Appears in collection : 2019 - T1 - WS1 - Variational methods and optimization in imaging

In this talk, a framework to remove parts of the systematic errors affecting popular restoration algorithms is presented, with a special focus on image processing tasks. Generalizing ideas that emerged for $\ell_1$ regularization, an approach re-fitting the results of standard methods towards the input data is developed. Total variation regularization and non-local means are special cases of interest. Important covariant information that should be preserved by the re-fitting method are identified, and the importance of preserving the Jacobian (w.r.t. the observed signal) of the original estimator is emphasized. Then, a numerical approach is proposed. It has a twicing flavor and allows re-fitting the restored signal by adding back a local affine transformation of the residual term. The benefits of the method are illustrated on numerical simulations for image restoration tasks. This a joint work with Charles-Alban. Deledalle (CNRS), Joseph Salmon (Univ. Montpellier) and Samuel Vaiter (CNRS)