  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.

Analysis of PDEs;Optimization and Control