Unsupervised image segmentation using triplet Markov fields

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.

Unsupervised Statistical Segmentation of Nonstationary Images Using Triplet Markov Fields

“…Anyway, it is important that T ¼ ðX; U; Y Þ is stationary and, thus, all of its parameters can be estimated from Y ¼ y (see [1]). As a consequence, nonstationary images can be segmented in an unsupervised way and, as shown below, such methods can significantly improve the efficiency of the classical method.…”

“…All parameters are estimated from Y ¼ y by a particular algorithm belonging to the so-called "ICE" family of methods, which is described in detail in [1].…”

“…This third field can have some physical interpretation or not; however, when the set of its values is not too large, analogous Bayesian processing can still be used to estimate X ¼ x from Y ¼ y. Different ways of defining such TMF are described in [1], along with a parameter estimation method making possible unsupervised Bayesian segmentation.…”

“…This implies that both conditional distributions pðyjxÞ and pðxjyÞ are Markovian: The former fact allows one to better model complex noises and the latter one still enables one to apply Bayesian segmentation. Afterward, triplet Markov fields (TMF) were proposed in which one introduces a third random field, U ¼ ðU s Þ s2S , and assumes the Markovianity of the triplet T ¼ ðX; U; Y Þ [1], [35]. This third field can have some physical interpretation or not; however, when the set of its values is not too large, analogous Bayesian processing can still be used to estimate X ¼ x from Y ¼ y.…”

“…Points 1 and 2 are considered simultaneously, resulting in a new nonstationary model with nonGaussian and correlated noise; 4. A new parameter estimation method which is based on the Pearson system and extends to the model in item 3 above, the methods proposed in [1] and [5] (some first results concerning this new model are presented in [3]). …”