Modeling the population covariance matrices of block-iterative expectation-maximization reconstructed images

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Abstract
We have analytically derived expressions which, for high signal-to-noise ratio (SNR), approximate the population mean images and covariance matrices of both ordered-subset expectation-maximization (OS-EM) and rescaled block- iterative expectation-maximization (RBI-EM) reconstructed images, using a theoretical-formulation strategy similar to that previously outlined for maximum-likelihood expectation- maximization (ML-EM). The approximate population mean images and approximate population covariance matrices were calculated at various iteration numbers for the two reconstruction methods. The theoretical formulations were verified by calculating the sample mean images and sample covariance matrices for the two reconstruction methods, at the same iteration numbers, using over 8000 noisy images per method. Subsequently, we compared the approximate population and sample mean images, the approximate population and sample variance images, as well as the approximate population and sample local covariance images for a pixel near the center of a uniformly emitting disk object, for each iteration number and reconstruction method, respectively. The results demonstrated that for each method iteration number, the image produced by reconstructing from noise-free data would be equal to the population mean image to a very close approximation. In addition, the theoretically calculated variance and local covariance images closely matched their respective sample counterparts. Thus the theoretical formulation is an accurate way to predict the population first- and second-order statistics of both OS-EM and RBI-EM reconstructed images, for high SNR.

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