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 (ML) estimate of the activity 'λ' and the attenuation factors 'a' in time-of-flight (TOF) positron emission tomography (PET), and this from emission data only. Hence sMLACF is an alternative to the MLACF algorithm. sMLACF is derived using the generalized expectation-maximization (GEM) 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 'ε ≤ a ≤ 1' where '0 < ε < 1' 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 (a,λ)- 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.