Statistical modeling and reconstruction of randoms pre-corrected PET data

Randoms pre-corrected PET data is formed as the difference of two Poisson random variables. The exact probability mass function (PMF) is not suitable for use in likelihood-based iterative image reconstruction as it contains an infinite summation. The shifted Poisson model is a tractable approximation to this PMF but requires that negative values are discarded, resulting in positively biased reconstructions in low count studies. Here we analyze the properties of the exact PMF and propose a new simple but accurate approximation that allows negative valued data. We investigate the properties of this approximation and demonstrate its application to penalized maximum likelihood image reconstruction