Document Improvement by using Bayesian Approach
N. H. Margam, Roger R. Dube · 2012
This paper deals with the recovery of clean images from a set of their noisy convolutive mixtures. In practice, this problem can be seen as the one of simultaneously separating and restoring source images that have been first degraded by un-known filters, then summed up and added with noise. We approach this problem in the framework of Blind Source Separation (BSS), where the unknown filters, in our case FIR filters in the form of blur kernels, must be estimated jointly with the sources. Assuming the statistical independence of the source images, we adopt Bayesian estimation for all the unknowns, and exploit information about local cor- relation within the individual sources through the use of suit- able Gibbs priors, accounting also for well-behaved edges in the images. We derive an algorithm for recovering the blur kernels that make the estimated sources fit the known properties of the original sources. The method is validated through numerical experiments in a simplified setting, which is how-ever related to real application scenarios. In this paper, we propose a general approach to deal with convolutive mixtures of images, when the unknown filtering operators are FIR filters, in the form of blur kernels, and the mixtures are affected by noise, while consider the document analysis application as our case study for the simulations. We adopt a Bayesian estimation formulation, which offers a flexible way to approach the integrated solution of two or more problems, and to account for prior knowledge which can be available. Thus, Bayesian estimation permits to formulate the convolutive BSS problem as the joint estimation of the mixing kernels and the sources. Furthermore, auto-correlation constraints of the individual sources can be naturally enforced through Markov Random Field (MRF) models, in the form of Gibbs priors. These constraints have been proved to be effective for achieving stable solutions in many inverse problems, and especially in those dealing with im-ages, where they correspond to natural features of real physi-cal maps and scenes. MRF models allow for retaining the in-dependence assumption of ICA, and the one we adopt herein has the property of being edge-preserving and of enforcing regularity constraints on the edges themselves. This is an important issue since edges, corresponding to intensity discontinuities due to object boundaries and textures, constitute essential features to be correctly preserved in an image, for analysis and understanding purposes. We propose an estimation strategy for recovering those kernels that, besides satisfying possible a priori information, make the estimated sources fit the known properties of the original sources. Thus, we reformulate the problem as the estimation of the mixing operator alone, based on the source and mixing priors, while the sources are kept clamped to their Maximum A Posteriori (MAP) estimate, for any status of the mixing. From the theoretical scheme, reasonable approximations are derived which allow for reducing the computational complexity, and finding a remedy to other drawbacks, such as the unavailability of analytical formulas for the sources viewed as functions of the kernels, and non-convexity of the priors. These will make the method computational efficient and still effective. In particular, our method is implemented through an iterative scheme where the Maximum Likelihood (ML) estimation of the mixing alternates with the MAP estimation of the sources. This scheme ensures stability of the solutions and employs GNC- like gradient ascent algorithms to update the sources and Simulated Annealing (SA) to esti-mate the blur coefficients.