaries for Sparse Signal Approximation

An efficient and flexible dictionary structure is proposed for sparse and redundant signal representation. The proposed sparse dictionary is based on a sparsity model of the dictionary atoms over a base dictionary, and takes the form D = Phi A, where Phi is a fixed base dictionary and A is sparse. The sparse dictionary provides efficient forward and adjoint operators, has a compact representation, and can be effectively trained from given example data. In this, the sparse structure bridges the gap between implicit dictionaries, which have efficient implementations yet lack adaptability, and explicit dictionaries, which are fully adaptable but non-efficient and costly to deploy. In this paper, we discuss the advantages of sparse dictionaries, and present an efficient algorithm for training them. We demonstrate the advantages of the proposed structure for 3-D image denoising.