Continuous-Domain Formulation of Inverse Problems for Composite Sparse Plus Smooth Signals
Thomas Debarre

Meeting • 08 March 2021

Abstract
We present a novel framework for reconstructing 1D composite signals, where one component is sparse and the other is smooth, based on a finite number of linear measurements. We formulate the reconstruction problem as a continuous-domain regularized inverse problem with multiple penalties, and we prove that solutions of this problem are of the desired form (i.e., the sum of a sparse and smooth components). We then discretize this problem using Riesz bases, which yields a discrete problem that can be solved using standard algorithms. This discretization is exact in the sense that we are solving the continuous-domain problem over the search space specified by our bases without any discretization error. We propose a complete algorithmic pipeline and demonstrate its feasibility on some simulated data.