A theoretical solution to MAP-EM partial volume segmentation of medical images

Voxels near tissue borders in medical images contain useful clinical information, but are subject to severe partial volume (PV) effect, which is a major cause of imprecision in quantitative volumetric and texture analysis. When modeling each tissue type as a conditionally independent Gaussian distribution, the tissue mixture fractions in each voxel via the modeled unobservable random processes of the underlying tissue types can be estimated by maximum a posteriori expectation-maximization (MAP-EM) algorithm in an iterative manner. This article presents, based on the assumption that PV effect could be fully described by a tissue mixture model, a theoretical solution to the MAP-EM segmentation algorithm, as opposed to our previous approximation which simplified the posteriori cost function as a quadratic term. It was found out that the theoretically-derived solution existed in a set of high-order nonlinear equations. Despite of the induced computational complexity when seeking for optimum numerical solutions to nonlinear equations, potential gains in robustness, consistency and quantitative precision were noticed. Results from both synthetic digital phantoms and real patient bladder MR images were presented, demonstrating the accuracy and efficiency of the presented theoretical MAP-EM solution.