Model-Based Deep Learning PET Image Reconstruction Using Forward–Backward Splitting Expectation–Maximization

We propose a forward-backward splitting algorithm to integrate deep learning into maximum-<i>a-posteriori</i> (MAP) positron emission tomography (PET) image reconstruction. The MAP reconstruction is split into regularization, expectation-maximization (EM), and a weighted fusion. For regularization, the use of either a Bowsher prior (using Markov-random fields) or a residual learning unit (using convolutional-neural networks) were considered. For the latter, our proposed forward-backward splitting EM (FBSEM), accelerated with ordered subsets (OS), was unrolled into a recurrent-neural network in which network parameters (including regularization strength) are shared across all states and learned during PET reconstruction. Our network was trained and evaluated using PET-only (FBSEM-p) and PET-MR (FBSEM-pm) datasets for low-dose simulations and short-duration <i>in-vivo</i> brain imaging. It was compared to OSEM, Bowsher MAPEM, and a post-reconstruction U-Net denoising trained on the same PET-only (Unet-p) or PET-MR (Unet-pm) datasets. For simulations, FBSEM-p(m) and Unet-p(m) nets achieved a comparable performance, on average, 14.4% and 13.4% normalized root-mean square error (NRMSE), respectively; and both outperformed OSEM and MAPEM methods (with 20.7% and 17.7% NRMSE, respectively). For <i>in-vivo</i> datasets, FBSEM-p(m), Unet-p(m), MAPEM, and OSEM methods achieved average root-sum-of-squared errors of 3.9%, 5.7%, 5.9%, and 7.8% in different brain regions, respectively. In conclusion, the studied U-Net denoising method achieved a comparable performance to a representative implementation of the FBSEM net.