A discrete-continuous Bayesian model for Emission Tomography

In this contribution, we propose a discrete-continuous reconstruction method for Positron Emission Tomography (PET). The goal is to reconstruct a continuous radiotracer activity distribution from a finite set of measurements (namely, the discrete projections of detected random emissions). Our approach can be viewed as an indirect density estimation problem, i.e, the problem of recovering a probability density function based on indirect observations. We cast the reconstruction problem in a Bayesian nonparametric estimation framework where regularization of the ill-posed inverse problem is achieved by putting a prior on the investigated radiotracer activity distribution. We propose a hierarchical model and use it for the MCMC schemes to generate samples from the posterior activity distribution and compute its functionals (mean, standard deviation etc.). Results will illustrate the performances of the proposed method and we compare our approach to another Bayesian method, the maximum a posteriori estimation (MAP), which is based on a fully discrete-discrete problem formulation.