Statistical image reconstruction methods for randoms-precorrected PET scans

Positron emission tomography (PET) measurements are usually precorrected for accidental coincidence events by real-time subtraction of the delayed-window coincidences. Randoms subtraction compensates on average for accidental coincidences but destroys the Poisson statistics. We propose and analyze two new approximations to the exact log-likelihood of the precorrected measurements, one based on a 'shifted Poisson' model, the other based on saddle-point approximations to the measurement of probability mass function (PMF). The methods apply to both emission and transmission tomography; however, in this paper we focus on transmission tomography. We compare the new models to conventional data-weighted least-squares (WLS) and conventional maximum-likelihood methods [based on the ordinary Poisson (OP) model] using simulations and analytic approximations. The results demonstrate that the proposed methods avoid the systematic bias of the WLS method, and lead to significantly lower variance than the conventional OP method. The saddle-point method provides a more accurate approximation to the exact log-likelihood than the WLS, OP and shifted Poisson alternatives. However, the simpler shifted Poisson method yielded comparable bias-variance performance to the saddle-point method in the simulations. The new methods offer improved image reconstruction in PET through more realistic statistical modeling, yet with negligible increase in computation time over the conventional OP method.