Second order elastic metrics on the shape space of curves

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.

Completeness properties of Sobolev metrics on the space of curves

“…Sobolev metrics have been generalized to manifold-valued curves [9,13,25] and to higherdimensional immersed manifolds [3,8]. Numerical discretizations are available for first order metrics [24,27] as well as for second order ones [4,5,6]. See [7] for an overview of Riemannian metrics on spaces of curves and related spaces.…”

A Numerical Framework for Sobolev Metrics on the Space of Curves

“…In the Riemannian setting the length of a path c : 4) and the geodesic distance between two curves c 0 , c 1 ∈ Imm(S 1 , R d ) is the infimum of the lengths of all paths connecting these curves, i.e.,…”