Shape Analysis on Homogeneous Spaces: A Generalised SRVT Framework

  • Elena Celledoni
  • Sølve Eidnes
  • Alexander Schmeding
Conference paper
Part of the Abel Symposia book series (ABEL, volume 13)


Shape analysis is ubiquitous in problems of pattern and object recognition and has developed considerably in the last decade. The use of shapes is natural in applications where one wants to compare curves independently of their parametrisation. One computationally efficient approach to shape analysis is based on the Square Root Velocity Transform (SRVT). In this paper we propose a generalised SRVT framework for shapes on homogeneous manifolds. The method opens up for a variety of possibilities based on different choices of Lie group action and giving rise to different Riemannian metrics.



This work was supported by the Norwegian Research Council, and by the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie, grant agreement No. 691070.


  1. 1.
    Absil, P.-A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton (2007)zbMATHGoogle Scholar
  2. 2.
    Amiri, H., Schmeding, A.: A Differentiable Monoid of Smooth Maps on Lie Groupoids (2017). arXiv:1706.04816v1Google Scholar
  3. 3.
    Bastiani, A.: Applications différentiables et variétés différentiables de dimension infinie. J. Anal. Math. 13, 1–114 (1964)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bauer, M., Bruveris, M., Harms, P., Michor, P.W.: Vanishing geodesic distance for the Riemannian metric with geodesic equation the KdV-equation. Ann. Global Anal. Geom. 41(4), 461–472 (2012)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bauer, M., Bruveris, M., Michor, P.W.: Overview of the geometries of shape spaces and diffeomorphism groups. J. Math. Imaging Vis. 50(1), 1–38 (2014)MathSciNetzbMATHGoogle Scholar
  6. 6.
    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)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bauer, M., Eslitzbichler, M., Grasmair, M.: Landmark-guided elastic shape analysis of human character motions. Inverse Prob. Imaging 11(4), 601–621 (2015). MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bruveris, M.: Optimal reparametrizations in the square root velocity framework. SIAM J. Math. Anal. 48(6), 4335–4354 (2016)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Celledoni, E., Eslitzbichler, M., Schmeding, A.: Shape analysis on Lie groups with applications in computer animation. J. Geom. Mech. 8(3), 273–304 (2016)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Celledoni, E., Owren, B.: On the implementation of Lie group methods on the Stiefel manifold. Numer. Algorithm. 32(2–4), 163–183 (2003)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Gallot, S., Hulin, D., Lafontaine, J.: Riemannian Geometry. Universitext, 3rd edn. Springer, Berlin (2004)CrossRefGoogle Scholar
  12. 12.
    Glöckner, H.: Fundamentals of Submersions and Immersions Between Infinite-Dimensional Manifolds (2015). arXiv:1502.05795v3 [math]Google Scholar
  13. 13.
    Glöckner, H.: Regularity Properties of Infinite-Dimensional Lie Groups, and Semiregularity (2015). arXiv:1208.0715v3Google Scholar
  14. 14.
    Huper, K., Leite, F.: On the geometry of rolling and interpolation curves on S n, SOn, and Grassmann manifolds. J. Dyn. Control. Syst. 13, 467–502 (2007)Google Scholar
  15. 15.
    Kriegl, A., Michor, P.W.: The convenient setting of global analysis. In: Mathematical Surveys and Monographs, vol. 53. American Mathematical Society, ProvidenceGoogle Scholar
  16. 16.
    Kobayashi, S., Nomizu, K. Foundations of Differential Geometry, vol. II. Interscience Tracts in Pure and Applied Mathematics, no. 15, vol. II. Interscience Publishers John Wiley, New York/London/Sydney (1969)Google Scholar
  17. 17.
    Knapp, A.W.: Lie groups beyond an introduction. In: Progress in Mathematics, vol. 140, 2nd edn. Birkhäuser, Boston (2002)Google Scholar
  18. 18.
    Le Brigant, A.: Computing distances and geodesics between manifold-valued curves in the SRV framework. J. Geom. Mech. 9(2) (2017). MathSciNetCrossRefGoogle Scholar
  19. 19.
    Michor, P.W.: Manifolds of Differentiable Mappings. In: Shiva Mathematics Series, vol. 3. Shiva Publishing Ltd., Nantwich (1980)Google Scholar
  20. 20.
    Munthe-Kaas, H., Verdier, O.: Integrators on homogeneous spaces: isotropy choice and connections. Found. Comput. Math. 16(4), 899–939 (2016)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Michor, P.W., Mumford, D.: Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms. Doc. Math. 10, 217–245 (2005)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Michor, P.W., Mumford, D.: Riemannian geometries on spaces of plane curves. J. Eur. Math. Soc. (JEMS) 8(1), 1–48 (2006)Google Scholar
  23. 23.
    Ortega, J.-P., Ratiu, T.S.: Momentum maps and Hamiltonian reduction. In: Progress in Mathematics, vol. 222. Birkhäuser Boston, Inc., Boston (2004)Google Scholar
  24. 24.
    Sharpe, R.W.: Differential geometry. In: Graduate Texts in Mathematics, vol. 166. Springer, New York (1997). Cartan’s generalization of Klein’s Erlangen program, With a foreword by S. S. ChernGoogle Scholar
  25. 25.
    Su, Z., Klassen, E., Bauer, M.: Comparing Curves in Homogeneous Spaces (2017). 1712.04586v1
  26. 26.
    Su, Z., Klassen, E., Bauer, M.: The square root velocity framework for curves in a homogeneous space. In: 2017 IEEE Conference on Computer Vision and Pattern Recognition Workshops (CVPRW), pp. 680–689. IEEE (2017)Google Scholar
  27. 27.
    Srivastava, A., Klassen, E., Joshi, S., Jermyn, I.: Shape analysis of elastic curves in euclidean spaces. IEEE Trans. Pattern Anal. Mach. Intell. 33, 1415–1428 (2011)CrossRefGoogle Scholar
  28. 28.
    Sebastian, T.B., Klein, P.N., Kimia, B.B.: On aligning curves. IEEE Trans. Pattern Anal. Mach. Intell. 25(1), 116–125 (2003)CrossRefGoogle Scholar
  29. 29.
    Su, J., Kurtek, S., Klassen, E., Srivastava, A.: Statistical analysis of trajectories on Riemmannian manifolds: bird migration, hurricane tracking and video surveillance. Ann. Appl. Stat. 8(2), 530–552 (2014)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Elena Celledoni
    • 1
  • Sølve Eidnes
    • 1
  • Alexander Schmeding
    • 1
  1. 1.NTNU TrondheimInstitutt for matematiske fagTrondheimNorway

Personalised recommendations