Image reconstruction via non-isotropic diffusion in Dubins/Reed-Shepp-like control systems

We compare the image inpainting results of two models of geometry of vision obtained through control theoretic considerations (the semi-discrete versions of the Citti-Petitot-Sarti and Mumford Elastica models). The main feature described by these models is the lifting of 2D images to the 3D group of translations and discrete rotations on the plane SE(2,N), done by the primary visual cortex. Corrupted images are then reconstructed by minimizing the energy necessary to activate neurons corresponding to the missing regions. This minimization procedure, which gives rise to Dubins/Reed-Shepp-like optimal control problems in the case of corrupted curves, is described by an hypoelliptic diffusion on SE(2,N). We present two numerical algorithms for the resolution of the diffusion equation in both models and then compare the results.