Non-stationary fuzzy Markov chain
Introduction
The fuzzy segmentation problem consists of estimating the hidden realization x = (xs)1⩽s⩽N, for a given set of D observations , where xs = (ε1(s), ε2(s), …, εK(s)). Each component εi(s) represents the contribution of each class ωi in a finite discrete set Ω = {ω1, …, ωK} of K hard classes. The fuzzy belonging of each pixel respects the normalization condition: ε1(s) + ε2(s) + ⋯ + εK(s) = 1. In the context of two “hard” classes, a set Ω = {0, 1} yields xs ∈ [0, 1]. Then, all values xs ∈ [0, 1] model the proportion of the class “0” in the pixel related to Xs, whereas 1 − xs corresponds to the proportion of the class “1”. The distribution at each random variable Xs is given by a density hs with respect to a measure ν including discrete components (Dirac functions δ0, δ1 on {0, 1}) and a continuous component (the Lebesgue measure μ on ]0, 1[) (Caillol et al., 1993):The discrete components of ν are associated with the hard classes, whereas the continuous component μ is associated with the fuzzy feature. In this paper, we will consider the case D = N (mono-spectral context). When X is a Markov chain called “fuzzy Markov chain” (FMC) and the variable Y is independent conditionally on X, it is possible to express the joint distribution p(x, y) with respect to a measure νN ⊗ μN, as follows:In particular, the posterior field X conditional on Y is Markovian. Thus, one can process the posterior realizations of the hidden variable X, called Hidden Fuzzy Markov Chain (HFMC). Generally the distribution p(x, y) depends on unknown parameter θ = (θX, θY) where the prior parameters θX define the prior density of the Markov chain and the parameters θY define the distribution parameters of the driven data conditional on X. Algorithms like “Expectation Maximization” (EM) (McLachlan and Krishnan, 1997) or its stochastic version (SEM) (Celeux and Diebolt, 1985) are efficient to estimate the hyper-parameter when θX does not vary locally, i.e., when the variable X is stationary. Recent studies have focused on unsupervised segmentation of Markov chain in the fuzzy context (Avrachenkov and Sanchez, 2002, Mohammed and Gader, 2000, Carincotte et al., 2004). In particular, we derive θX from the prior joint density p(xs, xs+1) at each neighbored sites. We present here a new model based on a parameterized joint density, which governs locally the attractiveness between two neighbored states. Unfortunately, when θX does not vary locally, these approaches are sometimes badly adapted. Thus one has to introduce a new fuzzy hidden Markov chain model which represents non-stationary data. In this work we model the non-stationarity by a third auxiliary process U, which governs the changing values of θX in the hidden process. A such method has been successfully applied in the hard context (Hughes et al., 1999, Lanchantin and Pieczynski, 2004), and we propose in this article to extend non-stationary Markov chain to the fuzzy case. The solution proposed in (Lanchantin and Pieczynski, 2004) is derived from a recent triplet Markov chain model (Pieczynski, 2002) which can be described in the following manner: the pairwise process Z = (X, U) is assumed to be Markovian, X and U separately are not necessary Markovian. The triplet process T = (X, U, Y) is then a particular triplet Markov chain. In Section 2, we present the stationary fuzzy Markov chain (SFMC) with and without a parameterized joint density (P-SFMC versus NP-SFMC). We briefly introduce the stationary fuzzy Markov field (SFMF) (Salzenstein and Pieczynski, 1997), which is used in the experimental part to enrich the comparisons. In the next Section3 we generalize the non-stationary model of Lanchantin and Pieczynski (2004) presenting a new fuzzy model in the context of non-stationary Markov chain (NSFMC) with a possibly joint parameterized density (P-NSFMC versus NP-NSFMC). We describe the noise model used (Section 4), the MPM segmentation procedure applied to the S/NS-FMC methods (Section 5) and the associated hyper-parameter estimation step (Section 6). Finally we show the efficiency of the new method though synthetic images (Section 7) and real images (Section 8).
Section snippets
The stationary fuzzy Markov chain (SFMC)
Let us consider now a Markov chain X = (xs)1⩽s⩽N with continuous statements, i.e., Xs ∈ [0, 1]. To define the distribution π(x) of the variable X, we need the density p(x1) of the initial distribution, and the transition densities p(xs∣xs−1)1⩽s⩽N:When the chain is stationary, all prior distributions can be deduced from a joint density. The prior joint density p(xs, xs+1) is defined on the pairwise (xs, xs+1) ∈ [0, 1]2. According to a measure ν ⊗ ν, the normalization
The non-stationary fuzzy Markov chain (NSFMC)
Authors (Lanchantin and Pieczynski, 2004) propose to add to an initial process X an additional process U, which takes its values in a finite set Λ = {λ1, λ2, …, λK}. The couple Z = (X, U) = {(x1, u1), (x2, u2), …, (xN, uN)} is supposed to be a stationary Markov chain, where X is an interested non-stationary process, and U models auxiliary states:In (Lanchantin and Pieczynski, 2004), X and U take their values into discrete classes. We propose to generalize this model by
Model of the observations in a non-stationary context
The joint process Z = (X, U) being assumed to be Markovian, the aim of our paper is to process multispectral data. We observe D realizations (y(1), y(2), …, y(D)) of the random vector Y = (Y(1), Y(2), …, Y(D)). They represent a single scene observed at different wavelengths or from different sensors. For each field Y(i), the variables are spatially independent conditionally on Z. One has the following relationships (Lanchantin and Pieczynski, 2004):
Segmentation of the SFMC
Given the set of observations Y = y, we wish to estimate one realization X = x ∈ [0, 1]N. It is possible to adapt the MPM criterion (Maroquin et al., 1987) to the fuzzy context (Salzenstein and Pieczynski, 1997). For a such approach, the final decision process is performed as following: given a realization Y = y, the bayesian decision such that , will involve minimizing a conditional expectation (27) at each location s, in order to obtain an optimal value of Xs
Hyper-parameter estimation
We focus in this section on the estimation of the parameter θ in the context of a non-stationary variable. Actually, the stationary context is a particular case, for which U owns one discrete state i.e., Card Λ = 1. The final segmentation step requires the parameter set θ = (θZ, θY) where the prior parameters θZ define the prior density of the Markov chain Z, which could be the set of parameters for the P-SFMC and P-NSFMC approaches. The parameters θY = ((μ0, μ1); (σ0, σ1
Results on synthetic images
We simulated a non-stationary fuzzy Markov chain on M = 10 discrete fuzzy levels, with two homogeneous states (Card Λ = 2) and r = 1. The variables X and U are represented in Fig. 1a and b. The class “0” (in black) of U corresponds to an hard-dominating area in X , where as the class “1” (in white) corresponds to a fuzzy area in X . A noisy version is represented in Fig. 1c. We give below the following corresponding prior and data driven parameters:
Results on real images
We wish to identify different homogeneous regions inside an image. We processed here our images in the mono-spectral context. We present in Fig. 4a and b two images of Oakland typically exhibiting a such situation. Fig. 4a contains a sea area and the city, which appear to be inhomogeneous on the picture, noticing that the distribution of this part of the image behaves differently from the distribution of the sea part. Fig. 4b contains a cloudy area and a town area. We processed first all images
Conclusion
We presented in this paper a new fuzzy Markov chain model based on a non-stationary approach. On one hand we modeled the prior parameters of a stationary chain using a parameterized joint density defined on neighbored sites. On the other hand, we used an intermediate field U in order to govern the switching in the distribution of X. Here the classes in U are discrete while the classes in X are continuous. The proposed method merges the fuzzy processing technique and a recent technique that has
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