Exploiting Sparsity in Tight-Dimensional Spaces for Piecewise Continuous Signal Recovery

Recovery of certain piecewise continuous signals from noisy observations has been a major challenge in sciences and engineering. In this paper, in a tight-dimensional representation space, we exploit sparsity hidden in a class of possibly discontinuous signals named finite-dimensional piecewise continuous (FPC) signals. More precisely, we propose a tight-dimensional linear transformation which reveals a certain sparsity in discrete samples of the FPC signals. This transformation is designed by exploiting the fact that most of the consecutive samples are contained in special subspaces. Numerical experiments on recovery of piecewise polynomial signals and piecewise sinusoidal signals show the effectiveness of the revealed sparsity.