Section: New Results

New results: neurophysiology

We gave new contributions to the developing theory of human locomotion modeled through optimal control problems. In this paradigm, the trajectories are assumed to be solutions of an optimal control problem whose cost has to be determined.

• The purpose of [6] has been to analyze the class of optimal control problems defined in this way. We proved strong convergence of their solutions, on the one hand for perturbations of the initial and final points (stability), and on the other hand for perturbations of the cost (robustness).

• In [5] we discussed the modeling of both the dynamical system and the cost to be minimized, and we analyzed the corresponding optimal synthesis. The main results describe the asymptotic behavior of the optimal trajectories as the target point goes to infinity.

In [10] we studied the model of geometry of vision due to Petitot, Citti and Sarti [81] . 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 $SE\left(2\right)$ or on the projective tangent bundle of the plane. We provided a mathematical proof of several important facts for this model. We first proved that the model is mathematically consistent only if directions are considered without orientation. We then proved that the convolution of a $L^2({ℝ}^2,ℝ)$ function (e.g. an image) with a 2D Gaussian is generically a Morse function. This fact is important since the lift of Morse functions to the projective tangent bundle of the plane is defined on a smooth manifold. We then provided the explicit expression of the hypoelliptic heat kernel on the projective tangent bundle of the plane in terms of Mathieu functions. Finally, we presented the main ideas of an algorithm which allows to perform image reconstruction on real non-academic images. The algorithm is massively parallelizable and needs no information on where the image is corrupted.