Co-fan-sum ratio algorithm for randoms smoothing and detector normalization in PET

We describe a new exact analytical algorithm for estimating the single event rates in detectors from random coincidence data (for randoms variance reduction), and detector efficiencies from true coincidence data (for normalization), in positron emission tomography (PET). The estimates are derived from the ratios of the co-fan sums of coincidence events between individual detectors. The co-fan for any two detectors is defined as the set of co-detectors they have in common among the measured lines of response. The estimates are unbiased and have noise properties similar to fan-sum estimates. The detector efficiency estimation algorithm additionally employs the concept of a mask-restricted co-fan sum that can be easily adapted to a known source distribution, and thus does not require uniform illumination of all lines of response, or a centered uniform cylindrical source, for accurate estimation. The algorithm is simple to implement and can be applied in 2D or 3D. Some examples are given.