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Intrinsic Riemannian Metrics on Spaces of Curves: Theory and Computation

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Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging

Abstract

This chapter reviews some past and recent developments in shape comparison and analysis of curves based on the computation of intrinsic Riemannian metrics on the space of curve modulo shape-preserving transformations. We summarize the general construction and theoretical properties of quotient elastic metrics for Euclidean as well as non-Euclidean curves before considering the special case of the square root velocity metric for which the expression of the resulting distance simplifies through a particular transformation. We then examine the different numerical approaches that have been proposed to estimate such distances in practice and in particular to quotient out curve reparametrization in the resulting minimization problems.

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Notes

  1. 1.

    To be mathematically exact, one should limit oneself to the slightly smaller set of free immersions in this definition,as the quotient space has some mild singularities without this restriction. We will, however, ignore this subtlety for thepurpose of this book chapter.

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Acknowledgements

M. Bauer was partially supported by NSF-grant 1912037 (collaborative research in connection with NSF-grant 1912030) and NSF-grant 1953244 (collaborative research in connection with NSF-grant 1953267). N. Charon was partially supported by NSF-grant 1945224 and NSF-grant 1953267 (collaborative research in connection with NSF-grant 1953244). Eric Klassen gratefully acknowledges the support of the Simons Foundation-grant 317865.

Author information

Authors and Affiliations

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

    Martin Bauer & Eric Klassen

  2. Department of Applied Mathematics and Statistics, Johns Hopkins University, Baltimore, MD, USA

    Nicolas Charon

  3. Department of Applied Mathematics, University Paris 1, Paris, France

    Alice Le Brigant

Authors
  1. Martin Bauer
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  2. Nicolas Charon
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  3. Eric Klassen
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  4. Alice Le Brigant
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Corresponding author

Correspondence to Nicolas Charon .

Editor information

Editors and Affiliations

  1. Department of Mathematical Sciences, The University of Liverpool, Liverpool, UK

    Ke Chen

  2. Dept Appl Math & Theoretical Phys, University of Cambridge, Cambridge, UK

    Carola-Bibiane Schönlieb

  3. Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong

    Xue-Cheng Tai

  4. Department of Applied Mathematics & Stat, Johns Hopkins University, Baltimore, MD, USA

    Laurent Younces

Section Editor information

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    Xue-Cheng Tai

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    Carola-Bibiane Schönlieb

  3. No affiliation provided

    Ke Chen

  4. No affiliation provided

    Laurent Younes

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Bauer, M., Charon, N., Klassen, E., Le Brigant, A. (2021). Intrinsic Riemannian Metrics on Spaces of Curves: Theory and Computation. In: Chen, K., Schönlieb, CB., Tai, XC., Younces, L. (eds) Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging. Springer, Cham. https://doi.org/10.1007/978-3-030-03009-4_87-1

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  • DOI: https://doi.org/10.1007/978-3-030-03009-4_87-1

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