On maximum entropy image reconstruction from projections via lagrange parameter analysis

A method for maximum-entropy image reconstruction from projections is presented. Two image entropies are studied under the assumption that image-density distributions over image-element array follow a multinomial random process. The multinomial distribution process reflects those physical aspects of images: (1) image density is non-negative, (2) image-density distributions are conserved, (3) image-density distributions are spatially correlated. One image entropy assumes a uniform spatial correlation and resembles Shannon's entropy that has been widely used in image-processing problems. The other assumes non-uniform spatial correlations among nearby image elements and resembles the cross entropy introduced by Kullback and Leibler. The non-uniform correlations among nearby elements are modeled in a similar manner as Markov random field does, i.e., image density changes slowly within a homogeneous region and rapidly at the boundaries of the homogeneous regions. The projections are assumed as measured by a linear imaging system. The measurement constraints are incorporated into the maximum-entropy method by employing a set of Lagrange parameters. Each Lagrange parameter is related to a constraint of a datum. The solutions that maximize the image entropies subject to the measurement constraints are given, as determined respectively by a superposition of the Lagrange parameters. The Lagrange parameters are then estimated iteratively using a steepest descent technique. Comparative study on the image entropies is carried out using computer generated noise-free and noisy projections. Improved results are obtained by use of the image entropy assuming the non-uniform correlations. The non-uniform correlated image entropy needs little more computation time and memory.