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Sub-Riemannian Methods in Shape Analysis

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Handbook of Variational Methods for Nonlinear Geometric Data

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Abstract

Because they reduce the search domains for geodesic paths in manifolds, sub-Riemannian methods can be used both as computational and modeling tools in designing distances between complex objects. This chapter provides a review of the methods that have been recently introduced in this context to study shapes, with a special focus on shape spaces defined as homogeneous spaces under the action of diffeomorphisms. It describes sub-Riemannian methods that are based on control points, possibly enhanced with geometric information, and their generalization to deformation modules. It also discusses the introduction of implicit constraints on geodesic evolution, and the associated computational challenges. Several examples and numerical results are provided as illustrations.

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Author information

Authors and Affiliations

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

    Laurent Younes

  2. Laboratoire Jacques-Louis Lions, Sorbonne Université, Paris, France

    Barbara Gris

  3. Centre de Mathématiques et leurs Applications, ENS Paris-Saclay, Cachan, France

    Alain Trouvé

Authors
  1. Laurent Younes
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  2. Barbara Gris
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  3. Alain Trouvé
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Corresponding author

Correspondence to Laurent Younes .

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Editors and Affiliations

  1. Faculty of Mathematics, University of Vienna, Wien, Austria

    Philipp Grohs

  2. Institute of Mathematics and Scientific Computing, University of Graz, Graz, Austria

    Martin Holler

  3. Department of Mathematics and Natural Sciences, Hochschule Darmstadt, Darmstadt, Germany

    Andreas Weinmann

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Younes, L., Gris, B., Trouvé, A. (2020). Sub-Riemannian Methods in Shape Analysis. In: Grohs, P., Holler, M., Weinmann, A. (eds) Handbook of Variational Methods for Nonlinear Geometric Data. Springer, Cham. https://doi.org/10.1007/978-3-030-31351-7_17

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