Geometrical optical illusion via sub-Riemannian geodesics in the roto-translation group

We present a neuro-mathematical model for geometrical optical illusions (GOIs), a class of illusory phenomena that consists in a mismatch of geometrical properties of the visual stimulus and its associated percept. They take place in the visual areas V1/V2 whose functional architecture have been modeled in previous works by Citti and Sarti as a Lie group equipped with a sub-Riemannian (SR) metric. Here we extend their model proposing that the metric responsible for the cortical connectivity is modulated by the modeled neuro-physiological response of simple cells to the visual stimulus, hence providing a more biologically plausible model that takes into account a presence of visual stimulus. Illusory contours in our model are described as geodesics in the new metric. The model is confirmed by numerical simulations, where we compute the geodesics via SR-Fast Marching

Vessel tracking via sub-riemannian geodesics on the projective line bundle

\u3cp\u3eWe study a data-driven sub-Riemannian (SR) curve optimization model for connecting local orientations in orientation lifts of images. Our model lives on the projective line bundle R\u3csup\u3e2\u3c/sup\u3e × P\u3csup\u3e1\u3c/sup\u3e, with P\u3csup\u3e1\u3c/sup\u3e = S\u3csup\u3e1\u3c/sup\u3e/~ with identification of antipodal points. It extends previous cortical models for contour perception on R\u3csup\u3e2\u3c/sup\u3e × P\u3csup\u3e1\u3c/sup\u3e to the data-driven case. We provide a complete (mainly numerical) analysis of the dynamics of the 1st Maxwell-set with growing radii of SR-spheres, revealing the cut-locus. Furthermore, a comparison of the cusp-surface in R\u3csup\u3e2\u3c/sup\u3e × P\u3csup\u3e1\u3c/sup\u3e to its counterpart in R\u3csup\u3e2\u3c/sup\u3e × S\u3csup\u3e1\u3c/sup\u3e of a previous model, reveals a general and strong reduction of cusps in spatial projections of geodesics. Numerical solutions of the model are obtained by a single wavefront propagation method relying on a simple extension of existing anisotropic fast-marching or iterative morphological scale space methods. Experiments show that the projective line bundle structure greatly reduces the presence of cusps. Another advantage of including R\u3csup\u3e2\u3c/sup\u3e × P\u3csup\u3e1\u3c/sup\u3e instead of R\u3csup\u3e2\u3c/sup\u3e × S\u3csup\u3e1\u3c/sup\u3e in the wavefront propagation is reduction of computational time.\u3c/p\u3