Sobolev Metrics on Spaces of Discrete Regular Curves

TL;DR

This paper investigates the properties of discretized Sobolev metrics on spaces of regular curves, focusing on their metric and geodesic completeness, and bridges the gap between infinite-dimensional theory and finite-dimensional applications.

Contribution

It provides a detailed study of the metric and geodesic completeness of finite-dimensional approximations of Sobolev metrics on curves, extending infinite-dimensional results to practical discretized settings.

Findings

Finite-dimensional Sobolev metric spaces are metric and geodesically complete.

Results mirror the properties of the infinite-dimensional case.

Provides theoretical foundation for practical shape analysis applications.

Abstract

Reparametrization invariant Sobolev metrics on spaces of regular curves have been shown to be of importance in the field of mathematical shape analysis. For practical applications, one usually discretizes the space of smooth curves and considers the induced Riemannian metric on a finite dimensional approximation space. Surprisingly, the theoretical properties of the corresponding finite dimensional Riemannian manifolds have not yet been studied in detail, which is the content of the present article. Our main theorem concerns metric and geodesic completeness and mirrors the results of the infinite dimensional setting as obtained by Bruveris, Michor and Mumford.