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Intrinsic Polynomials for Regression on Riemannian Manifolds

Authors:

P. Thomas Fletcher,

Sarang JoshiAuthors Info & Claims

Journal of Mathematical Imaging and Vision, Volume 50, Issue 1-2

Pages 32 - 52

https://doi.org/10.1007/s10851-013-0489-5

Published: 01 September 2014 Publication History

Abstract

We develop a framework for polynomial regression on Riemannian manifolds. Unlike recently developed spline models on Riemannian manifolds, Riemannian polynomials offer the ability to model parametric polynomials of all integer orders, odd and even. An intrinsic adjoint method is employed to compute variations of the matching functional, and polynomial regression is accomplished using a gradient-based optimization scheme. We apply our polynomial regression framework in the context of shape analysis in Kendall shape space as well as in diffeomorphic landmark space. Our algorithm is shown to be particularly convenient in Riemannian manifolds with additional symmetry, such as Lie groups and homogeneous spaces with right or left invariant metrics. As a particularly important example, we also apply polynomial regression to time-series imaging data using a right invariant Sobolev metric on the diffeomorphism group. The results show that Riemannian polynomials provide a practical model for parametric curve regression, while offering increased flexibility over geodesics.

References

[1]

Bandulasiri, A., Gunathilaka, A., Patrangenaru, V., Ruymgaart, F., Thompson, H.: Nonparametric shape analysis methods in glaucoma detection. I. J. Stat. Sci. 9, 135---149 (2009)

[2]

Bookstein, F.L.: Morphometric Tools for Landmark Data: Geometry and Biology. Cambridge Univ. Press, Cambridge (1991)

[3]

Bou-Rabee, N.: Hamilton-Pontryagin integrators on Lie groups. Ph.D. thesis, California Institute of Technology (2007)

[4]

Bruveris, M., Gay-Balmaz, F., Holm, D., Ratiu, T.: The momentum map representation of images. J. Nonlinear Sci. 21(1), 115---150 (2011)

[5]

Burnham, K., Anderson, D.: Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach. Springer, New York (2002)

[6]

Cates, J., Fletcher, P.T., Styner, M., Shenton, M., Whitaker, R.: Shape modeling and analysis with entropy-based particle systems. In: Proceedings of Information Processing in Medical Imaging (IPMI) (2007)

[7]

Cheeger, J., Ebin, D.G.: Comparison Theorems in Riemannian Geometry, vol. 365. AMS Bookstore, Providence (1975)

[8]

Davis, B.C., Fletcher, P.T., Bullitt, E., Joshi, S.C.: Population shape regression from random design data. Int. J. Comput. Vis. 90(2), 255---266 (2010)

Digital Library

[9]

do Carmo, M.P.: Riemannian Geometry, 1st edn. Birkhäuser, Boston (1992)

[10]

Driesen, N, Raz, N: The influence of sex, age, and handedness on corpus callosum morphology: A meta-analysis. Psychobiology (1995).

[11]

Dryden, I.L., Kume, A., Le, H., Wood, A.T.: A multi-dimensional scaling approach to shape analysis. Biometrika 95(4), 779---798 (2008)

[12]

Fletcher, PT: Geodesic regression and the theory of least squares on Riemannian manifolds. Int. J. Comput. Vis. (2012).

[13]

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. Imaging 23(8), 995---1005 (2004)

[14]

Giambò, R., Giannoni, F., Piccione, P.: An analytical theory for Riemannian cubic polynomials. IMA J. Math. Control Inf. 19(4), 445---460 (2002)

[15]

Hinkle, J., Muralidharan, P., Fletcher, P.T., Joshi, S.C.: Polynomial regression on Riemannian manifolds. In: ECCV, Florence, Italy, vol. 3, pp. 1---14 (2012)

[16]

Holm, D.D., Marsden, J.E., Ratiu, T.S.: The Euler-Poincaré equations and semidirect products with applications to continuum theories. Adv. Math. 137, 1---81 (1998)

[17]

Huckemann, S., Hotz, T., Munk, A.: Intrinsic shape analysis: geodesic principal component analysis for Riemannian manifolds modulo Lie group actions. Discussion paper with rejoinder. Stat. Sin. 20, 1---100 (2010)

[18]

Jupp, P.E., Kent, J.T.: Fitting smooth paths to spherical data. Appl. Stat. 36(1), 34---46 (1987)

[19]

Kendall, D.G.: Shape manifolds, procrustean metrics, and complex projective spaces. Bull. Lond. Math. Soc. 16(2), 81---121 (1984)

[20]

Kendall, D.G.: A survey of the statistical theory of shape. Stat. Sci. 4(2), 87---99 (1989)

[21]

Kenobi, K., Dryden, I.L., Le, H.: Shape curves and geodesic modelling. Biometrika 97(3), 567---584 (2010)

[22]

Kume, A., Dryden, I.L., Le, H.: Shape-space smoothing splines for planar landmark data. Biometrika 94(3), 513---528 (2007).

[23]

Le, H., Kendall, D.G.: The Riemannian structure of Euclidean shape spaces: a novel environment for statistics. Ann. Stat. 21(3), 1225---1271 (1993)

[24]

Silva Leite, F, Krakowski, K: Covariant differentiation under rolling maps. Departamento de Matemática, Universidade of Coimbra, Portugal (2008), No. 08-22, 1---8

[25]

Lewis, A., Murray, R.: Configuration controllability of simple mechanical control systems. SIAM J. Control Optim. 35(3), 766---790 (1997)

Digital Library

[26]

Machado, L., Leite, F.S., Krakowski, K.: Higher-order smoothing splines versus least squares problems on Riemannian manifolds. J. Dyn. Control Syst. 16(1), 121---148 (2010)

Digital Library

[27]

Marcus, D., Wang, T., Parker, J., Csernansky, J., Morris, J., Buckner, R.: Open access series of imaging studies (OASIS): cross-sectional MRI data in young, middle aged, nondemented, and demented older adults. J. Cogn. Neurosci. 19(9), 1498---1507 (2007)

Digital Library

[28]

Marsden, J., Ratiu, T.: Introduction to Mechanics and Symmetry: a Basic Exposition of Classical Mechanical Systems, vol. 17. Springer, Berlin (1999)

[29]

Micheli, M., Michor, P., Mumford, D.: Sectional curvature in terms of the cometric, with applications to the Riemannian manifolds of landmarks. SIAM J. Imaging Sci. 5(1), 394---433 (2012)

Digital Library

[30]

Miller, M.I., Trouvé, A., Younes, L.: Geodesic shooting for computational anatomy. J. Math. Imaging Vis. 24(2), 209---228 (2006).

Digital Library

[31]

Niethammer, M., Huang, Y., Vialard, F.X.: Geodesic regression for image time-series. In: Proceedings of Medical Image Computing and Computer Assisted Intervention (MICCAI) (2011)

[32]

Noakes, L., Heinzinger, G., Paden, B.: Cubic splines on curved surfaces. IMA J. Math. Control Inf. 6, 465---473 (1989)

[33]

O'Neill, B.: The fundamental equations of a submersion. Mich. Math. J. 13(4), 459---469 (1966)

[34]

Shi, X., Styner, M., Lieberman, J., Ibrahim, J.G., Lin, W., Zhu, H.: Intrinsic regression models for manifold-valued data. In: Medical Image Computing and Computer-Assisted Intervention--MICCAI 2009, pp. 192---199. Springer, Berlin (2009)

[35]

Singh, N., Wang, A., Sankaranarayanan, P., Fletcher, P., Joshi, S.: Genetic, structural and functional imaging biomarkers for early detection of conversion from MCI to AD. In: Proceedings of Medical Image Computing and Computer Assisted Intervention (MICCAI), pp. 132---140. Springer, Berlin (2012)

[36]

Turaga, P., Veeraraghavan, A., Srivastava, A., Chellappa, R.: Statistical computations on Grassmann and Stiefel manifolds for image and video-based recognition. IEEE Trans. Pattern Anal. Mach. Intell. 33(11), 2273---2286 (2011)

Digital Library

[37]

Vaillant, M., Glaunes, J.: Surface matching via currents. In: Information Processing in Medical Imaging, pp. 1---5. Springer, Berlin (2005)

[38]

Younes, L., Qiu, A., Winslow, R., Miller, M.: Transport of relational structures in groups of diffeomorphisms. J. Math. Imaging Vis. 32(1), 41---56 (2008)

Digital Library

[39]

Yushkevich, P.A., Piven, J., Cody Hazlett, H., Gimpel Smith, R., Ho, S., Gee, J.C., Gerig, G.: User-guided 3D active contour segmentation of anatomical structures: significantly improved efficiency and reliability. NeuroImage 31(3), 1116---1128 (2006)

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Intrinsic Polynomials for Regression on Riemannian Manifolds
1. Computing methodologies
  1. Artificial intelligence
  2. Computer graphics
    1. Shape modeling
2. Theory of computation

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Published In

cover image Journal of Mathematical Imaging and Vision

Journal of Mathematical Imaging and Vision Volume 50, Issue 1-2

September 2014

177 pages

ISSN:0924-9907

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Copyright © Copyright © 2014 Springer Science+Business Media New York.

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Kluwer Academic Publishers

United States

Publication History

Published: 01 September 2014

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Hanik MNava-Yazdani Evon Tycowicz C(2024)De Casteljau's algorithm in geometric data analysisComputer Aided Geometric Design10.1016/j.cagd.2024.102288110:COnline publication date: 1-May-2024
https://dl.acm.org/doi/10.1016/j.cagd.2024.102288
Nava-Yazdani EAmbellan FHanik Mvon Tycowicz C(2023)Sasaki metric for spline models of manifold-valued trajectories▪Computer Aided Geometric Design10.1016/j.cagd.2023.102220104:COnline publication date: 1-Jul-2023
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Chau Jvon Sachs R(2022)Time-varying spectral matrix estimation via intrinsic wavelet regression for surfaces of Hermitian positive definite matricesComputational Statistics & Data Analysis10.1016/j.csda.2022.107477174:COnline publication date: 1-Oct-2022
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Adouani ISamir C(2021)An Optimization Method for Accurate Nonparametric Regressions on Stiefel ManifoldsMachine Learning, Optimization, and Data Science10.1007/978-3-030-95470-3_25(325-337)Online publication date: 4-Oct-2021
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