An Expectation Maximization Method for Joint Estimation of Emission Activity Distribution and Photon Attenuation Map in PET

A maximum likelihood expectation maximization (MLEM) method is proposed for joint estimation of emission activity distribution and photon attenuation map from positron emission tomography (PET) emission data alone. The method is appealing since: (i) it guarantees monotonic likelihood increase to a local extremum, (ii) does not require arbitrary parameters, and (iii) guarantees the positivity of the estimated distributions. Moreover, we propose a discrete Poisson data acquisition model and numerical algorithm for: (i) efficient graphics processing unit (GPU) based formulation, and (ii) a closed form exact solution for the MLEM update equations, which is essential for accurate and robust estimation. Numerical experiments indicate that in the presence of noise, joint EMAA estimation converges to the true emission activity distribution with root mean square errors of 4% and 0.5% respectively in estimation of lung-and myocardial emission activity distributions for a computational XCAT thorax phantom.