The estimation of probabilistic deformable template models in computer vision or of probabilistic atlases in Computational Anatomy are core issues in both fields. A first coherent statistical framework where the geometrical variability is modelled as a hidden random variable has been given by [S. Allassonnière et al., J. Roy. Stat. Soc. 69 (2007) 3-29]. They introduce a bayesian approach and mixture of them to estimate deformable template models. A consistent stochastic algorithm has been introduced in [S. Allassonnière et al. (in revision)] to face the problem encountered in [S. Allassonnière et al., J. Roy. Stat. Soc. 69 (2007) 3-29] for the convergence of the estimation algorithm for the one component model in the presence of noise. We propose here to go on in this direction of using some “SAEM-like” algorithm to approximate the MAP estimator in the general bayesian setting of mixture of deformable template models. We also prove the convergence of our algorithm toward a critical point of the penalised likelihood of the observations and illustrate this with handwritten digit images and medical images.
Classification: 60J22, 62F10, 62F15, 62M40
Keywords: stochastic approximations, non rigid-deformable templates, shapes statistics, MAP estimation, bayesian method, mixture models
@article{PS_2010__14__382_0,
author = {Allassonni\`ere, St\'ephanie and Kuhn, Estelle},
title = {Stochastic algorithm for bayesian mixture effect template estimation},
journal = {ESAIM: Probability and Statistics},
publisher = {EDP-Sciences},
volume = {14},
year = {2010},
pages = {382-408},
doi = {10.1051/ps/2009001},
zbl = {pre05873002},
language = {en},
url = {http://www.numdam.org/item/PS_2010__14__382_0}
}
Allassonnière, Stéphanie; Kuhn, Estelle. Stochastic algorithm for bayesian mixture effect template estimation. ESAIM: Probability and Statistics, Volume 14 (2010) , pp. 382-408. doi : 10.1051/ps/2009001. http://www.numdam.org/item/PS_2010__14__382_0/
[1] , and , Toward a coherent statistical framework for dense deformable template estimation. J. Roy. Stat. Soc. 69 (2007) 3-29.
[2] , and , Map estimation of statistical deformable templates via nonlinear mixed effects models: Deterministic and stochastic approaches. In Proc. Int. Workshop on the Mathematical Foundations of Computational Anatomy (MFCA-2008), edited by X. Pennec and S. Joshi (2008).
[3] , and , Construction of Bayesian deformable models via a stochastic approximation algorithm: A convergence study. Bernoulli 16 (2010) 641-678.
[4] , and , Structural image restoration through deformable templates. J. Am. Statist. Assoc. 86 (1989) 376-387.
[5] , and , Stability of stochastic approximation under verifiable conditions. SIAM J. Control Optim. 44 (2005) 283-312 (electronic). | Zbl 1083.62073
[6] , and , Actives appearance models. In 5th Eur. Conf. on Computer Vision, Berlin, Vol. 2, edited by H. Burkhards and B. Neumann. Springer (1998) 484-498.
[7] , and , Convergence of a stochastic approximation version of the EM algorithm. Ann. Statist. 27 (1999) 94-128. | Zbl 0932.62094
[8] , and , Maximum likelihood from incomplete data via the EM algorithm. J. Roy. Statist. Soc. 1 (1977) 1-22. | Zbl 0364.62022
[9] and , Nonparametric density estimation in hidden Markov models. Statist. Inf. Stoch. Process. 5 (2002) 55-64. | Zbl 1005.62032
[10] and , A penalised likelihood approach to image warping. J. Roy. Statist. Soc., Ser. B 63 (2001) 465-492. | Zbl 1040.62054
[11] and , Template estimation form unlabeled point set data and surfaces for computational anatomy. In Proc. Int. Workshop on the Mathematical Foundations of Computational Anatomy (MFCA-2006), edited by X. Pennec and S. Joshi (2006) 29-39.
[12] , General Pattern Theory. Oxford Science Publications (1993). | Zbl 0827.68098
[13] and , Martingale limit theory and its application. Probab. Math. Statist. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York (1980). | Zbl 0462.60045
[14] and , Coupling a stochastic approximation version of EM with an MCMC procedure. ESAIM: PS 8 (2004) 115-131 (electronic). | Zbl 1155.62420
[15] and , Stochastic approximation methods for constrained and unconstrained systems, volume 26 of Appl. Math. Sci. Springer-Verlag, New York (1978). | Zbl 0381.60004
[16] , and , A minimum description length objective function for groupwise non rigid image registration. Image and Vision Computing (2007).
[17] and , Markov chains and stochastic stability. Communications and Control Engineering Series. Springer-Verlag, London Ltd. (1993). | Zbl 0925.60001
[18] , T.A. and L. Younes, On the metrics and Euler-Lagrange equations of computational anatomy. Ann. Rev. Biomed. Eng. 4 (2002) 375-405.
[19] , Méthodes de Monte Carlo par chaînes de Markov. Statistique Mathématique et Probabilité. [Mathematical Statistics and Probability]. Éditions Économica, Paris (1996). | Zbl 0917.60007
[20] , , and , Statistics on diffeomorphisms via tangent space representations. Neuroimage 23 (2004) S161-S169.
