Polynomial Regression on Riemannian Manifolds

  • Jacob Hinkle
  • Prasanna Muralidharan
  • P. Thomas Fletcher
  • Sarang Joshi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7574)


In this paper we develop the theory of parametric polynomial regression in Riemannian manifolds. The theory enables parametric analysis in a wide range of applications, including rigid and non-rigid kinematics as well as shape change of organs due to growth and aging. We show application of Riemannian polynomial regression to shape analysis in Kendall shape space. Results are presented, showing the power of polynomial regression on the classic rat skull growth data of Bookstein and the analysis of the shape changes associated with aging of the corpus callosum from the OASIS Alzheimer’s study.


Riemannian Manifold Corpus Callosum Polynomial Regression Parallel Transport Adjoint Equation 
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.


  1. 1.
    Dryden, I.L., Mardia, K.V.: Statistical Shape Analysis. Wiley (1998)Google Scholar
  2. 2.
    Davis, B.C., Fletcher, P.T., Bullitt, E., Joshi, S.C.: Population shape regression from random design data. Int. J. Comp. Vis. 90, 255–266 (2010)CrossRefGoogle Scholar
  3. 3.
    Jupp, P.E., Kent, J.T.: Fitting smooth paths to spherical data. Appl. Statist. 36, 34–46 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Fletcher, P.T.: Geodesic regression on Riemannian manifolds. In: International Workshop on Mathematical Foundations of Computational Anatomy, MFCA (2011)Google Scholar
  5. 5.
    Niethammer, M., Huang, Y., Vialard, F.-X.: Geodesic Regression for Image Time-Series. In: Fichtinger, G., Martel, A., Peters, T. (eds.) MICCAI 2011, Part II. LNCS, vol. 6892, pp. 655–662. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  6. 6.
    Arnol’d, V.I.: Mathematical Methods of Classical Mechanics, 2nd edn. Springer (1989)Google Scholar
  7. 7.
    Cootes, T.F., Twining, C.J., Taylor, C.J.: Diffeomorphic statistical shape models. In: BMVC (2004)Google Scholar
  8. 8.
    Vaillant, M., Glaunés, J.: Surface matching via currents. In: IPMI (2005)Google Scholar
  9. 9.
    Miller, M.I., Trouvé, A., Younes, L.: Geodesic shooting for computational anatomy. Journal of Mathematical Imaging and Vision 24, 209–228 (2006)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Turaga, P., Veeraraghavan, A., Srivastava, A., Chellappa, R.: Statistical computations on Grassmann and Stiefel manifolds for image and video-based recognition 33, 2273–2286 (2011)Google Scholar
  11. 11.
    Bookstein, F.L.: Morphometric Tools for Landmark Data: Geometry and Biology. Cambridge Univ. Press (1991)Google Scholar
  12. 12.
    Kent, J.T., Mardia, K.V., Morris, R.J., Aykroyd, R.G.: Functional models of growth for landmark data. In: Proceedings in Functional and Spatial Data Analysis, pp. 109–115 (2001)Google Scholar
  13. 13.
    do Carmo, M.P.: Riemannian Geometry, 1st edn. Birkhäuser, Boston (1992)zbMATHGoogle Scholar
  14. 14.
    Leite, F.S., Krakowski, K.A.: Covariant differentiation under rolling maps. Centro de Matemática da Universidade de Coimbra (2008) (preprint)Google Scholar
  15. 15.
    Fletcher, P.T., Liu, C., Pizer, S.M., Joshi, S.C.: Principal geodesic analysis for the study of nonlinear statistics of shape. IEEE Trans. Med. Imag. 23, 995–1005 (2004)CrossRefGoogle Scholar
  16. 16.
    Holm, D.D., Marsden, J.E., Ratiu, T.S.: The Euler-Poincaré equations and semidirect products with applications to continuum theories. Adv. in Math. 137, 1–81 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Kendall, D.G.: A survey of the statistical theory of shape. Statistical Science 4, 87–99 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Le, H., Kendall, D.G.: The Riemannian structure of Euclidean shape spaces: A novel environment for statistics. Ann. Statist. 21, 1225–1271 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    O’Neill, B.: The fundamental equations of a submersion. Michigan Math J. 13, 459–469 (1966)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Cates, J., Fletcher, P.T., Styner, M., Shenton, M., Whitaker, R.: Shape modeling and analysis with entropy-based particle systems. In: IPMI (2007)Google Scholar
  21. 21.
    Davis, B.C., Fletcher, P.T., Bullitt, E., Joshi, S.: Population shape regression from random design data. In: ICCV (2007)Google Scholar
  22. 22.
    Noakes, L., Heinzinger, G., Paden, B.: Cubic splines on curved surfaces. IMA J. Math. Control Inform. 6, 465–473 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Giambò, R., Giannoni, F., Piccione, P.: An analytical theory for Riemannian cubic polynomials. IMA J. Math. Control Inform. 19, 445–460 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Machado, L., Leite, F.S.: Fitting smooth paths on Riemannian manifolds. Int. J. App. Math. Stat. 4, 25–53 (2006)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Samir, C., Absil, P.A., Srivastava, A., Klassen, E.: A gradient-descent method for curve fitting on Riemannian manifolds. Found. Comp. Math. 12, 49–73 (2010)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Su, J., Dryden, I.L., Klassen, E., Le, H., Srivastava, A.: Fitting smoothing splines to time-indexed, noisy points on nonlinear manifolds. Image and Vision Computing 30, 428–442 (2012)CrossRefGoogle Scholar
  27. 27.
    Moussa, M.A.A., Cheema, M.Y.: Non-parametric regression in curve fitting. The Statistician 41, 209–225 (1992)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Jacob Hinkle
    • 1
  • Prasanna Muralidharan
    • 1
  • P. Thomas Fletcher
    • 1
  • Sarang Joshi
    • 1
  1. 1.SCI InstituteUniversity of UtahSalt Lake CityUSA

Personalised recommendations