Bayesian expectation maximization algorithm by using B-splines functions: Application in image segmentation

The Bayesian implementation of finite mixtures of distributions has been an area of considerable interest within the literature. Given a sample of independent identically distributed real-valued random variables with a common unknown probability density function ff, the considered problem here is to estimate the probability density function ff from the sample set. In our work, we suppose that the density ff is expressed as a finite linear combination of second order B-splines functions. The problem of estimating the density ff leads to the estimation of the coefficients of B-splines. In order to solve this problem, we suppose that the prior distribution of the B-splines coefficients is a Dirichlet distribution. The estimation of these coefficients allowed us to introduce a new algorithm called Bayesian expectation maximization. In fact, this algorithm, which is the combination of the Bayesian approach and the expectation maximization algorithm, attempts to directly optimize the posterior Bayesian distribution. This algorithm has been generalized to the case of mixing distributions. We have studied the asymptotic properties of the Bayesian estimator. Then, the performance of our algorithm has been evaluated and compared by making a simulation study, followed by a real image segmentation. In both cases, our proposed Bayesian algorithm is shown to give better results.