Fast Sparse Coding Algorithms for Piece-wise Smooth Signals

The problem of computing a proper sparse representation matrix for a signal matrix that obeys some local smoothness property, given an over-complete dictionary, is considered. The focus is on piece-wise smooth signals, defined as signals that comprise a number of blocks that each fulfills the considered smoothness property. A computationally efficient sparse coding algorithm is derived by limiting the number of times that a new support set of dictionary atoms is computed, exploiting the smoothness of the signal. Furthermore, a new, total-variation regularized problem is proposed for computing the required sparse coding coefficients, exploiting further the smoothness priors of the signals. The considered problem is solved using the alternating direction method of multipliers. Finally, numerical results considering hyperspectral images are provided, that demonstrate the applicability and complexity -denoising performance benefits of the novel algorithms.