Image Reconstruction Via Hypoelliptic Diffusion on the Bundle of Directions of the Plane

In this paper we present a model of geometry of vision which generalizes one due to Petitot, Citti and Sarti. One of its main features is that the primary visual cortex VI lifts the image from R-2 to the bundle of directions of the plane PTR2 = R-2 x P-1. Neurons are grouped into orientation columns, each of them corresponding to a point of the bundle PTR2.
In this model a corrupted image is reconstructed by minimizing the energy necessary for the activation of the orientation columns corresponding to regions in which the image is corrupted. The minimization process gives rise to an hypoelliptic heat equation on PTR2. The hypoelliptic heat equation is studied using the generalized Fourie: transform. It transforms the hypoelliptic equation into a 1-d heat equation with Mathieu potential, which one can solve numerically. Preliminary examples of image reconstruction are hereby provided.