Non-local Regularization of Inverse Problems

Gabriel Peyré (1) , Sébastien Bougleux (2) , Laurent D. Cohen (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 Equipe Image - Laboratoire GREYC - UMR6072

This article proposes a new framework to regularize imaging lin- ear inverse problems using an adaptive non-local energy. A non-local graph is optimized to match the structures of the image to recover. This allows a better reconstruction of geometric edges and textures present in natural images. A fast algorithm computes iteratively both the solution of the regularization pro- cess and the non-local graph adapted to this solution. The graph adaptation is efficient to solve inverse problems with randomized measurements such as inpainting random pixels or compressive sensing recovery. Our non-local regularization gives state-of-the-art results for this class of inverse problems. On more challenging problems such as image super-resolution, our method gives results comparable to sparse regularization in a translation invariant wavelet frame.

Mots clés: Non-local regularization; inpainting; super-resolution; compressive sensing

Domaines: Traitement du signal et de l'image [eess.SP]; Traitement du signal et de l'image [eess.SP]

Gabriel Peyré, Sébastien Bougleux, Laurent D. Cohen. Non-local Regularization of Inverse Problems. Inverse Problems and Imaging , 2011, 5 (2), pp.511-530. ⟨10.3934/ipi.2011.5.511⟩. ⟨hal-00419791v2⟩