NOISE PROPERTIES OF THE EM ALGORITHM .1. THEORY

The expectation-maximization (EM) algorithm is an important tool for m aximum-likelihood (ML) estimation and image reconstruction, especially in medical imaging. It is a nonlinear iterative algorithm that attemp ts to find the ML estimate of the object that produced a data set. The convergence of the algorithm and other deterministic properties are w ell established, but relatively little is known about how noise in the data influences noise in the final reconstructed image. In this paper we present a detailed treatment of these statistical properties. The specific application we have in mind is image reconstruction in emissi on tomography, but the results are valid for any application of the EM algorithm in which the data set can be described by Poisson statistic s. We show that the probability density function for the grey level at a pixel in the image is well approximated by a log-normal law. An exp ression is derived for the variance of the grey level and for pixel-to -pixel covariance. The variance increases rapidly with iteration numbe r at first, but eventually saturates as the ML estimate is approached. Moreover, the variance at any iteration number has a factor proportio nal to the square of the mean image (though other factors may also dep end on the mean image), so a map of the standard deviation resembles t he object itself. Thus low-intensity regions of die image tend to have low noise. By contrast, linear reconstruction methods, such as filter ed back-projection in tomography, show a much more global noise patter n, with high-intensity regions of the object contributing to noise at rather distant low-intensity regions. The theoretical results of this paper depend on two approximations, but in the second paper in this se ries we demonstrate through Monte Carlo simulation that the approximat ions are justified over a wide range of conditions in emission tomogra phy. The theory can, therefore, be used as a basis for calculation of objective figures of merit for image quality.