Intrinsic polynomials for regression on riemannian manifolds

Journal of Mathematical Imaging and Vision

Volumn 50, Issue 1, 2014, Pages 32-52

  Hinkle, Jacob ^a   Fletcher, P Thomas ^a   Joshi, Sarang ^a  

^a UNIVERSITY OF UTAH   (United States)

Author keywords

Lie groups; Polynomial; Regression; Riemannian geometry; Rolling maps; Shape space

Indexed keywords

GEOMETRY; POLYNOMIALS; REGRESSION ANALYSIS;

GRADIENT-BASED OPTIMIZATION; INCREASED FLEXIBILITY; POLYNOMIAL REGRESSION; REGRESSION; RIEMANNIAN GEOMETRY; RIEMANNIAN MANIFOLD; ROLLING MAPS; SHAPE SPACE;

LIE GROUPS;

EID: 85028148412     PISSN: 09249907     EISSN: 15737683     Source Type: Journal    
DOI: 10.1007/s10851-013-0489-5     Document Type: Article

References (39)

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)
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). doi:10.3758/BF03332028
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). doi:10.1007/s11263-012-0591-y
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). doi:10.1093/biomet/asm047
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)
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)
27. Marcus, D., Wang, T., Parker, J., Csernansky, J., Morris, J., Buckner, R.: Open access series of imaging studies (OASIS): crosssectional MRI data in young, middle aged, nondemented, and demented older adults. J. Cogn. Neurosci. 19(9), 1498–1507 (2007)
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)
30. Miller, M. I., Trouvé, A., Younes, L.: Geodesic shooting for computational anatomy. J. Math. Imaging Vis. 24(2), 209–228 (2006). doi:10.1007/s10851-005-3624-0
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 fromMCI 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)
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)
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)