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Elastic Shape Analysis of Surfaces and Images

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Riemannian Computing in Computer Vision

Abstract

We describe two Riemannian frameworks for statistical shape analysis of parameterized surfaces. These methods provide tools for registration, comparison, deformation, averaging, statistical modeling, and random sampling of surface shapes. A crucial property of both of these frameworks is that they are invariant to reparameterizations of surfaces. Thus, they result in natural shape comparisons and statistics. The first method we describe is based on a special representation of surfaces termed square-root functions (SRFs). The pullback of the \(\mathbb{L}^{2}\) metric from the SRF space results in the Riemannian metric on the space of surfaces. The second method is based on the elastic surface metric. We show that a restriction of this metric, which we call the partial elastic metric, becomes the standard \(\mathbb{L}^{2}\) metric under the square-root normal field (SRNF) representation. We show the advantages of these methods by computing geodesic paths between highly articulated surfaces and shape statistics of manually generated surfaces. We also describe applications of this framework to image registration and medical diagnosis.

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Notes

  1. 1.

    It is interesting to note that the first two terms in the full elastic metric form the unique family of ultralocal metrics on the space of Riemannian metrics on which diffeomorphisms act by isometries [14]. The \(\lambda = 0\) case of this metric on Riemannian metrics has been studied in e.g., [10, 16, 18].

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Acknowledgements

This work was partially supported by a grant from the Simons Foundation (#317865 to Eric Klassen).

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Authors and Affiliations

  1. Department of Statistics, The Ohio State University, Columbus, OH, USA

    Sebastian Kurtek

  2. Department of Mathematical Sciences, Durham University, Durham, UK

    Ian H. Jermyn

  3. Department of Statistics, Florida State University, Tallahassee, FL, USA

    Qian Xie

  4. Department of Mathematics, Florida State University, Tallahassee, FL, USA

    Eric Klassen

  5. Phenomics and Bioinformatics Research Centre, University of South Australia, Adelaide, SA, Australia

    Hamid Laga

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  1. Sebastian Kurtek
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Correspondence to Sebastian Kurtek .

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Editors and Affiliations

  1. Arizona State University, Tempe, Arizona, USA

    Pavan K. Turaga

  2. Florida State University, Tallahassee, Florida, USA

    Anuj Srivastava

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Kurtek, S., Jermyn, I.H., Xie, Q., Klassen, E., Laga, H. (2016). Elastic Shape Analysis of Surfaces and Images. In: Turaga, P., Srivastava, A. (eds) Riemannian Computing in Computer Vision. Springer, Cham. https://doi.org/10.1007/978-3-319-22957-7_12

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  • DOI: https://doi.org/10.1007/978-3-319-22957-7_12

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-22956-0

  • Online ISBN: 978-3-319-22957-7

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