Penalized-likelihood estimators and noise analysis for randoms-precorrected PET transmission scans

This paper analyzes and compares image reconstruction methods based on practical approximations to the exact log likelihood of randoms-precorrected positron emission tomography (PET) measurements. The methods apply to both emission and transmission tomography, however, in this paper we focus on transmission tomography. The results of experimental PET transmission scans and variance approximations demonstrate that the shifted Poisson (SP) method avoids the systematic bias of the conventional data-weighted least squares (WLS) method and leads to significantly lower variance than conventional statistical methods based on the log likelihood of the ordinary Poisson (OP) model. We develop covariance approximations to analyze the propagation of noise from attenuation maps into emission images via the attenuation correction factors (ACF's). Empirical pixel and region variances from real transmission data agree closely with the analytical predictions. Both the approximations and the empirical results show that the performance differences between the OP model and SP model are even larger, when considering noise propagation from the transmission images into the final emission images, than the differences in the attenuation maps themselves.