Bayesian Approaches in Image Source Seperation

In this thesis, a general solution to the component separation problem in images is introduced. Unlike most existing works, the spatial dependencies of images are modelled in the separation process with the use of Markov random fields (MRFs). In the MRFs model, Cauchy density is used for the gradient images. We provide a general Bayesian framework for the estimation of the parameters of this model. Due to the intractability of the problem we resort to numerical solutions for the joint maximization of the a posteriori distribution of the sources, the mixing matrix and the noise variances. For numerical solution, four different methods are proposed. In first method, the difficulty of working analytically with general Gibbs distributions of MRF is overcome by using an approximate density. In this approach, the Gibbs distribution is modelled by the product of directional Gaussians. The sources are estimated by Maximum-a-Posteriori (MAP) estimation using the approximate density as the prior. The second method that uses the Markov Chain Monte Carlo (MCMC) is a fully Bayesian method. In this method, modified-Gibbs embedded with the Metropolis steps is used to find the joint estimate of sources, mixing matrix and noise variances. The third method is improved version of the second method by adding learning steps of the MRF parameters. In the last method, importance sampling is used to find point estimates of source pixels iteratively. The proposed methods are contrasted to approximate Bayesian solutions such as Iterated Conditional Modes (ICM) and to non-Bayesian solutions of Independent Component Analysis (ICA) variety namely, Fixed Point Independent Component Analysis (FPICA Second Order Blind Identification (SOBI) and Spectral Matching ICA (SMICA). The performance of the method is tested on synthetic mixtures of texture images and astrophysical images under various noise scenarios. The techniques have been exploited in yet unexplored issues for the astrophysical source separation problem which has important actuality due to the WMAP satellite results published and the PLANCK satellite measurements anticipated. The proposed methods are shown to outperform significantly both its approximate Bayesian and non-Bayesian competitors.