An EM Approach to MAP Solution of Segmenting Tissue Mixtures: A Numerical Analysis

This work presents an iterative expectation-maximization (EM) approach to the maximum a posteriori (MAP) solution of segmenting tissue mixtures inside each image voxel. Each tissue type is assumed to follow a normal distribution across the field-of-view (FOV). Furthermore, all tissue types are assumed to be independent from each other. Under these assumptions, the summation of all tissue mixtures inside each voxel leads to the image density mean value at that voxel. The summation of all the tissue mixtures´ unobservable random processes leads to the observed image density at that voxel, and the observed image density value also follows a normal distribution (image data are observed to follow a normal distribution in many applications). By modeling the underlying tissue distributions as a Markov random field across the FOV, the conditional expectation of the posteriori distribution of the tissue mixtures inside each voxel is determined, given the observed image data and the current-iteration estimation of the tissue mixtures. Estimation of the tissue mixtures at next iteration is computed by maximizing the conditional expectation. The iterative EM approach to a MAP solution is achieved by a finite number of iterations and reasonable initial estimate. This MAP-EM framework provides a theoretical solution to the partial volume effect, which has been a major cause of quantitative imprecision in medical image processing. Numerical analysis demonstrated its potential to estimate tissue mixtures accurately and efficiently.