Construction of Bayesian deformable models via a stochastic approximation algorithm: a convergence study. (English) Zbl 1220.62101

Summary: The problem of 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 by S. Allassonnière, Y. Amit and A. Trouvé [J. R. Stat. Soc., Ser. B 69, 3–29 (2007)]. 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.


62L20 Stochastic approximation
62F15 Bayesian inference
62H35 Image analysis in multivariate analysis
Full Text: DOI arXiv


[1] 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.
[2] 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.
[3] Amit, Y. (1996). Convergence properties of the Gibbs sampler for perturbations of Gaussians., Ann. Statist. 24 122-140. · Zbl 0854.60066 · doi:10.1214/aos/1033066202
[4] Amit, Y., Grenander, U. and Piccioni, M. (1991). Structural image restoration through deformable template., J. Amer. Statist. Assoc. 86 376-387.
[5] Andrieu, C. and Moulines, É. (2006). On the ergodicity properties of some adaptive MCMC algorithms., Ann. Appl. Probab. 16 1462-1505. · Zbl 1114.65001 · doi:10.1214/105051606000000286
[6] Andrieu, C., Moulines, É. and Priouret, P. (2005). Stability of stochastic approximation under verifiable conditions., SIAM J. Control Optim. 44 283-312 (electronic). · Zbl 1083.62073 · doi:10.1137/S0363012902417267
[7] Chef d’Hotel, C., Hermosillo, G. and Faugeras, O. (2002). Variational methods for multimodal image matching., International Journal of Computer Vision 50 329-343. · Zbl 1012.68788
[8] Delyon, B., Lavielle, M. and Moulines, E. (1999). Convergence of a stochastic approximation version of the EM algorithm., Ann. Statist. 27 94-128. · Zbl 0932.62094 · doi:10.1214/aos/1018031103
[9] 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. JSTOR: · Zbl 0364.62022
[10] Douc, R., Moulines, E. and Rosenthal, J.S. (2004). Quantitative bounds on convergence of time-inhomogeneous Markov chains., Ann. Appl. Probab. 14 1643-1665. · Zbl 1072.60059 · doi:10.1214/105051604000000620
[11] Glasbey, C.A. and Mardia, K.V. (2001). A penalised likelihood approach to image warping., J. Roy. Statist. Soc. Ser. B 63 465-492. JSTOR: · Zbl 1040.62054 · doi:10.1111/1467-9868.00295
[12] Grenander, U. and Miller, M.I. (1998). Computational anatomy: An emerging discipline., Quart. Appl. Math. LVI 617-694. · Zbl 0952.92016
[13] Kuhn, E. and Lavielle, M. (2004). Coupling a stochastic approximation version of EM with an MCMC procedure., ESAIM Probab. Stat. 8 115-131 (electronic). · Zbl 1155.62420 · doi:10.1051/ps:2004007
[14] 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.
[15] Meyn, S.P. and Tweedie, R.L. (1993)., Markov Chains and Stochastic Stability. Communications and Control Engineering Series . London: Springer. · Zbl 0925.60001
[16] 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.
[17] Robert, C. (1996)., Méthodes de Monte Carlo par chaînes de Markov. Statistique Mathématique et Probabilité. [Mathematical Statistics and Probability] . Paris: Éditions Économica. · Zbl 0917.60007
[18] Vaillant, M., Miller, I., Trouvé, A. and Younes, L. (2004). Statistics on diffeomorphisms via tangent space representations., Neuroimage 23 S161-S169.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.