On the Gamma-convergence of some polygonal curvature functionals

Abstract

We study the convergence of polygonal approximations of two variational problems for curves in the plane. These are classical Euler’s elastica and a linear growth model which has connections to minimizing length in a space of positions and orientations. The geometry of these minimizers plays a role in several image-processing tasks, and also in modelling certain processes in visual perception. We prove Gamma-convergence for the linear growth model in a natural topology, and existence of cluster points for sequences of discrete minimizers. Combining the technique for cluster points with a previous Gamma-convergence result for elastica, we also give a proof of convergence of discrete minimizers to continuous minimizers in that case, when a length penalty is present in the functional. Finally, some numerical experiments with these approximations are presented, and a scale invariant modification is proposed for practical applications.

Keywords:

• elastic curves
• finite curvature
• variational approximation
• polygonal curves
• Gamma-convergence

AMS Subject Classifications:

• 49M25
• 65K10
• 68U10

Acknowledgements

We would like to thank Arpan Ghosh for fruitful discussions about the relevance and geometry of the continuous minimizing curves for the sub-Riemannian problem in and , and for providing references about them.

Additional information

Funding

This work was supported by the Seventh Framework Programme of the European Commission under the Initial Training Network-FIRST [agreement number PITN-GA-2009-238702].