Processing stationary noise: model and parameter selection in variational methods

TL;DR

This paper analyzes variational denoising methods for stationary noise, demonstrating that such noise can often be modeled as colored Gaussian noise and providing parameter selection strategies for effective denoising.

Contribution

It offers a statistical analysis of stationary noise in variational denoising, introduces analytical parameter selection, and compares constrained and Lagrangian formulations.

Findings

Stationary noise can be approximated by colored Gaussian noise.

Analytical regularization parameters are derived for Morozov's discrepancy principle.

Lagrangian formulation enables more efficient denoising with strong convexity.

Abstract

Additive or multiplicative stationary noise recently became an important issue in applied fields such as microscopy or satellite imaging. Relatively few works address the design of dedicated denoising methods compared to the usual white noise setting. We recently proposed a variational algorithm to tackle this issue. In this paper, we analyze this problem from a statistical point of view and provide deterministic properties of the solutions of the associated variational problems. In the first part of this work, we demonstrate that in many practical problems, the noise can be assimilated to a colored Gaussian noise. We provide a quantitative measure of the distance between a stationary process and the corresponding Gaussian process. In the second part, we focus on the Gaussian setting and analyze denoising methods which consist of minimizing the sum of a total variation term and an $l^2$ data fidelity term. While the constrained formulation of this problem allows to easily tune the parameters, the Lagrangian formulation can be solved more efficiently since the problem is strongly convex. Our second contribution consists in providing analytical values of the regularization parameter in order to approximately satisfy Morozov's discrepancy principle.