Abstract

The expectation-maximization (EM) algorithm is a nonlinear iterative algorithm that attempts to find the ML estimate of the object that produced a data set. The convergence of the algorithm and other deterministic properties are well established, but relatively little is known about how noise in the data influences noise in the final processed image. In this paper we present a detailed treatment of these statistical properties. The specific application we choose for this study is image reconstruction in emission tomography, but the results are valid for any application of the EM algorithm in which the data set can be described by Poisson statistics. We show that the probability density function for the gray level at a pixel in the reconstructed image is well approximated by a log-normal law. An expression is derived for the mean and variance of the gray level and for pixel-to-pixel covariance. All of these quantities vary with iteration number and with the original object distribution in a manner predicted by the theory. We also report an extensive Monte Carlo study that validates the theory over a wide range of experimental conditions.

© 1993 Optical Society of America