Why Use Sobolev Metrics on the Space of Curves

TL;DR

This paper explores reparametrization invariant Sobolev metrics on spaces of regular curves, analyzing their mathematical properties and emphasizing the need for efficient numerical methods for higher order metrics in shape analysis.

Contribution

It provides a detailed study of the completeness and applicability of Sobolev metrics in shape analysis, highlighting the importance of developing numerical methods for higher order metrics.

Findings

Sobolev metrics are reparametrization invariant and have desirable mathematical properties.

Completeness properties of these metrics are analyzed.

Efficient numerical methods for higher order Sobolev metrics are identified as a crucial future goal.

Abstract

We study reparametrization invariant Sobolev metrics on spaces of regular curves. We discuss their completeness properties and the resulting usability for applications in shape analysis. In particular, we will argue, that the development of efficient numerical methods for higher order Sobolev type metrics is an extremely desirable goal.