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

This paper analyzes and compares image reconstruction methods based on prac tical approximations to the exact log likelihood of randoms-precorrected po sitron emission tomography (PET) measurements. The methods apply to both em ission and transmission tomography, however, in this paper we focus on tran smission tomography. The results of experimental PET transmission scans and variance approximations demonstrate that the shifted Poisson (SP) method a voids the systematic bias of the conventional data-weighted least squares ( WLS) method and leads to significantly lower variance than conventional sta tistical methods based on the log likelihood of the ordinary Poisson (OP) m odel. We develop covariance approximations to analyze the propagation of no ise from attenuation maps into emission images via the attenuation correcti on factors (ACF's), Empirical pixel and region variances from real transmis sion data agree closely with the analytical predictions. Both the approxima tions and the empirical results show that the performance differences betwe en the OP model and SP model are even larger, when considering noise propag ation from the transmission images into the final emission images, than the differences in the attenuation maps themselves.