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.
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.