Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-25T13:04:52.700Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastic algorithm for Bayesian mixture effect template estimation

Published online by Cambridge University Press:  22 December 2010

Stéphanie Allassonnière  and
Estelle Kuhn
Stéphanie Allassonnière
Affiliation:
CMAP - École polytechnique, Route de Saclay, 91128 Palaiseau, France; [email protected]
Estelle Kuhn
Affiliation:
LAGA - Université Paris 13, 99 av. J.-B. Clément, 93430 Villetaneuse, France and INRA - Unité MIA, Domaine de Vilvert, 78352 Jouy-en-Josas, France
Get access

Abstract

The estimation of probabilistic deformable template models incomputer vision or of probabilistic atlases in Computational Anatomyare core issues in both fields.A first coherent statistical framework where the geometrical variability ismodelled as a hiddenrandom 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 theconvergence of the estimation algorithm for the one component model inthe 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 ofmixture of deformable template models.We also prove the convergence of our algorithm toward a criticalpoint of the penalised likelihood of the observations andillustrate this with handwritten digit images and medical images.

Keywords

Stochastic approximationsnon rigid-deformable templatesshapes statisticsMAP estimationBayesian methodmixture models
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 14 , 2010 , pp. 382 - 408
Copyright
© EDP Sciences, SMAI, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Allassonnière, S., Amit, Y. and Trouvé, A., Toward a coherent statistical framework for dense deformable template estimation. J. Roy. Stat. Soc. 69 (2007) 3–29.
S. Allassonnière, E. Kuhn and A. Trouvé, 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).
Allassonnière, S., Kuhn, E. and Trouvé, A., Construction of Bayesian deformable models via a stochastic approximation algorithm: A convergence study. Bernoulli 16 (2010) 641–678. CrossRef
Amit, Y., Grenander, U. and Piccioni, M., Structural image restoration through deformable templates. J. Am. Statist. Assoc. 86 (1989) 376–387. CrossRef
Andrieu, C., Moulines, R. and Priouret, P., Stability of stochastic approximation under verifiable conditions. SIAM J. Control Optim. 44 (2005) 283–312 (electronic). CrossRef
T.F. Cootes, G.J. Edwards and C.J. Taylor, Actives appearance models. In 5th Eur. Conf. on Computer Vision, Berlin, Vol. 2, edited by H. Burkhards and B. Neumann. Springer (1998) 484–498.
Delyon, B., Lavielle, M. and Moulines, E., Convergence of a stochastic approximation version of the EM algorithm. Ann. Statist. 27 (1999) 94–128.
Dempster, A.P., Laird, N.M. and Rubin, D.B., Maximum likelihood from incomplete data via the EM algorithm. J. Roy. Statist. Soc. 1 (1977) 1–22.
Dorea, C. and Zhao, L., Nonparametric density estimation in hidden Markov models. Statist. Inf. Stoch. Process. 5 (2002) 55–64. CrossRef
C.A. Glasbey and K.V. Mardia, A penalised likelihood approach to image warping. J. Roy. Statist. Soc., Ser. B 63 (2001) 465–492.
J. Glaunès and S. Joshi, 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.
U. Grenander, General Pattern Theory. Oxford Science Publications (1993).
P. Hall and C.C. Heyde, Martingale limit theory and its application. Probab. Math. Statist. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York (1980).
Kuhn, E. and Lavielle, M., Coupling a stochastic approximation version of EM with an MCMC procedure. ESAIM: PS 8 (2004) 115–131 (electronic). CrossRef
H.J. Kushner and D.S. Clark, Stochastic approximation methods for constrained and unconstrained systems, volume 26 of Appl. Math. Sci. Springer-Verlag, New York (1978).
S. Marsland, C. Twining and C. Taylor, A minimum description length objective function for groupwise non rigid image registration. Image and Vision Computing (2007).
S.P. Meyn and R.L. Tweedie, Markov chains and stochastic stability. Communications and Control Engineering Series. Springer-Verlag, London Ltd. (1993).
Miller, M.I., T.A. and L. Younes, On the metrics and Euler-Lagrange equations of computational anatomy. Ann. Rev. Biomed. Eng. 4 (2002) 375–405. CrossRef
C. Robert, Méthodes de Monte Carlo par chaînes de Markov. Statistique Mathématique et Probabilité. [Mathematical Statistics and Probability]. Éditions Économica, Paris (1996).
Vaillant, M., Miller, I., Trouvé, M.A. and Younes, L., Statistics on diffeomorphisms via tangent space representations. Neuroimage 23 (2004) S161–S169. CrossRef