Image reconstruction via optimal control 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 V1 lifts the image from ℝ2 to the bundle of directions of the plane PTℝ2 = ℝ2 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 Fourier 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