Curvature-dependent energies minimizers and visual curve completion

Abstract

Geometrical actions often used to describe elastic properties of elastic rods and fluid membranes have been proposed recently to explain functional mechanism of the primary visual cortex V1. These energies are defined in terms of functionals depending on the Frenet–Serret curvatures of a curve (profile curve, for axisymmetric membranes) and are relevant in image restoration by curve completion. In this context, extremals of length, total squared curvature (bending energy) and total squared torsion, acting on spaces of curves of the unit tangent bundle of the plane, are studied here. We first see that Sub-Riemannian geodesics in \({\mathbb {R}}^2\times {\mathbb {S}}^1\) project down to minimizers of a total curvature type energy in the plane. This motivates us to analyze the associated variational problem in Euclidean space under different boundary conditions. Although, as we show, parametrized extremals can be obtained by quadratures, their concrete explicit determination faces technical difficulties which can be overcome numerically. We use a numerical approach, based on a gradient descent method, to determine both critical trajectories for these three energies and their projection into the image plane under different boundary and isoperimetric constraints.