A nonparametric bayesian approach for PET reconstruction

We introduce a PET reconstruction algorithm following a nonparametric Bayesian (NPB) approach. In contrast with expectation maximization (EM), the proposed technique does not rely on any space discretization. Namely, the activity distribution - normalized emission intensity of the spatial Poisson process - is considered as a spatial probability density and observations are the projections of random emissions whose distribution has to be estimated. This approach is nonparametric in the sense that the quantity of interest belongs to the set of probability measures on R^k (for reconstruction in k-dimensions) and it is Bayesian in the sense that we define a prior directly on this spatial measure and infer on the posterior distribution of the activity distribution. In this context, we propose to model the nonparametric probability density as an infinite mixture of multivariate normal distributions. As a prior for this mixture we consider a Dirichlet process mixture (DPM) with a normal-inverse wishart (NTW) model as base distribution of the Dirichlet process. As in EM-family reconstruction, we use a data augmentation scheme where the set of hidden variables are the emission locations in the continuous object space for each observed coincidence. Thanks to the data augmentation, we propose a Markov chain Monte Carlo (MCMC) algorithm (Gibbs sampler) which is able to generate draws from the posterior distribution of the spatial intensity. A difference with EM is that hidden variables involved in the Gibbs sampler correspond to generated emission locations while the number of emissions per pixel detected on a projection line is used for complete data in EM. Another key difference is that the estimated spatial intensity is a continuous function - such that there is no need to compute a projection matrix - while parameters in EM are given by the mean intensity per pixel. Finally, draws from the intensity posterior distribution allow the estimation of posterior function- als like the mean and variance or confidence intervals. The nonparametric behavior is characterized by an increase of DPM components (clusters) and consequently a resolution improvement with the number of recorded events. Results are presented for simulated data based on a 2D brain phantom and compared to ML-EM and Bayesian MAP-EM.