Abstract. 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 P T R 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 P T R 2 is defined on a smooth manifold. We then provide the explicit expression of the hypoelliptic heat kernel on P T R 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.Keywords: sub-Riemannian geometry, image reconstruction, hypoelliptic diffusion 1. Introduction. In this paper we study a model of geometry of vision due to Petitot, Citti and Sarti. The main reference for the model is the paper [15]. Its first version can be found in [37,39]. This model was also studied by the authors of the present paper in [10], by Hladky and Pauls [24] and, independently, by Duits et al. in a series of papers mostly for contour completion [17] and contour enhancement [18,19]. This model has been called the pinwheel model by Petitot himself, see [40]. See also [38,45] and references therein.To start with, assume that a grey-level image is represented by a function I ∈ L 2 (D, R), where D is an open bounded domain of R 2 . The algorithm that we present here is based on three crucial ideas coming from neurophysiology:1. It is widely accepted that the retina approximately smoothes the images by making the convolution with a Gaussian function (see for instance [28,33,36] and references therein), equivalently solving a certain isotropic heat equation. Moreover, smoothing by the same technique is a widely used method in image processing. Then, it is an interesting question in itself to understand generic properties of these smoothed images. Our first result (proved in Appendix A) is that, given G(σ x , σ y ) the two dimensional Gaussian centered in (0, 0)
