An algorithm for maximum entropy image reconstruction from noisy data

Abstract

A Maximum Aposteriori Probability (MAP-) approach is considered for reconstructing an image from noisy projections. This approach needs, according to Bayes rule, two specified distributions. The entropy concept is used to specify the à priori distribution of the unknown image vector. The second distribution (which reflects measurement errors) is taken as an uncorrelated normal distribution. The MAP approach then leads to a discrete, strictly convex, nonquadratic optimization problem. The model contains a real parameter which controls the degree of smoothing imposed on the solution. We propose the use of cross-validation for picking the value of this parameter. We derive an algorithm to solve the optimization problem based on Bregman's convex programming scheme. First a simple generalization of Bregman's method is made which leads to increased efficiency for general linear constraints. Based on this generalization we write down the corresponding Bregman method for our reconstruction problem. The method requires that in each iteration a single nonlinear equation is solved. We propose to use one step of Newton's method to approximate the root. Finally a convergence proof is given for the new algorithm.