Statistical image segmentation using triplet Markov fields

Hidden Markov fields (HMF) are widely used in image processing. In such models, the hidden random field of interest X = (X_s)_sεS is a Markov field, and the distribution of the observed random field Y = (Y_s)_sεS (conditional on X) is given by p(y|x) = ∏_sεS p(y_s|X_s). The posterior distribution p(x|y) is then a Markov distribution, which affords different Bayesian processing. However, when dealing with the segmentation of images containing numerous classes with different textures, the simple form of the distribution p(y|x) above is insufficient and has to be replaced by a Markov field distribution. This poses problems, because taking p(y|x) Markovian implies that the posterior distribution p(x|y), whose Markovianity is needed to use Bayesian techniques, may no longer be a Markov distribution, and so different model approximations must be made to remedy this. This drawback disappears when considering directly the Markovianity of (X,Y); in these recent "Pairwise Markov Fields (PMF) models, both p(y|x) and p(x|y) are then Markovian, the first one allowing us to model textures, and the second one allowing us to use Bayesian restoration without model approximations. In this paper we generalize the PMF to Triplet Markov Fields (TMF) by adding a third random field U = (U_s)_sεS and considering the Markovianity of (X,U,Y). We show that in TMF X is still estimable from Y by Bayesian methods. The parameter estimation with Iterative Conditional Estimation (ICE) is specified and we give some numerical results showing how the use of TMF can improve the classical HMF based segmentation.