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

Abstract

We present a novel framework for the reconstruction of 1D composite signals assumed to be a mixture of two additive components, one sparse and the other smooth, given a finite number of linear measurements. We formulate the reconstruction problem as a continuous-domain regularized inverse problem with multiple penalties. We prove that these penalties induce reconstructed signals that indeed take the desired form of the sum of a sparse and a smooth component. We then discretize this problem using Riesz bases, which yields a discrete problem that can be solved by standard algorithms. Our 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 simulated data.

TV-based Spline Reconstruction with Fourier Measurements: Uniqueness and Convergence of Grid-Based Methods

“…To achieve this, we leverage our uniqueness result of Theorem 1. On the algorithmic side, grid-based methods to solve optimization problems with TV-based regularization have been proposed in [16,23,32,33,34].…”