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

arXiv: https://arxiv.org/abs/1606.05158 · HTML: https://mlsanity.com/paper/1606.05158/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Jun 2016.

## Built on by (most cited first, 4 of 4)

- [Variational Regularization in Inverse Problems and Machine Learning](https://mlsanity.com/paper/2112.04591/) (2021)
- [Derivatives of Stochastic Gradient Descent in parametric optimization](https://mlsanity.com/paper/2405.15894/) (2024)
- [Proximal Operators of Sorted Nonconvex Penalties](https://mlsanity.com/paper/2506.15315/) (2025)
- [Preference-Based Dynamic Ranking Structure Recognition](https://mlsanity.com/paper/2509.24493/) (2025)

## Abstract (arXiv)

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 ℓ1 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.