Logic Of Triplet Markov Fields

Unsupervised image segmentation using triplet Markov fields
by
Dalila Benboudjema, Wojciech Pieczynski

Abstract
Hidden Markov fields (HMF) models are widely applied to various problems arising in
image processing. In these models, the hidden process of interest X is a Markov field and must be estimated from its observable noisy version Y. The success of HMF is mainly due to the fact that the conditional probability distribution of the hidden process with respect to the observed one remains Markovian, which facilitates different processing strategies such as Bayesian restoration. HMF have been recently generalized to ‘‘pairwise’’ Markov fields (PMF), which offer similar processing advantages and superior modeling capabilities. In PMF one directly assumes the Markovianity of the pair (X,Y). Afterwards, ‘‘triplet’’ Markov fields (TMF), in which the distribution of the pair (X,Y) is the marginal distribution of a Markov field (X,U,Y), where U is an auxiliary process, have been proposed and still allow restoration processing. The aim of this paper is to propose a new parameter estimation method adapted to TMF, and to study the corresponding unsupervised image segmentation methods. The latter are validated via experiments and real image processing.
@ 2005 Elsevier Inc. All rights reserved.

Keywords: Hidden Markov fields; Pairwise Markov fields; Triplet Markov fields; Bayesian classification; Mixture estimation; Iterative conditional estimation; Stochastic gradient; Unsupervised image segmentation

1. Introduction
Hidden Markov fields (HMF) are widely used in solving various problems, comprising two stochastic processes X = (Xs)s2S and Y = (Ys)s2S, in which X = x is unobservable and must be estimated from the observed Y = y. This wide use is due to the fact that standard Bayesian restoration methods can be used in spite of the large size of S: see [3,12,19] for seminal papers and [14,33], among others, for general books. The qualifier ‘‘hidden Markov’’ means that the hidden process X has a Markov law. When the distributions p (y|x) of Y conditional on X = x are simple enough, the pair (X,Y) then retains the Markovian structure, and likewise for the distribution p(x|y) of X conditional on Y = y. The Markovianity of p(x|y) is crucial because it allows one to estimate the unobservable X = x from the observed Y = y, even in the case of very rich sets S. However, the simplicity of p (y|x) required in standard HMF to ensure the Markovianity of p(x|y) can pose problems; in particular, such situations occur in textured images segmentation [21]. To remedy this, the use of pairwise Markov fields (PMF), in which one directly assumes the Markovianity of (X,Y), has been discussed in [26]. Both p(y|x) and p(x|y) are then Markovian, the former ensuring possibilities of modeling textures without approximations, and the latter allowing Bayesian processing, similar to those provided by HMF. PMF have then been generalized to ‘‘triplet’’ Markov fields (TMF), in which the distribution of the pair Z = (X,Y) is the marginal distribution of a Markov field T = (X,U,Y), where U = (Us)s2S is an auxiliary random field [27]. Once the space K of possible values of each Us is simple enough, TMF still allow one to estimate the unobservable X = x from the observed Y = y. Given that in TMF T = (X,U,Y) the distribution of Z = (X,Y) is its marginal distribution, the Markovianity of T does not necessarily imply the Markovianity of Z; and thus a TMF model is not necessarily a PMF one. Therefore, TMF are more general than PMF and thus are likely to be able to model more complex situations. Conversely, a PMF model can be seen as a particular TMF model in which X = U. There are some studies concerning triplet Markov chains [18,28], where general ideas somewhat similar to those discussed in the present paper, have been investigated. However, as Markov fields based processing is quite different from the Markov chains based one, we will concentrate here on Markov fields with no further reference to Markov chains.