Continuous space-time reconstruction in 4D PET

The aim of this work is to propose a method for reconstructing space-time 4D PET images directly from the data without any discretization, neither in space nor in time. To accomplish this, we cast the reconstruction problem in the context of Bayesian nonparametrics (BNP). The 4D activity distribution is viewed as an entire probability density on R-3 x R+ and inferred directly. The regularization of the inverse problem is done in the Bayesian framework. We put a prior on the random probability measure of interest and compute its posterior. The random activity distribution is modeled as a dependent Dirichlet process mixture (DPM). By assuming independence between space and time random distributions in each component of the mixture for brain functional imaging, we use a Normal-Inverse Wishart (NIW) model as base distribution for the marginalized spatial Dirichlet process. The time dependency is taken into account through a nested DPM of Polya Trees. The resulting hierarchical nonparametric model allows inference on the so-called functional volumes which define regions of brain whose activity follows a particular kinetic. A challenging task is to tackle the infinite distributions without truncation of models. We approximate the targeted posterior distribution of the space-time distribution with a Markov Chain Monte-Carlo (MCMC) inference scheme for which we make use of a particular update formula combined with a strategy called slice sampling that allows to deal with a finite number of components at each sweep of the sampler. The Bayesian nature of the proposed method gives access to posterior uncertainty. This ability will be used to explore the behavior of the reconstruction algorithm in a situation of low injected doses. To assess our results in this context, we furnish a statistical validation based on synthetic replicates in 3D. An application to space-time PET reconstruction is presented for simulated data from a 4D digital phantom and preliminary results on real data are provided.