A Numerical Framework for Sobolev Metrics on the Space of Curves
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.
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© 2017, Society for Industrial and Applied Mathematics.
History
Submitted: 17 March 2016
Accepted: 13 October 2016
Published online: 12 January 2017
Keywords
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Funding Information
Austrian Science Fund http://dx.doi.org/10.13039/501100002428 : P246251
Funding Information
Brunel University London http://dx.doi.org/10.13039/501100007914 : BRIEF award
Funding Information
Erwin Schrödinger International Institute for Mathematical Physics http://dx.doi.org/10.13039/501100003066
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