Abstract—We propose a new model called a Pairwise Markov Chain (PMC), which generalizes the classical Hidden Markov Chain (HMC) model. The generalization, which allows one to model more complex situations, in particular implies that in PMC the hidden process is not necessarily a Markov process. However, PMC allows one to use the classical Bayesian restoration methods like Maximum A Posteriori (MAP), or Maximal Posterior Mode (MPM). So, akin to HMC, PMC allows one to restore hidden stochastic processes, with numerous applications to signal and image processing, such as speech recognition, image segmentation, and symbol detection or classification, among others. Furthermore, we propose an original method of parameter estimation, which generalizes the classical Iterative Conditional Estimation (ICE) valid for of classical hidden Markov chain model, and whose extension to possibly non-Gaussian and correlated noise is briefly treated. Some preliminary experiments validate the interest of the new model. Index Terms—Bayesian restoration, hidden data, image segmentation, iterative conditional estimation, hidden Markov chain, pairwise Markov chain, unsupervised classification. 1

Hidden Markov chains (HMC) are widely applied in various problems occurring in different areas like Biosciences, Climatology, Communications, Ecology, Econometrics and Finances, Image or Signal processing. In such models, the hidden process of interest X is a Markov chain, which must be estimated from an observable Y, interpretable as being a noisy version of X. The success of HMC is mainly due to the fact that the conditional probability distribution of the hidden process with respect to the observed process remains Markov, which makes possible different processing strategies such as Bayesian restoration. HMC have been recently generalized to ‘‘Pairwise’ ’ Markov chains (PMC) and ‘‘Triplet’ ’ Markov chains (TMC), which offer similar processing advantages and superior modeling capabilities. In PMC, one directly assumes the Markovianity of the pair (X, Y) and in TMC, the distribution of the pair (X, Y) is the marginal distribution of a Markov process (X, U, Y), where U is an auxiliary process, possibly contrived. Otherwise, the Dempster–Shafer fusion can offer interesting extensions of the calculation of the ‘‘a posteriori’ ’ distribution of the hidden data. The aim of this paper is to present different possibilities of using the Dempster–Shafer fusion in the context of different multisensor Markov models. We show that the posterior distribution remains calculable in different general situations and present some examples of their applications in remote sensing area.

Abstract—Let � a � sx be a hidden process, � a � sx an observed process, and � a � sx some additional process. We assume that � a @ � � �A is a (so-called “Triplet”) vector Markov chain (TMC). We first show that the linear TMC model encompasses and generalizes, among other models, the classical state-space systems with colored process and/or measurement noise(s). We next propose restoration Kalman-like filters for arbitrary linear Gaussian (LG) TMC. Index Terms—Bayesian signal restoration, hidden Markov chains, Kalman filtering, Markovian models, triplet Markov chains.

Hidden Markov fields (HMF) are widely used in image processing. In such models, the hidden random field of interest S s s X X = ) ( is a Markov field, and the distribution of the observed random field S s s Y Y = ) ( (conditional on X ) is given by s s x y p x y p ) ( ) ( . The posterior distribution ) ( y x p 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 ) ( x y p above is insufficient and has to be replaced by a Markov field distribution. This poses problems, because taking ) ( x y p Markovian implies that the posterior distribution ) ( y x p , 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 ) , ( Y X ; in these recent "Pairwise Markov Fields (PMF) models, both ) ( x y p and ) ( y x p are then Markovian, the first one allowing us to model textures, and the second one allowing us to use Bayesian restoration without model approximations.

Abstract—Recent developments in statistical theory and associated computational techniques have opened new avenues for image modeling as well as for image segmentation techniques. Thus, a host of models have been proposed and the ones which have probably received considerable attention are the hidden Markov fields (HMF) models. This is due to their simplicity of handling and their potential for providing improved image quality. Although these models provide satisfying results in the stationary case, they can fail in the nonstationary one. In this paper, we tackle the problem of modeling a nonstationary hidden random field and its effect on the unsupervised statistical image segmentation. We propose an original approach, based on the recent triplet Markov field (TMF) model, which enables one to deal with nonstationary class fields. Moreover, the noise can be correlated and possibly non-Gaussian. An original parameter estimation method which uses the Pearson system to find the natures of the noise margins, which can vary with the class, is also proposed and used to perform unsupervised segmentation of such images. Experiments indicate that the new model and related processing algorithm can improve the results obtained with the classical ones. Index Terms—Triplet Markov fields, statistical image segmentation, paramater estimation, Pearson system, iterative conditional estimation, nonstationary images, textures classification.

An important problem in signal processing consists in estimating an unobservable process x = {xn}n∈IN from an observed process y = {yn}n∈IN. 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 = {rn}n∈IN is some additional process) is Markovian and Gaussian. We address fixedinterval smoothing algorithms, and we extend to LGTMC the RTS algorithm by Rauch, Tung and Striebel, as well as the Two-Filter algorithm by Mayne and Fraser and Potter. 1.

Bayesian smoothing in conditionally linear Gaussian models, also called jump-Markov state-space systems, is an NP-hard problem. As a result, a number of approximate methods- either deterministic or Monte Carlo based- have been developed. In this paper we address the Bayesian smoothing problem in another triplet Markov chain model, in which the switching process R is not necessarily Markovian and the additive noises do not need to be Gaussian. We show that in this model the smoothing posterior mean and covariance matrix can be computed exactly with complexity linear in time.

Abstract. In the classical hidden Markov chain (HMC) model we have a hidden chain X, which is a Markov one and an observed chain Y. HMC are widely used; however, in some situations they have to be replaced by the more general “hidden semi-Markov chains ” (HSMC), which are particular “triplet Markov chains ” (TMC) T = ( X, U, Y) , where the auxiliary chain U models the semi-Markovianity of X. Otherwise, non stationary classical HMC can also be modeled by a triplet Markov stationary chain with, as a consequence, the possibility of parameters ' estimation. The aim of this paper is to use simultaneously both properties. We

The problem considered in this paper is the problem of the exact calculation of smoothing in hidden switching state-space systems. There is a hidden state-space chain X, the switching Markov chain R, and the observed chain Y. In the classical, widely used “conditionally Gaussian state-space linear model” (CGSSLM) the exact calculation with complexity linear in time is not feasible and different approximations have to be made. Different alternative models, in which the exact calculations are feasible, have been recently proposed (2008). The core difference between these models and the classical ones is that the couple (R, Y) is a Markov one in the recent models, while it is not in the classical ones. Here we propose a further extension of these recent models by relaxing the hypothesis of the Markovianity of X conditionally on (R, Y). In fact, in all classical models and as well as in the recent ones, the hidden chain X is always a Markov one conditionally on (R, Y). In the proposed model it can be of any form. In particular, different “long memory ” processes can be considered. In spite of this larger generality, we show that the smoothing formulae are still calculable exactly with the complexity polynomial in time.

In this work, we propose to improve the neighboring relationship ability of the Hidden Markov Chain (HMC) model, by extending the memory lengthes of both the Markov chain process and the data-driven densities arising in the model. The new model is able to learn more complex noise structures, with respect to the configuration of several previous states and observations. Model parameters estimation is performed from an extension of the general Iterative Conditional Estimation (ICE) method to take into account memories, which makes the classification algorithm unsupervised. The higher-order HMC model is then evaluated in the image segmentation context. A comparative study conducted on a simulated image is carried out according to the order of the chain. Experimental results on a Synthetic Aperture Radar (SAR) image show that higher-order model can provide more homogeneous segmentations than the classical model, but to the cost of higher memory and computing time requirements.