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

TL;DR

This paper introduces a convergent stochastic EM algorithm with an efficient anisotropic Langevin MCMC sampler for high-dimensional deformable template estimation, providing theoretical guarantees and practical validation on medical images and handwritten digits.

Contribution

It develops a new anisotropic Langevin MCMC within a stochastic EM framework, with proven ergodicity and convergence properties for high-dimensional deformable model estimation.

Findings

The new sampler is geometrically uniformly ergodic.

Parameters converge almost surely and are asymptotically Gaussian.

Validated on medical images and handwritten digits.

Abstract

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 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. We propose to use an optimized Monte Carlo Markov Chain method into a stochastic Expectation Maximization algorithm in order to estimate the model parameters by maximizing the likelihood. In this paper, we present a new Anisotropic Metropolis Adjusted Langevin Algorithm which we use as transition in the MCMC method. We first prove that this new sampler leads to a geometrically uniformly ergodic Markov chain. We prove 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.