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

TL;DR

This paper introduces a continuous-domain inverse problem framework for reconstructing 1D signals composed of sparse and smooth components from limited measurements, with an exact discretization approach and demonstrated feasibility.

Contribution

It formulates a novel continuous-domain regularized inverse problem for composite signals, proving the penalties induce the desired signal structure, and provides an exact discretization method with a complete algorithmic pipeline.

Findings

Successfully reconstructs composite signals from limited measurements.

Proves penalties induce sparse-plus-smooth signal structure.

Demonstrates the approach's feasibility on simulated data.

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.