Hypoelliptic Diffusion and Human Vision: A Semidiscrete New Twist

Abstract

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