The efficiency of Markov models in the context of SAR image segmentation mainly relies on their spatial regularity constraint. However, a pixel may have a rather different visual aspect when it is located near a boundary or inside a large set of pixels of the same class. According to the classical hypothesis in Hidden Markov Chain (HMC) models, this fact can not be taken into consideration. This is the very reason of the recent Pairwise Markov Chains (PMC) model which relies on the hypothesis that the pairwise process (X,Y) is
Markovian and stationary, but not necessarily X. The main interest of the PMC model in SAR image segmentation is to not assume that the speckle is spatially uncorrelated. Hence, it is possible to take into
account the difference between two successive pixels that belong to the same region or that overlap a boundary. Both PMC and HMC parameters are learnt from a variant of the Iterative Conditional Estimation method. This allows to apply the Bayesian Maximum Posterior Marginal criterion for the restoration of X in
an unsupervised manner. We will compare the PMC model with respect to the HMC one for the unsupervised segmentation of SAR images, for both Gaussian distributions and Pearson system of distributions.
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=(X1, ., Xn) we look for is a Markov chain, and the distribution p(y/x) of the observed process Y=(Y1, ., Yn), conditional on X, is given by p(y/x)=p(y1/x1). p(yn/xn). 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.
Hidden Markov fields (HMF) are widely used in image processing. In such models, the hidden random field of interest X=(Xs) is a Markov field, and the distribution p(y/x) of the observed random field Y=(Ys) conditional on X is given by the product of p(ys/xs), with s in the set of pixels. 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=(Us) 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.
Due to the enormous quantity of radar images acquired by satellites and through shuttle missions, there is an evident need for efficient automatic analysis tools. This article describes unsupervised classification of radar images in the framework of hidden Markov models and generalised mixture estimation. In particular, we show that hidden Markov chains, based on a Hilbert-Peano scan of the radar image, are a fast and efficient alternative to hidden Markov random fields for parameter estimation and unsupervised classification. We also describe how the distribution families and parameters of classes with homogeneous or textured radar reflectivity can be determined through generalised mixture estimation. Sample results obtained on real and simulated radar images are presented.
Mixture estimation has been widely applied to unsupervised contextual Bayesian segmentation. We present at first the algorithms which estimate distribution mixtures prior to contextual segmentation, such as estimation-maximization (EM), iterative conditional estimation (ICE), and their adaptive versions valid for nonstationary class fields. Upon removing the stationarity hypothesis, contextual segmentation can give much better results in certain cases. Results obtained attest to the suitability of adaptive versions of EM, ICE valid in the case of nonstationary random class fields. Then we present our experiences on the application of the unsupervised contextual Bayesian segmentation to images of blood vessel.