Provably convergent OSEM-like reconstruction algorithm for emission tomography

Abstract

We investigate a new, provably convergent OSEM-like (ordered-subsets expectation-maximization) reconstruction algorithm for emission tomography. The new algorithm, which we term C-OSEM (complete-data OSEM), can be shown to monotonically increase the log-likelihood at each iteration. The familiar ML-EM reconstruction algorithm for emission tomography can be derived in a novel way. One may write a single objective function with complete, incomplete data and the reconstruction variables as in the EM approach. But in the objective function approach, there is no E-step. Instead, a suitable alternating descent on the complete data and then the reconstruction variables results in two update equations that can be shown to be equivalent to the familiar EM algorithm. Hence, minimizing this objective becomes equivalent to maximizing the likelihood. We derive our C-OSEM algorithm by modifying the above approach to update the complete data only along ordered subsets. The resulting update equation is quite different from OSEM, but still retains the speed-enhancing feature of the updates due to the limited backprojection facilitated by the ordered subsets. Despite this modification, we are able to show that the objective function decreases at each iteration, and (given a few more mild assumptions regarding the number of fixed points) conclude that the C-OSEM algorithm provides a monotonic convergence toward the maximum likelihood solution. We simulated noisy and noiseless emission projection data, and reconstructed them using the ML-EM, and the proposed C-OSEM with 4 subsets. We also reconstruct the data using the OSEM method. Anecdotal results show that the C-OSEM algorithm is much faster than ML-EM though slower than OSEM.