Positron emission tomography by Markov chain Monte Carlo with auxiliary variables

We propose a new algorithm for positron emission tomography (PET) image reconstruction. The algorithm belongs to the family of Markov chain Monte Carlo methods with auxiliary variables. The idea is to iteratively generate hidden variables at one step and use them for image restoration at another step. The well-known model of Vardi et al. (J. Amer. Statist. Assoc. 80 (1985) 8) for PET is combined with the Bayesian model of Lasota and Niemiro (Pattern Recognition 36 (2003) 931) for the underlying images. This latter model takes advantage of the fact that medical images often consist of relatively few grey-levels of unknown intensity. The algorithm of Lasota and Niemiro (Pattern Recognition 36 (2003) 931) is used in the image restoration part of the PET algorithm, essentially as a noise-filtering and smoothing device. It is now equipped with an additional data reconstruction step. We include simulation results which suggest that the method is truly reliable. We also describe a version of the basic algorithm, in which a random simulation step is replaced by computation of expected value, similarly as in the EM algorithm.