• Bernoulli
  • Volume 16, Number 3 (2010), 641-678.

Construction of Bayesian deformable models via a stochastic approximation algorithm: A convergence study

Stéphanie Allassonnière, Estelle Kuhn, and Alain Trouvé

Full-text: Open access


The problem of the definition and estimation of generative models based on deformable templates from raw data is of particular importance for modeling non-aligned data affected by various types of geometric variability. This is especially true in shape modeling in the computer vision community or in probabilistic atlas building in computational anatomy. A first coherent statistical framework modeling geometric variability as hidden variables was described in Allassonnière, Amit and Trouvé [J. R. Stat. Soc. Ser. B Stat. Methodol. 69 (2007) 3–29]. The present paper gives a theoretical proof of convergence of effective stochastic approximation expectation strategies to estimate such models and shows the robustness of this approach against noise through numerical experiments in the context of handwritten digit modeling.

Article information

Bernoulli, Volume 16, Number 3 (2010), 641-678.

First available in Project Euclid: 6 August 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Bayesian modeling MAP estimation non-rigid deformable templates shape statistics stochastic approximation algorithms


Allassonnière, Stéphanie; Kuhn, Estelle; Trouvé, Alain. Construction of Bayesian deformable models via a stochastic approximation algorithm: A convergence study. Bernoulli 16 (2010), no. 3, 641--678. doi:10.3150/09-BEJ229.

Export citation


  • Allassonière, S., Amit, Y., Kuhn, E. and Trouvé, A. (2006). Generative model and consistent estimation algorithms for non-rigid deformable models. In, IEEE Intern. Conf. on Acoustics, Speech, and Signal Processing 5. IEEE.
  • Allassonnière, S., Amit, Y. and Trouvé, A. (2007). Toward a coherent statistical framework for dense deformable template estimation., J. R. Stat. Soc. Ser. B Stat. Methodol. 69 3–29.
  • Amit, Y. (1996). Convergence properties of the Gibbs sampler for perturbations of Gaussians., Ann. Statist. 24 122–140.
  • Amit, Y., Grenander, U. and Piccioni, M. (1991). Structural image restoration through deformable template., J. Amer. Statist. Assoc. 86 376–387.
  • Andrieu, C. and Moulines, É. (2006). On the ergodicity properties of some adaptive MCMC algorithms., Ann. Appl. Probab. 16 1462–1505.
  • Andrieu, C., Moulines, É. and Priouret, P. (2005). Stability of stochastic approximation under verifiable conditions., SIAM J. Control Optim. 44 283–312 (electronic).
  • Chef d’Hotel, C., Hermosillo, G. and Faugeras, O. (2002). Variational methods for multimodal image matching., International Journal of Computer Vision 50 329–343.
  • Delyon, B., Lavielle, M. and Moulines, E. (1999). Convergence of a stochastic approximation version of the EM algorithm., Ann. Statist. 27 94–128.
  • Dempster, A.P., Laird, N.M. and Rubin, D.B. (1977). Maximum likelihood from incomplete data via the EM algorithm., J. Roy. Statist. Soc. Ser. B 1 1–22.
  • Douc, R., Moulines, E. and Rosenthal, J.S. (2004). Quantitative bounds on convergence of time-inhomogeneous Markov chains., Ann. Appl. Probab. 14 1643–1665.
  • Glasbey, C.A. and Mardia, K.V. (2001). A penalised likelihood approach to image warping., J. Roy. Statist. Soc. Ser. B 63 465–492.
  • Grenander, U. and Miller, M.I. (1998). Computational anatomy: An emerging discipline., Quart. Appl. Math. LVI 617–694.
  • Kuhn, E. and Lavielle, M. (2004). Coupling a stochastic approximation version of EM with an MCMC procedure., ESAIM Probab. Stat. 8 115–131 (electronic).
  • Marsland, S., Twining, C. and Taylor, C. (2007). A minimum description length objective function for groupwise non rigid image registration., Image and Vision Computing 26 333–346.
  • Meyn, S.P. and Tweedie, R.L. (1993)., Markov Chains and Stochastic Stability. Communications and Control Engineering Series. London: Springer.
  • Richard, F., Samson, A. and Cuénod, C. (2009). A saem algorithm for the estimation of template and deformation parameters in medical image sequences., Statist. Comput. 19 465–478.
  • Robert, C. (1996)., Méthodes de Monte Carlo par chaînes de Markov. Statistique Mathématique et Probabilité. [Mathematical Statistics and Probability]. Paris: Éditions Économica.
  • Vaillant, M., Miller, I., Trouvé, A. and Younes, L. (2004). Statistics on diffeomorphisms via tangent space representations., Neuroimage 23 S161–S169.