Iterative image reconstruction algorithms for computed tomography are able to incorporate highly accurate physical models for the measured data. While such algorithms provide a high degree of accuracy, their large computational cost currently makes them infeasible for clinical practice. Clinical scanners instead use a linear model that provides fast reconstruction times but limited accuracy in some situations. Using a variety of approaches, the iterative image reconstruction algorithms can be modified to run faster and more accurately. Although the algorithms based on linear models can also be made faster, they cannot provide the same level of accuracy. We describe a parallelized implementation of an alternating minimization algorithm for fully three-dimensional image reconstruction. Various performance results are shown for the reconstruction of simulation data, a phantom scan, and a large clinical scan.