# Refitting solutions promoted by $\ell_{12}$ sparse analysis regularization with block penalties

## TL;DR

This paper introduces a novel refitting framework that removes bias from $\\ell_{12}$ analysis regularized solutions in inverse problems, using block penalties that act on the co-support to improve estimation accuracy.

## Contribution

It proposes a new refitting block penalty and an efficient algorithm for bias removal in $\\ell_{12}$ regularized inverse problems, enhancing solution accuracy.

## Key findings

- The new block penalty effectively reduces bias in estimations.
- The algorithm efficiently computes both biased and refitted solutions.
- Experiments demonstrate improved estimation quality with the proposed method.

## Abstract

In inverse problems, the use of an $\ell_{12}$ analysis regularizer induces a bias in the estimated solution. We propose a general refitting framework for removing this artifact while keeping information of interest contained in the biased solution. This is done through the use of refitting block penalties that only act on the co-support of the estimation. Based on an analysis of related works in the literature, we propose a new penalty that is well suited for refitting purposes. We also present an efficient algorithmic method to obtain the refitted solution along with the original (biased) solution for any convex refitting block penalty. Experiments illustrate the good behavior of the proposed block penalty for refitting.