Impact of PSF modelling on the convergence rate and edge behaviour of EM images in PET

EM reconstructions with point-spread-function (PSF) modelling is performed to increase the spatial resolution in PET images. These images exhibit slower initial convergence compared to reconstructions without PSF modelling. Furthermore, they exhibit more pronounced ringing around the edges of sharp features. We investigate the effect of different objects and PSF modelling on the convergence rate and edge behaviour of the EM algorithm in two stages: (i) at the initial iterations where the updates are large and (ii) at the later iterations where the updates are small. For the initial iterations, we compare the sharpness of the EM updates with and without PSF modelling. We show via simulations that the PSF modelling during the backprojection step causes smoother updates and consequently smoother images in the early stages of the EM algorithm. For the later iterations, we approximate the image as the ML image plus a perturbation term and develop an approximate update equation for the perturbation, which depends on the Hessian (H) of the log-likelihood. Based on this equation and the spectral analysis of H, we demonstrate how edges with ringing are preserved in the later stages of the algorithm and eliminated only for the case of noiseless data reconstruction with an unrealistically high number of iterations. In addition, we provide an intuitive explanation for the creation of the edge artefacts in terms of the PSF modelling during the backprojection step.