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 [Formula: see text] where [Formula: see text] 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 [Formula: see text]-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.