Periodic Splines and Gaussian Processes for the Resolution of Linear Inverse Problems

TL;DR

This paper compares variational and statistical methods for reconstructing periodic signals from noisy data, showing they often produce equivalent solutions as periodic splines, thus broadening the applicability of variational techniques.

Contribution

It provides a unified framework for both approaches, characterizes conditions for their equivalence, and demonstrates this in simulations for inverse problems.

Findings

Both approaches often yield the same periodic spline solution.

The equivalence extends the practical use of variational methods.

Conditions for equivalence are fully characterized.

Abstract

This paper deals with the resolution of inverse problems in a periodic setting or, in other terms, the reconstruction of periodic continuous-domain signals from their noisy measurements. We focus on two reconstruction paradigms: variational and statistical. In the variational approach, the reconstructed signal is solution to an optimization problem that establishes a tradeoff between fidelity to the data and smoothness conditions via a quadratic regularization associated to a linear operator. In the statistical approach, the signal is modeled as a stationary random process defined from a Gaussian white noise and a whitening operator; one then looks for the optimal estimator in the mean-square sense. We give a generic form of the reconstructed signals for both approaches, allowing for a rigorous comparison of the two.We fully characterize the conditions under which the two formulations yield the same solution, which is a periodic spline in the case of sampling measurements. We also show that this equivalence between the two approaches remains valid on simulations for a broad class of problems. This extends the practical range of applicability of the variational method.