Spline-based Reconstruction of Periodic Signals with Sparse Innovations

TL;DR

This paper introduces spline-based methods and algorithms for reconstructing periodic signals with sparse innovations, providing theoretical insights and practical algorithms with promising results.

Contribution

It presents three algorithms for periodic signal reconstruction using total variation penalization and analyzes their theoretical properties, including conditions for solution uniqueness.

Findings

The Frank-Wolfe algorithm shows promising greedy reconstruction performance.

Theoretical conditions for solution uniqueness are derived based on differential operators.

Periodic exponential splines are characterized, aiding in understanding solution shapes.

Abstract

Optimization-based problems have become of great interest for signal approximation purposes, as they achieved good accuracy results while being extremely flexible and versatile. In this work, we put our focus on the context of periodic signals sampled with spatial measurements. The optimization problems are penalized thanks to the total variation norm, using a specific class of (pseudo-)differential operators to use well-chosen reconstruction functions. We introduce three algorithms and their adaptation to this specific use case. The first one is a discrete grid-based $\ell_1$ method, the second is called CPGD, which relies on the estimation of discrete innovations within the FRI framework, and the last one is the Frank-Wolfe algorithm. We put the emphasis on that later algorithm as we underline its greedy behavior. We consider a refined version of this algorithm, that leads to very encouraging outputs. We pursue the algorithmic treatment with a theoretical analysis of the optimization problem, based on convexity and duality theory. We draw away two conditions that, if verified by the differential operator, would inform on the shape of the solutions and even ensure uniqueness, in some cases. Moreover, we demonstrate the general expression of periodic one-dimensional exponential splines, as a first step to verify the latter conditions for exponential operators.