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Sub-Riemannian Methods in Shape Analysis

In: Handbook of Variational Methods for Nonlinear Geometric Data

Author

Listed:
  • Laurent Younes

    (Johns Hopkins University, Department of Applied Mathematics and Statistics)

  • Barbara Gris

    (Sorbonne Université, Laboratoire Jacques-Louis Lions)

  • Alain Trouvé

    (ENS Paris-Saclay, Centre de Mathématiques et leurs Applications)

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.

Suggested Citation

  • Laurent Younes & Barbara Gris & Alain Trouvé, 2020. "Sub-Riemannian Methods in Shape Analysis," Springer Books, in: Philipp Grohs & Martin Holler & Andreas Weinmann (ed.), Handbook of Variational Methods for Nonlinear Geometric Data, chapter 0, pages 463-495, Springer.
    • Handle: RePEc:spr:sprchp:978-3-030-31351-7_17
    DOI: 10.1007/978-3-030-31351-7_17
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