Hypoelliptic diffusion and human vision: a semi-discrete new twist

TL;DR

This paper introduces a semi-discrete model for neurogeometry of vision using the group of translations and discrete rotations, leading to efficient inpainting algorithms based on Mathieu-type diffusions.

Contribution

It proposes a novel semi-discrete approach on the group $SE(2,N)$, simplifying harmonic analysis and enabling efficient image inpainting with Mathieu diffusions.

Findings

The method effectively inpaints deeply corrupted images.

Harmonic analysis on $SE(2,N)$ simplifies computations.

Mathieu-type diffusions are efficiently integrated using standard numerical methods.

Abstract

This paper presents a semi-discrete alternative to the theory of neurogeometry of vision, due to Citti, Petitot and Sarti. We propose a new ingredient, namely working on the group of translations and discrete rotations $SE(2,N)$. The theoretical side of our study relates the stochastic nature of the problem with the Moore group structure of $SE(2,N)$. Harmonic analysis over this group leads to very simple finite dimensional reductions. We then apply these ideas to the inpainting problem which is reduced to the integration of a completely parallelizable finite set of Mathieu-type diffusions (indexed by the dual of $SE(2,N)$ in place of the points of the Fourier plane, which is a drastic reduction). The integration of the the Mathieu equations can be performed by standard numerical methods for elliptic diffusions and leads to a very simple and efficient class of inpainting algorithms. We illustrate the performances of the method on a series of deeply corrupted images.