CLEAR: Covariant LEAst-square Re-fitting with applications to image restoration

TL;DR

This paper introduces CLEAR, a re-fitting framework that reduces systematic errors in image restoration algorithms by preserving covariant information and the Jacobian, improving restoration quality.

Contribution

We develop a covariant re-fitting approach that generalizes existing regularization techniques, emphasizing Jacobian preservation for enhanced image restoration.

Findings

Improved image restoration results in numerical simulations

Effective reduction of systematic errors in standard algorithms

Preservation of covariant information enhances re-fitting accuracy

Abstract

In this paper, we propose a new framework to remove parts of the systematic errors affecting popular restoration algorithms, with a special focus for image processing tasks. Generalizing ideas that emerged for $\ell_1$ regularization, we develop an approach re-fitting the results of standard methods towards the input data. Total variation regularizations and non-local means are special cases of interest. We identify important covariant information that should be preserved by the re-fitting method, and emphasize the importance of preserving the Jacobian (w.r.t. the observed signal) of the original estimator. Then, we provide an approach that has a "twicing" flavor and allows re-fitting the restored signal by adding back a local affine transformation of the residual term. We illustrate the benefits of our method on numerical simulations for image restoration tasks.