Second order elastic metrics on the shape space of curves

TL;DR

This paper develops algorithms to compute geodesics and averages on the shape space of planar curves using second order Sobolev metrics, enabling more complete and flexible shape analysis.

Contribution

It introduces numerical methods for geodesic and mean computation under second order Sobolev metrics, enhancing shape analysis capabilities.

Findings

Algorithms successfully compute geodesics and Karcher means.

The framework allows flexible weighting of metric terms.

Effective analysis demonstrated on physical shape data.

Abstract

Second order Sobolev metrics on the space of regular unparametrized planar curves have several desirable completeness properties not present in lower order metrics, but numerics are still largely missing. In this paper, we present algorithms to numerically solve the initial and boundary value problems for geodesics. The combination of these algorithms allows to compute Karcher means in a Riemannian gradient-based optimization scheme. Our framework has the advantage that the constants determining the weights of the zero, first, and second order terms of the metric can be chosen freely. Moreover, due to its generality, it could be applied to more general spaces of mapping. We demonstrate the effectiveness of our approach by analyzing a collection of shapes representing physical objects.