Cortical-inspired image reconstruction via sub-Riemannian geometry and hypoelliptic diffusion

Ugo Boscain (1, 2) , Roman Chertovskih (3) , Jean-Paul Gauthier (4) , Dario Prandi (5, 1) , Alexey Remizov (6)

1 CNRS - Centre National de la Recherche Scientifique
2 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
3 IEPT - Institute of Earthquake Prediction Theory and Mathematical Geophysics
4 LSIS - Laboratoire des Sciences de l'Information et des Systèmes
5 L2S - Laboratoire des signaux et systèmes
6 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique

In this paper we review several algorithms for image inpainting based on the hypoelliptic diffusion naturally associated with a mathematical model of the primary visual cortex. In particular, we present one algorithm that does not exploit the information of where the image is corrupted, and others that do it. While the first algorithm is able to reconstruct only images that our visual system is still capable of recognize, we show that those of the second type completely transcend such limitation providing reconstructions at the state-of-the-art in image inpainting. This can be interpreted as a validation of the fact that our visual cortex actually encodes the first type of algorithm.

Domaines: Optimisation et contrôle [math.OC]; Analyse numérique [cs.NA]

Ugo Boscain, Roman Chertovskih, Jean-Paul Gauthier, Dario Prandi, Alexey Remizov. Cortical-inspired image reconstruction via sub-Riemannian geometry and hypoelliptic diffusion. SMAI 2017 - 8e Biennale Française des Mathématiques Appliquées et Industrielles, Jun 2017, La Tremblade, France. pp.37 - 53, ⟨10.1051/proc/201864037⟩. ⟨hal-01721718⟩