  In this paper we study a model of geometry of vision due to Petitot, Citti
and Sarti. One of the main features of this model is that the primary visual
cortex V1 lifts an image from $R^2$ to the bundle of directions of the plane.
Neurons are grouped into orientation columns, each of them corresponding to a
point of this bundle.
  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 intrinsically
defines an hypoelliptic heat equation on the bundle of directions of the plane.
  In the original model, directions are considered both with and without
orientation, giving rise respectively to a problem on the group of
rototranslations of the plane SE(2) or on the projective tangent bundle of the
plane $PTR^2$.
  We provide a mathematical proof of several important facts for this model. We
first prove that the model is mathematically consistent only if directions are
considered without orientation. We then prove that the convolution of a
$L^2(R^2,R)$ function (e.g. an image) with a 2-D Gaussian is generically a
Morse function. This fact is important since the lift of Morse functions to
$PTR^2$ is defined on a smooth manifold. We then provide the explicit
expression of the hypoelliptic heat kernel on $PTR^2$ in terms of Mathieu
functions.
  Finally, we present the main ideas of an algorithm which allows to perform
image reconstruction on real non-academic images. A very interesting point is
that this algorithm is massively parallelizable and needs no information on
where the image is corrupted.
