In this paper, we present a new model for deformations of shapes. A pseudo-likelihood is based on the statistical distribution of the gradient vector field of the gray level. The prior distribution is based on the Probabilistic Principal Component Analysis (PPCA). We also propose a new model based on mixtures of PPCA that is useful in the case of greater variability in the shape. A criterion of global or local object specificity based on a preliminary color segmentation of the image, is included into the model. The localization of a shape in an image is then viewed as minimizing the corresponding Gibbs field. We use the Exploration/Selection (E/S) stochastic algorithm in order to find the optimal deformation. This yields a new unsupervised statistical method for localization of shapes. In order to estimate the statistical parameters for the gradient vector field of the gray level, we use an Iterative Conditional Estimation (ICE) procedure. The color segmentation of the image can be computed with an Exploration/Selection/Estimation (ESE) procedure.

In this paper, we present a Hidden Markov Random Field (HMRF) data-fusion model. The proposed model is applied to the segmentation of natural images based on the fusion of colors and textons into Julesz ensembles. The corresponding Exploration/ Selection/Estimation (ESE) procedure for the estimation of the parameters is presented. This method achieves the estimation of the parameters of the Gaussian kernels, the mixture proportions, the region labels, the number of regions, and the Markov hyper-parameter. Meanwhile, we present a new proof of the asymptotic convergence of the ESE procedure, based on original finite time bounds for the rate of convergence.

We propose a new stochastic algorithm for computing useful Bayesian estimators of hidden Markov random field (HMRF) models that we call exploration/selection/estimation (ESE) procedure. The algorithm is based on an optimization algorithm of O. François, called the exploration/selection (E/S) algorithm. The novelty consists of using the a posteriori distribution of the HMRF, as exploration distribution in the E/S algorithm. The ESE procedure computes the estimation of the likelihood parameters and the optimal number of region classes, according to global constraints, as well as the segmentation of the image. In our formulation, the total number of region classes is fixed, but classes are allowed or disallowed dynamically. This framework replaces the mechanism of the split-and-merge of regions that can be used in the context of image segmentation. The procedure is applied to the estimation of a HMRF color model for images, whose likelihood is based on multivariate distributions, with each component following a Beta distribution. Meanwhile, a method for computing the maximum likelihood estimators of Beta distributions is presented. Experimental results performed on 100 natural images are reported. We also include a proof of convergence of the E/S algorithm in the case of nonsymmetric exploration graphs.