Parameter Estimation in Hidden Fuzzy Markov Random Fields and Image Segmentation

tion problem is to decide, from the observed image, in This paper proposes a new unsupervised fuzzy Bayesian which class each pixel lies. In the first case we speak of image segmentation method using a recent model using hidden fuzzy segmentation, and in the second case of hard segmenfuzzy Markov fields. The originality of this model is to use tation. As we can see, fuzzy and hard segmentations are Dirac and Lebesgue measures simultaneously at the class field not competing but correspond to two different situations. level, which allows the coexistence of hard and fuzzy pixels in Now, if we wish to use some statistical method we have a same picture. We propose to solve the main problem of to introduce random variables and probability distribuparameter estimation by using of a recent general method of tions. We insist that from the viewpoint we adopt there is estimation in the case of hidden data, called iterative condino connection between fuzziness and the stochastic modeltional estimation (ICE), which has been successfully applied ing that can be used, although as suggested by some authors in classical segmentation based on hidden Markov fields. The [42], probability measures can be considered as modeling first part of our work involves estimating the parameters definfuzziness. The aim of this work is to propose a Markovianing the Markovian distribution of the noise-free fuzzy picture. We then combine this algorithm with the ICE method in order model-based unsupervised method of satistical fuzzy segto estimate all the parameters of the fuzzy picture corrupted mentation that is able to cope with situations such as those with noise. Last, we combine the parameter estimation step in the first example above. with two segmentation methods, resulting in two unsupervised Thus fuzzy segmentation of images consists in allowing statistical fuzzy segmentation methods. The efficiency of the each pixel to belong to numerous classes simultaneously. proposed methods is tested numerically on synthetic images and Let V 5 hg1, . . . , gkj be the set of classes. The problem a fuzzy segmentation of a real image of clouds is studied.  1997 is to associate to each pixel a vector (x1, . . . , xk) [ [0,1] Academic Press with x1 1 ? ? ? 1 xk 5 1. Classical, or ‘‘hard,’’ segmentation then appears as a particular case: all xi are null except one, which is 1. This generalization turns out to be very