Triplet Markov chains in hidden signal restoration

Hidden Markov Chain (HMC) models are widely used in various signal or image restoration problems. In such models, one considers that the hidden process X = (X_l,..., X_n) we look for is a Markov chain, and the distribution p(y|x) of the observed process Y = (Y_l,..., Y_n), conditional on X, is given by p(y|x) = ∏_i=lⁿ p(y_i|x_i). The "a posteriori" distribution p(x|y) of X given Y = y is then a Markov chain distribution, which makes possible the use of different Bayesian restoration methods. Furthermore, all parameters can be estimated by the general "Expectation-Maximization" algorithm, which renders Bayesian restoration unsupervised. This paper is devoted to an extension of the HMC model to a "Triplet Markov Chain" (TMC) model, in which a third auxiliary process U is introduced and the triplet (X,U,Y) is considered as a Markov chain. Then a more general model is obtained, in which X can still be restored from Y = y. Moreover, the model parameters can be estimated with Expectation-Maximization (EM) or Iterative Conditional Estimation (ICE), making the TMC based restoration methods unsupervised. We present a short simulation study of image segmentation, where the bi- dimensional set of pixels is transformed into a mono-dimensional set via a Hilbert-Peano scan, that shows that using TMC can improve the results obtained with HMC.