Abstract
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.
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Dryden, I.L., Mardia, K.V.: Statistical Shape Analysis. Wiley (1998)
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)
Jupp, P.E., Kent, J.T.: Fitting smooth paths to spherical data. Appl. Statist. 36, 34–46 (1987)
Fletcher, P.T.: Geodesic regression on Riemannian manifolds. In: International Workshop on Mathematical Foundations of Computational Anatomy, MFCA (2011)
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)
Arnol’d, V.I.: Mathematical Methods of Classical Mechanics, 2nd edn. Springer (1989)
Cootes, T.F., Twining, C.J., Taylor, C.J.: Diffeomorphic statistical shape models. In: BMVC (2004)
Vaillant, M., Glaunés, J.: Surface matching via currents. In: IPMI (2005)
Miller, M.I., Trouvé, A., Younes, L.: Geodesic shooting for computational anatomy. Journal of Mathematical Imaging and Vision 24, 209–228 (2006)
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)
Bookstein, F.L.: Morphometric Tools for Landmark Data: Geometry and Biology. Cambridge Univ. Press (1991)
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)
do Carmo, M.P.: Riemannian Geometry, 1st edn. Birkhäuser, Boston (1992)
Leite, F.S., Krakowski, K.A.: Covariant differentiation under rolling maps. Centro de Matemática da Universidade de Coimbra (2008) (preprint)
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)
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)
Kendall, D.G.: A survey of the statistical theory of shape. Statistical Science 4, 87–99 (1989)
Le, H., Kendall, D.G.: The Riemannian structure of Euclidean shape spaces: A novel environment for statistics. Ann. Statist. 21, 1225–1271 (1993)
O’Neill, B.: The fundamental equations of a submersion. Michigan Math J. 13, 459–469 (1966)
Cates, J., Fletcher, P.T., Styner, M., Shenton, M., Whitaker, R.: Shape modeling and analysis with entropy-based particle systems. In: IPMI (2007)
Davis, B.C., Fletcher, P.T., Bullitt, E., Joshi, S.: Population shape regression from random design data. In: ICCV (2007)
Noakes, L., Heinzinger, G., Paden, B.: Cubic splines on curved surfaces. IMA J. Math. Control Inform. 6, 465–473 (1989)
Giambò, R., Giannoni, F., Piccione, P.: An analytical theory for Riemannian cubic polynomials. IMA J. Math. Control Inform. 19, 445–460 (2002)
Machado, L., Leite, F.S.: Fitting smooth paths on Riemannian manifolds. Int. J. App. Math. Stat. 4, 25–53 (2006)
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)
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)
Moussa, M.A.A., Cheema, M.Y.: Non-parametric regression in curve fitting. The Statistician 41, 209–225 (1992)
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Hinkle, J., Muralidharan, P., Fletcher, P.T., Joshi, S. (2012). Polynomial Regression on Riemannian Manifolds. In: Fitzgibbon, A., Lazebnik, S., Perona, P., Sato, Y., Schmid, C. (eds) Computer Vision – ECCV 2012. ECCV 2012. Lecture Notes in Computer Science, vol 7574. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33712-3_1
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DOI: https://doi.org/10.1007/978-3-642-33712-3_1
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