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On the consistency of Fréchet means in deformable models for curve and image analysis. (English) Zbl 1274.62276

Summary: A new class of statistical deformable models is introduced to study high-dimensional curves or images. In addition to the standard measurement error term, these deformable models include an extra error term modeling the individual variations in intensity around a mean pattern. It is shown that an appropriate tool for statistical inference in such models is the notion of sample Fréchet means, which leads to estimators of the deformation parameters and the mean pattern. The main contribution of this paper is to study how the behavior of these estimators depends on the number \(n\) of design points and the number \(J\) of observed curves (or images). Numerical experiments are given to illustrate the finite sample performances of the procedure.

MSC:

62G08 Nonparametric regression and quantile regression
62M40 Random fields; image analysis
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[1] S. Allassonière, 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 Afsari. Riemannian, l p center of mass: existence, uniqueness, and convexity. Proceedings of the American Mathematical Society , 139(2):655-673, 2011. · Zbl 1220.53040 · doi:10.1090/S0002-9939-2010-10541-5
[3] S. Allassonière, 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. · Zbl 1202.62049 · doi:10.1214/10-AOS800
[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. · Zbl 1490.68272 · doi:10.1007/s10851-009-0146-1
[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. · Zbl 1175.62067 · doi:10.1137/070691231
[7] J. Bigot. Landmark-based registration of curves via the continuous wavelet transform., Journal of Computational and Graphical Statistics , 15(3):542-564, 2006. · doi:10.1198/106186006X133023
[8] J. Bigot, J.-M. Loubes, and M. Vimond. Semiparametric estimation of shifts on compact lie groups for image registration., Probability Theory and Related Fields , to be published, 2010. · Zbl 1318.62076 · doi:10.1007/s00440-010-0327-2
[9] L. Birgé and P. Massart. Minimum contrast estimators on sieves: exponential bounds and rates of convergence., Bernoulli , 4(3):329-375, 1998. · Zbl 0954.62033 · doi:10.2307/3318720
[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. · Zbl 1020.62026 · doi:10.1214/aos/1046294456
[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. · Zbl 1072.62033 · doi:10.1214/009053605000000093
[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. · Zbl 0901.62072
[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. · Zbl 0035.20802
[14] T. Gasser and A. Kneip. Statistical tools to analyze data representing a sample of curves., Annals of Statistics , 20(3) :1266-1305, 1992. · Zbl 0785.62042 · doi:10.1214/aos/1176348769
[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. · Zbl 0830.62035 · doi:10.2307/3318681
[16] F. Gamboa, J.-M. Loubes, and E. Maza. Semi-parametric estimation of shifts., Electron. J. Stat. , 1:616-640, 2007. · Zbl 1141.62313 · doi:10.1214/07-EJS026
[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. · Zbl 1040.62054 · doi:10.1111/1467-9868.00295
[18] U. Grenander and M. Miller., Pattern Theory: From Representation to Inference . Oxford Univ. Press, Oxford, 2007. · Zbl 1259.62089
[19] C. Goodall. Procrustes methods in the statistical analysis of shape., J. Roy. Statist. Soc. Ser. B , 53(2):285-339, 1991. · Zbl 0800.62346
[20] U. Grenander., General pattern theory - A mathematical study of regular structures . Clarendon Press, Oxford, 1993. · Zbl 0827.68098
[21] S. Helgason., Differential geometry, Lie groups, and symmetric spaces , volume 34 of Graduate Studies in Mathematics . American Mathematical Society, Providence, RI, 2001. · Zbl 0993.53002
[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. · Zbl 1180.62087
[24] R.A. Horn and C.R. Johnson., Matrix analysis . Cambridge University Press, Cambridge, 1990. · Zbl 0704.15002
[25] S Huckemann. Inference on 3d procrustes means: Tree bole growth, rank deficient diffusion tensors and perturbation models., Scand. J. Statist. , To appear, 2010. · Zbl 1246.62120
[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. · Zbl 1216.62084 · doi:10.1214/10-AOS862
[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. · Zbl 0354.57005 · doi:10.1002/cpa.3160300502
[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. · Zbl 0940.60006
[30] D.G. Kendall. Shape manifolds, procrustean metrics, and complex projective spaces., Bull. London Math Soc. , 16:81-121, 1984. · Zbl 0579.62100 · doi:10.1112/blms/16.2.81
[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. · Zbl 0675.58042 · doi:10.1112/plms/s3-61.2.371
[32] A. Kneip and T. Gasser. Convergence and consistency results for self-modelling regression., Annals of Statistics , 16:82-112, 1988. · Zbl 0725.62060 · doi:10.1214/aos/1176350692
[33] J. T. Kent and K. V. Mardia. Consistency of Procrustes estimators., J. Roy. Statist. Soc. Ser. B , 59(1):281-290, 1997. · Zbl 0890.62041 · doi:10.1111/1467-9868.00069
[34] H. Le. On the consitency of procrustean mean shapes., Advances in Applied Probability , 30:53-63, 1998. · Zbl 0906.60007 · doi:10.1239/aap/1035227991
[35] H. Le and A. Kume. The fréchet mean shape and the shape of the means., Advances in Applied Probability , 32:101-113, 2000. · Zbl 0961.60020 · doi:10.1239/aap/1013540025
[36] X. Liu and H.G. Muller. Functional convex averaging and synchronization for time-warped random curves., Journal of the American Statistical Association , 99(467):687-699, 2004. · Zbl 1117.62392 · doi:10.1198/016214504000000999
[37] P. Massart., Concentration inequalities and model selection , volume 1896 of Lecture Notes in Mathematics . Springer, Berlin, 2007. · Zbl 1170.60006 · doi:10.1007/978-3-540-48503-2
[38] M. I. Miller and L. Younes. Group actions, homeomorphisms, and matching: A general framework., International Journal of Computer Vision , 41:61-84, 2001. · Zbl 1012.68714 · doi:10.1023/A:1011161132514
[39] J. M. Oller and J. M. Corcuera. Intrinsic analysis of statistical estimation., Ann. Statist. , 23(5) :1562-1581, 1995. · Zbl 0843.62027 · doi:10.1214/aos/1176324312
[40] X. Pennec. Intrinsic statistics on riemannian manifolds: Basic tools for geometric measurements., J. Math. Imaging Vis. , 25:127-154, July 2006. · Zbl 1478.94072 · doi:10.1007/s10851-006-6228-4
[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. · Zbl 0979.62018 · doi:10.1111/1467-9868.00283
[43] T. Trigano, U. Isserles, and Y. Ritov. Semiparametric curve alignment and shift density estimation for biological data., Preprint , 2010. · Zbl 1392.94491
[44] A. Trouvé and L. Younes. Metamorphoses through lie group action., Foundations of Computational Mathematics , 5(2):173-198, 2005. · Zbl 1099.68116 · doi:10.1007/s10208-004-0128-z
[45] A. W. van der Vaart., Asymptotic statistics , volume 3 of Cambridge Series in Statistical and Probabilistic Mathematics . Cambridge University Press, Cambridge, 1998. · Zbl 0910.62001 · doi:10.1017/CBO9780511802256
[46] M. Vimond. Efficient estimation for a subclass of shape invariant models., Annals of statistics , 38(3) :1885-1912, 2010. · Zbl 1189.62057 · doi:10.1214/07-AOS566
[47] K. Wang and T. Gasser. Alignment of curves by dynamic time warping., Annals of Statistics , 25(3) :1251-1276, 1997. · Zbl 0898.62051 · doi:10.1214/aos/1069362747
[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. · Zbl 0413.60024
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