A fast fully 4D incremental gradient reconstruction algorithm for list mode PET data

We present a fully four-dimensional, globally convergent, incremental gradient algorithm to estimate the continuous-time tracer density from list mode positron emission tomography (PET) data. The rate function in each voxel is modeled as an inhomogeneous Poisson process whose rate function can be reconstructed using a cubic B-spline basis. The rate functions are then estimated by maximizing the objective function formed by the sum of the likelihood of arrival times and spatial and temporal smoothness penalties. We first provide a computable bound for the norms of the optimal temporal basis function coefficients, and based on this bound we construct an incremental gradient algorithm that converges to the solution. Fully four-dimensional simulations demonstrate the convergence of the algorithm for a high count dataset on a 4-ring scanner.