Unsupervised Signal Restoration in Partially Observed Markov Chains

An important problem in signal processing consists in estimating an unobservable process x = {x_n}_nisinN from an observed process y = {y_n}_nisinN. In linear Gaussian hidden Markov chains (LGHMC), recursive solutions are given by Kalman-like Bayesian restoration algorithms. In this paper, we consider the more general framework of linear Gaussian triplet Markov chains (LGTMC), i.e. of models in which the triplet (x, r, y) (where r = {r_n})_nisinN is some additional process) is Markovian and Gaussian. We address unsupervised restoration in LGTMC by extending to LGTMC the EM parameter estimation algorithm which was already developed in classical state-space models