Anthropomorphic image reconstruction via hypoelliptic diffusion

TL;DR

This paper analyzes a geometric model of vision for image reconstruction using hypoelliptic diffusion on the bundle of directions, providing mathematical proofs, explicit kernels, and a parallelizable algorithm for real images.

Contribution

It proves the model's consistency only without orientation, derives the hypoelliptic heat kernel on PTR^2, and introduces a parallelizable reconstruction algorithm.

Findings

The model is consistent only when directions are considered without orientation.

The hypoelliptic heat kernel on PTR^2 is explicitly expressed via Mathieu functions.

The proposed algorithm effectively reconstructs real images without prior corruption location information.

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 $PTR^2$. 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.