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Reparameterization Invariant Metric on the Space of Curves

  • Alice Le BrigantEmail author
  • Marc Arnaudon
  • Frédéric Barbaresco
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9389)

Abstract

This paper focuses on the study of open curves in a manifold M, and its aim is to define a reparameterization invariant distance on the space of such paths. We use the square root velocity function (SRVF) introduced by Srivastava et al. in [11] to define a reparameterization invariant metric on the space of immersions \(\mathcal {M}=\text {Imm}([0,1],M)\) by pullback of a metric on the tangent bundle \(\text {T}\mathcal {M}\) derived from the Sasaki metric. We observe that such a natural choice of Riemannian metric on \(\text {T}\mathcal {M}\) induces a first-order Sobolev metric on \(\mathcal {M}\) with an extra term involving the origins, and leads to a distance which takes into account the distance between the origins and the distance between the image curves by the SRVF parallel transported to a same vector space, with an added curvature term. This provides a generalized theoretical SRV framework for curves lying in a general manifold M.

Keywords

Vector Field Tangent Bundle Tangent Plane Geodesic Distance Parallel Transport 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

This research was supported by Thales Air Systems and the french MoD DGA.

References

  1. 1.
    Bauer, M., Bruveris, M., Marsland, S., Michor, P.W.: Constructing reparameterization invariant metrics on spaces of plane curves. Differ. Geom. Appl. 34, 139–165 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bauer, M., Bruveris, M., Michor, P.: Why use Sobolev metrics on the space of curves. In: Riemannian Computing in Computer Vision. Springer (to appear). arXiv:1502.03229
  3. 3.
    Bauer, M., Harms, P., Michor, P.: Sobolev metrics on shape space of surfaces. J. Geom. Mech. 3(4), 389–438 (2011)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Laga, H., Kurtek, S., Srivastava, A., Miklavcic, S.J.: Landmark-free statistical analysis of the shape of plant leaves. J. Theoret. Biol. 363, 41–52 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Le Brigant, A., Arnaudon, M., Barbaresco, F.: Reparameterization invariant distance on the space of curves in the hyperbolic plane. AIP Conf. Proc. 1641, 504 (2015)CrossRefGoogle Scholar
  6. 6.
    Michor, P., Mumford, D.: Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms. Documenta Math. 10, 217–245 (2005)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Michor, P., Mumford, D.: Riemannian geometries on spaces of plane curves. J. Eur. Math. Soc. (JEMS) 8, 1–48 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Michor, P., Mumford, D.: An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach. Appl. Comput. Harmonic Anal. 23, 74–113 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Sasaki, S.: On the differential geometry of tangent bundles of Riemannian manifolds. Tohoku Math. J. 10, 338–354 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Sasaki, S.: On the differential geometry of tangent bundles of Riemannian manifolds II. Tohoku Math. J. 14, 146–155 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Srivastava, A., Klassen, E., Joshi, S.H., Jermyn, I.H.: Shape analysis of elastic curves in Euclidean spaces. IEEE T. Pattern Anal. 33(7), 1415–1428 (2011)CrossRefGoogle Scholar
  12. 12.
    Su, J., Kurtek, S., Klassen, E., Srivastava, A.: Statistical analysis of trajectories on Riemannian manifolds: bird migration, hurricane tracking and video surveillance. Ann. Appl. Stat. 8(1), 530–552 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Younes, L., Michor, P., Shah, J., Mumford, D.: A metric on shape space with explicit geodesics. Rend. Lincei Mat. Appl. 9, 25–57 (2008)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Zhang, Z., Su, J., Klassen, E., Le, H., Srivastava, A.: Video-based action recognition using rate-invariant analysis of covariance trajectories (2015). arXiv:1503.06699 [cs.CV]

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Alice Le Brigant
    • 1
    • 2
    Email author
  • Marc Arnaudon
    • 1
  • Frédéric Barbaresco
    • 2
  1. 1.Institut Mathématique de Bordeaux, UMR 5251Université de Bordeaux and CNRSTalenceFrance
  2. 2.Thales Air System, Surface Radar Domain, Technical DirectorateLimoursFrance

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