CLEAR: Covariant LEAst-square Re-fitting with applications to image restoration
This work addresses incremental improvements in image processing by reducing errors in existing restoration techniques.
The paper tackles systematic errors in image restoration algorithms by proposing a covariant re-fitting framework that adjusts outputs of standard methods to better align with input data, demonstrating benefits through numerical simulations.
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.