Construction of Bayesian Deformable Models via Stochastic Approximation Algorithm: A Convergence Study

TL;DR

This paper introduces a stochastic approximation algorithm for Bayesian deformable models, demonstrating its convergence to a critical point and applying it to handwritten digit images for shape modeling.

Contribution

It develops a stochastic SAEM-based algorithm for Bayesian deformable models and proves its convergence, extending previous deterministic methods.

Findings

Algorithm converges to a critical point of the likelihood

Effective application to handwritten digit images

Provides theoretical convergence guarantees

Abstract

The problem of the definition and the estimation of generative models based on deformable templates from raw data is of particular importance for modelling non aligned data affected by various types of geometrical variability. This is especially true in shape modelling in the computer vision community or in probabilistic atlas building for Computational Anatomy (CA). A first coherent statistical framework modelling the geometrical variability as hidden variables has been given by Allassonni\`ere, Amit and Trouv\'e (JRSS 2006). Setting the problem in a Bayesian context they proved the consistency of the MAP estimator and provided a simple iterative deterministic algorithm with an EM flavour leading to some reasonable approximations of the MAP estimator under low noise conditions. In this paper we present a stochastic algorithm for approximating the MAP estimator in the spirit of the SAEM algorithm. We prove its convergence to a critical point of the observed likelihood with an illustration on images of handwritten digits.