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.
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Acknowledgments
This research was supported by MINE CO-FEDER Grant MTM2014-54804-P and Gobierno Vasco Grant IT1094-16, Spain. A. Pámpano has been supported by Programa Predoctoral de Formación de Personal Investigador No Doctor, Dpto de Educación, Política Lingüística y Cultura del Gobierno Vasco, 2015. We also thank the referees for many useful comments which have improved the original version. Finally, we are very grateful to one of the referees for bringing the works [10] and [12] to our attention.
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Arroyo, J., Garay, O.J. & Pámpano, A. Curvature-dependent energies minimizers and visual curve completion. Nonlinear Dyn 86, 1137–1156 (2016). https://doi.org/10.1007/s11071-016-2953-4
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DOI: https://doi.org/10.1007/s11071-016-2953-4