Anthropomorphic Image Reconstruction via Hypoelliptic Diffusion

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 R² 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 a 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 PTR². 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 an L²(R²,R) function (e.g., an image) with a two-dimensional (2D) Gaussian is generically a Morse function. This fact is important since the lift of Morse functions to PTR² is defined on a smooth manifold. We then provide the explicit expression of the hypoelliptic heat kernel on PTR² in terms of Mathieu functions. Finally, we present the main ideas of an algorithm which allows us to perform image reconstruction on real nonacademic images. A very interesting point is that this algorithm is massively parallelizable and needs no information on where the image is corrupted.