On the consistency of fre ́chet means in deformable models for curve and image analysis

Electronic Journal of Statistics

Volumn 5, Issue , 2011, Pages 1054-1089

  Bigot, Jéŕemie ^a   Charlier, Benjamin ^a  

^a UNIVERSITÉ DE TOULOUSE   (France)

Author keywords

Curve registration; Deformable models; Fr chet mean; Geometric variability; High dimensional data; Image warping; Mean pattern estimation; Shape analysis

Indexed keywords

EID: 84859863105     PISSN: 19357524     EISSN: None     Source Type: Journal    
DOI: 10.1214/11-EJS633     Document Type: Article

References (48)

1. S. Allassoniére, Y. Amit, and A. Trouvé. Toward a coherent statistical framework for dense deformable template estimation. Journal of the Statistical Royal Society (B), 69:3-29, 2007.
2. Bijan A. Riemannian lp center of mass: existence, uniqueness, and convexity. Proceedings of the American Mathematical Society, 139(2):655-673, 2011.
3. S. Allassoniére, E. Kuhn, and A. Trouvé. Bayesian deformable models building via stochastic approximation algorithm: a convergence study. Bernoulli, To be published, 2009.
4. J. Bigot and S. Gadat. A deconvolution approach to estimation of a common shape in a shifted curves model. Annals of statistics, to be published, 2010.
5. J. Bigot, S. Gadat, and J.-M. Loubes. Statistical M-estimation and consistency in large deformable models for image warping. J. Math. Imaging Vision, 34(3):270-290, 2009.
6. J. Bigot, F. Gamboa, and M. Vimond. Estimation of translation, rotation and scaling between noisy images using the fourier mellin transform. SIAM Journal on Imaging Sciences, 2(2):614-645, 2009.
7. J. Bigot. Landmark-based registration of curves via the continuous wavelet transform. Journal of Computational and Graphical Statistics, 15(3):542-564, 2006.
8. J. Bigot, J.-M. Loubes, and M. Vimond. Semiparametric estimation of shifts on compact lie groups for image registration. Prob-ability Theory and Related Fields, to be published, 2010.
9. L. Birgé and P. Massart. Minimum contrast estimators on sieves: exponential bounds and rates of convergence. Bernoulli, 4(3):329-375, 1998.
10. R. Bhattacharya and V. Patrangenaru. Large sample theory of intrinsic and extrinsic sample means on manifolds (i). Annals of statistics, 31(1):1-29, 2003.
11. R. Bhattacharya and V. Patrangenaru. Large sample theory of intrinsic and extrinsic sample means on manifolds (ii). Annals of statistics, 33:1225-1259, 2005.
12. I. L. Dryden and K. V. Mardia. Statistical shape analysis. Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley & Sons Ltd., Chichester, 1998.
13. M. Fréchet. Les éléments aléatoires de nature quelconque dans un espace distancié. Ann. Inst. H.Poincaré, Sect. B, Prob. et Stat., 10:235-310, 1948.
14. T. Gasser and A. Kneip. Statistical tools to analyze data representing a sample of curves. Annals of Statistics, 20(3):1266-1305, 1992.
15. R. D. Gill and B. Y. Levit. Applications of the Van Trees inequality: a Bayesian Cramér-Rao bound. Bernoulli, 1(1-2):59-79, 1995.
16. F. Gamboa, J.-M. Loubes, and E. Maza. Semi-parametric estimation of shifts. Electron. J. Stat., 1:616-640, 2007.
17. C. A. Glasbey and K. V. Mardia. A penalized likelihood approach to image warping. J. R. Stat. Soc. Ser. B Stat. Methodol., 63(3):465-514, 2001.
18. U. Grenander and M. Miller. Pattern Theory: From Represen-tation to Inference. Oxford Univ. Press, Oxford, 2007.
19. C. Goodall. Procrustes methods in the statistical analysis of shape. J. Roy. Statist. Soc. Ser. B, 53(2):285-339, 1991.
20. U. Grenander. General pattern theory-A mathematical study of regular structures. Clarendon Press, Oxford, 1993.
21. S. Helgason. Differential geometry, Lie groups, and symmetric spaces, volume 34 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2001.
22. S. Huckemann, T. Hotz, and A. Munk. Intrinsic manova for riemannian manifolds with an application to kendalls spaces of planar shapes. IEEE Trans. Patt. Anal. Mach. Intell. Special Section on Shape Analysis and its Applications in Image Understanding, 32(4):593-603, 2010.
23. S. Huckemann, T. Hotz, and A. Munk. Intrinsic shape analysis: Geodesic principal component analysis for riemannian manifolds modulo lie group actions. discussion paper with rejoinder. Statistica Sinica, 20:1-100, 2010.
24. R.A. Horn and C.R. Johnson. Matrix analysis. Cambridge University Press, Cambridge, 1990.
25. S Huckemann. Inference on 3d procrustes means: Tree bole growth, rank deficient diffusion tensors and perturbation models. Scand. J. Statist., To appear, 2010.
26. S Huckemann. Intrinsic inference on the mean geodesic of planar shapes and tree discrimination by leaf growth. Ann. Statist., 39(2):1098-1124, 2011.
27. S. Joshi, B. Davis, B. Jomier, and Guido G. Unbiased diffeomorphic atlas construction for computational anatomy. Neuroimage, 23:151-160, 2004.
28. H. Karcher. Riemannian center of mass and mollifier smoothing. Comm. Pure Appl. Math., 30(5):509-541, 1977.
29. D. G. Kendall, D. Barden, T. K. Carne, and H. Le. Shape and shape theory. Wiley Series in Probability and Statistics. John Wiley & Sons Ltd., Chichester, 1999.
30. D.G. Kendall. Shape manifolds, procrustean metrics, and complex projective spaces. Bull. London Math Soc., 16:81-121, 1984.
31. W.S. Kendall. Probability, convexity and harmonic maps with small image i: uniqueness and fine existence. Proc. London Math. Soc., 3(61):371-406, 1990.
32. A. Kneip and T. Gasser. Convergence and consistency results for self-modelling regression. Annals of Statistics, 16:82-112, 1988.
33. J. T. Kent and K. V. Mardia. Consistency of Procrustes estimators. J. Roy. Statist. Soc. Ser. B, 59(1):281-290, 1997.
34. H. Le. On the consitency of procrustean mean shapes. Advances in Applied Probability, 30:53-63, 1998.
35. H. Le and A. Kume. The fréchet mean shape and the shape of the means. Advances in Applied Probability, 32:101-113, 2000.
36. X. Liu and H.G. Muller. Functional convex averaging and synchronization for time-warped random curves. Journal of the Amer-ican Statistical Association, 99(467):687-699, 2004.
37. P. Massart. Concentration inequalities and model selection, volume 1896 of Lecture Notes in Mathematics. Springer, Berlin, 2007.
38. M. I. Miller and L. Younes. Group actions, homeomorphisms, and matching: A general framework. International Journal of Com-puter Vision, 41:61-84, 2001.
39. J. M. Oller and J. M. Corcuera. Intrinsic analysis of statistical estimation. Ann. Statist., 23(5):1562-1581, 1995.
40. X. Pennec. Intrinsic statistics on riemannian manifolds: Basic tools for geometric measurements. J. Math. Imaging Vis., 25:127-154, July 2006.
41. J.O. Ramsay and X. Li. Curve registration. Journal of the Royal Statistical Society (B), 63:243-259, 2001.
42. B. B. Rønn. Nonparametric maximum likelihood estimation for shifted curves. J. R. Stat. Soc. Ser. B Stat. Methodol., 63(2):243-259, 2001.
43. T. Trigano, U. Isserles, and Y. Ritov. Semiparametric curve alignment and shift density estimation for biological data. Preprint, 2010.
44. A. Trouvé and L. Younes. Metamorphoses through lie group action. Foundations of Computational Mathematics, 5(2):173-198, 2005.
45. A. W. van der Vaart. Asymptotic statistics, volume 3 of Cam-bridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 1998.
46. M. Vimond. Efficient estimation for a subclass of shape invariant models. Annals of statistics, 38(3):1885-1912, 2010.
47. K. Wang and T. Gasser. Alignment of curves by dynamic time warping. Annals of Statistics, 25(3):1251-1276, 1997.
48. H. Ziezold. On expected figures and a strong law of large numbers for random elements in quasi-metric spaces. In Transactions of the Seventh Prague Conference on Information Theory, Statistical Decision Functions, Random Processes and of the Eighth European Meeting of Statisticians (Tech. Univ. Prague, Prague, 1974), Vol. A, pages 591-602. Reidel, Dordrecht, 1977.