Shape Analysis of Surfaces Using General Elastic Metrics
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Shape Analysis of Surfaces Using General Elastic Metrics Zhe Su1 · Martin Bauer1 · Stephen C. Preston2 · Hamid Laga3,4 · Eric Klassen1 Received: 15 October 2019 / Accepted: 9 April 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this article, we introduce a family of elastic metrics on the space of parametrized surfaces in 3D space using a corresponding family of metrics on the space of vector-valued one-forms. We provide a numerical framework for the computation of geodesics with respect to these metrics. The family of metrics is invariant under rigid motions and reparametrizations; hence, it induces a metric on the “shape space” of surfaces. This new class of metrics generalizes a previously studied family of elastic metrics and includes in particular the Square Root Normal Field (SRNF) metric, which has been proven successful in various applications. We demonstrate our framework by showing several examples of geodesics and compare our results with earlier results obtained from the SRNF framework. Keywords Shape spaces · Vector valued one-forms · Elastic metrics · SRNF metric · Surface registration
1 Introduction Shape analysis of surfaces in R3 has been motivated by many applications in bioinformatics, computer graphics and M. Bauer was partially supported by NSF-Grant 1912037 (collaborative research in connection with NSF-Grant 1912030). S. C. Preston was partially supported by Simons Foundation Collaboration Grant for Mathematicians No. 318969. E. Klassen was partially supported by Simons Foundation Collaboration Grant for Mathematicians No. 317865.
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Martin Bauer [email protected] Zhe Su [email protected] Stephen C. Preston [email protected] Hamid Laga [email protected] Eric Klassen [email protected]
1
Department of Mathematics, Florida State University, Tallahassee, USA
2
Department of Mathematics, Brooklyn College and CUNY Graduate Center, New York, USA
3
Information Technology, Mathematics and Statistics, Murdoch University, Perth, Australia
4
The Phenomics and Bioinformatics Research Centre, University of South Australia, Adelaide, Australia
medical imaging, see, e.g., [2,14,16,19,22,32]. In most applications, the actual parametrization of the surfaces under consideration is unknown and one is only able to observe the “shape” of the object, i.e., a priori the point correspondences between the surfaces are unknown and should be an output of the performed analysis. Furthermore, we will often identify surfaces that only differ by a rigid motion. Thus, we define the shape space of surfaces as the quotient space of all parametrized surfaces modulo the group of reparametrizations and/or the group of rigid motions. One goal in shape analysis is to quantify the differences and find the optimal deformations between the given objects; see Fig. 1 for two examples of optimal deformations between distinct surfaces. The main challenge in the context of shape analysis of surfaces consists in the registration problem, i.e., finding the (optimal) po
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