Résumé en anglais

This article proposes a new framework to regularize linear inverse problems using the total variation on non-local graphs. This non-local graph allows to adapt the penalization to the geometry of the underlying function to recover. A fast algorithm computes iteratively both the solution of the regularization process and the non-local graph adapted to this solution. We show numerical applications of this method to the resolution of image processing inverse problems such as inpainting, super-resolution and compressive sampling.