  In many linear regression problems, including ill-posed inverse problems in
image restoration, the data exhibit some sparse structures that can be used to
regularize the inversion. To this end, a classical path is to use $\ell_{12}$
block based regularization. While efficient at retrieving the inherent sparsity
patterns of the data -- the support -- the estimated solutions are known to
suffer from a systematical bias. We propose a general framework for removing
this artifact by refitting the solution towards the data while preserving key
features of its structure such as the support. This is done through the use of
refitting block penalties that only act on the support of the estimated
solution. Based on an analysis of related works in the literature, we introduce
a new penalty that is well suited for refitting purposes. We also present a new
algorithm 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 solutions of Total
Variation and Total GeneralizedVariation models.
