Noise properties of the EM algorithm. I. Theory

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Abstract
The expectation-maximization (EM) algorithm is an important tool for maximum-likelihood (ML) estimation and image reconstruction, especially in medical imaging. The authors present a detailed treatment of the algorithm's statistical properties. The specific application they have in mind 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. They authors 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 expression is derived for the variance of the grey level and for pixel-to-pixel covariance. The variance increases rapidly with iteration number at first, but eventually saturates as the ML estimate is approached. Moreover, the variance at any iteration number has a factor proportional to the square of the mean image (though other factors may also depend on the mean image), so a map of the standard deviation resembles the object itself. Thus low-intensity regions of the image tend to have low noise. By contrast, linear reconstruction methods, such as filtered back-projection in tomography, show a much more global noise pattern, 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 pt.II the authors demonstrate through Monte Carlo simulation that the approximations are justified over a wide range of conditions in emission tomography. The theory can, therefore, be used as a basis for calculation of objective figures of merit for image quality.