Transmission-less attenuation correction in time-of-flight PET: analysis of a discrete iterative algorithm

The maximum likelihood attenuation correction factors (MLACF) algorithm has been developed to calculate the maximum-likelihood estimate of the activity image and the attenuation sinogram in time-of-flight (TOF) positron emission tomography, using only emission data without prior information on the attenuation. We consider the case of a Poisson model of the data, in the absence of scatter or random background. In this case the maximization with respect to the attenuation factors can be achieved in a closed form and the MLACF algorithm works by updating the activity. Despite promising numerical results, the convergence of this algorithm has not been analysed. In this paper we derive the algorithm and demonstrate that the MLACF algorithm monotonically increases the likelihood, is asymptotically regular, and that the limit points of the iteration are stationary points of the likelihood. Because the problem is not convex, however, the limit points might be saddle points or local maxima. To obtain some empirical insight into the latter question, we present data obtained by applying MLACF to 2D simulated TOF data, using a large number of iterations and different initializations.