Section: New Results
New results: neurophysiology

[3] presents a semidiscrete 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 Mathieutype 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.

In [4] and [7] we consider the problem of minimizing ${\int}_{0}^{l}\sqrt{{\xi}^{2}+K{\left(s\right)}^{2}}ds$ for a planar curve having fixed initial and final positions and directions. The total length $l$ is free. Here $s$ is the arclength parameter, $K\left(s\right)$ is the curvature of the curve and $\xi >0$ is a fixed constant. This problem comes from a model of geometry of vision due to Petitot, Citti and Sarti. We study existence of local and global minimizers for this problem. In [7] we characterize subRiemannian geodesics and the range of the exponential map. In [4] we prove that if for a certain choice of boundary conditions there is no global minimizer, then there is neither a local minimizer nor a geodesic. We finally give properties of the set of boundary conditions for which there exists a solution to the problem.