A Numerical Framework for Sobolev Metrics on the Space of Curves

TL;DR

This paper develops numerical algorithms for Sobolev metrics on shape spaces of curves, enabling geodesic computation, mean estimation, and shape analysis, with applications demonstrated on cell nuclei shapes.

Contribution

It introduces algorithms for discretizing and solving geodesic problems for higher-order Sobolev metrics on curves, facilitating advanced shape analysis tasks.

Findings

Algorithms successfully compute geodesics for Sobolev metrics.

Framework enables principal component analysis and clustering of shapes.

Effective analysis demonstrated on biological cell nuclei shapes.

Abstract

Statistical shape analysis can be done in a Riemannian framework by endowing the set of shapes with a Riemannian metric. Sobolev metrics of order two and higher on shape spaces of parametrized or unparametrized curves have several desirable properties not present in lower order metrics, but their discretization is still largely missing. In this paper, we present algorithms to numerically solve the geodesic initial and boundary value problems for these metrics. The combination of these algorithms enables one to compute Karcher means in a Riemannian gradient-based optimization scheme and perform principal component analysis and clustering. Our framework is sufficiently general to be applicable to a wide class of metrics. We demonstrate the effectiveness of our approach by analyzing a collection of shapes representing HeLa cell nuclei.