Visual Curve Completion and Rotational Surfaces of Constant Negative Curvature

Abstract

If a piece of the contour of a picture is missing to the eye vision, then the brain tends to complete it using some kind of sub-Riemannian geodesics of the unit tangent bundle of the plane, R2xS1. These geodesics can be obtained by lifting extremal curves of a total curvature type energy in the plane. We completely solve this variational problem, geometrically. Moreover, we also show a way of constructing rotational surfaces of constant negative curvature in R3 by evolving these extremal curves under their associated binormal flow with prescribed velocity. Finally, we prove that, locally, all rotational constant negative curvature surfaces of R3 are foliated by extremal curves of these energies. Therefore, we conclude that there exists a one-to-one correspondence between the sub-Riemannian geodesics used by the brain for visual curve completion and these rotational surfaces of R3.