sMLACF: a generalized expectation-maximization algorithm for TOF-PET to reconstruct the activity and attenuation simultaneously

The 'simultaneous maximum-likelihood attenuation correction factors' (sMLACF) algorithm presented here, is an iterative algorithm to calculate the maximum-likelihood estimate of the activity λ and the attenuation factors a in time-of-flight positron emission tomography, and this from emission data only. Hence sMLACF is an alternative to the MLACF algorithm. sMLACF is derived using the generalized expectation-maximization principle by introducing an appropriate set of complete data. The resulting iteration step yields a simultaneous update of λ and a which, in addition, enforces in a natural way the constraints where is a fixed lower bound that ensures the boundedness of the reconstructed activities. Some properties—like the monotonic increase of the likelihood and the asymptotic regularity of the estimated —of sMLACF are proven. Comparison of sMLACF with MLACF for two data sets reveals that both algorithms show very similar results, although sMLACF converges slower.