Convergent stochastic Expectation Maximization algorithm with efficient sampling in high dimension. Application to deformable template model estimation

Estimation in the deformable template model is a big challenge in image analysis. The issue is to estimate an atlas of a population. This atlas contains a template and the corresponding geometrical variability of the observed shapes. The goal is to propose an accurate estimation algorithm with low computational cost and with theoretical guaranties of relevance. This becomes very demanding when dealing with high dimensional data, which is particularly the case of medical images. The use of an optimized Monte Carlo Markov Chain method for a stochastic Expectation Maximization algorithm, is proposed to estimate the model parameters by maximizing the likelihood. A new Anisotropic Metropolis Adjusted Langevin Algorithm is used as transition in the MCMC method. First it is proven that this new sampler leads to a geometrically uniformly ergodic Markov chain. Furthermore, it is proven also that under mild conditions, the estimated parameters converge almost surely and are asymptotically Gaussian distributed. The methodology developed is then tested on handwritten digits and some 2D and 3D medical images for the deformable model estimation. More widely, the proposed algorithm can be used for a large range of models in many fields of applications such as pharmacology or genetic. The technical proofs are detailed in an appendix.1