Analysis and Control of the Accuracy and Convergence of the ML-EM Iteration

In inverse problems like tomography reconstruction we need to solve an over-determined linear system corrupted with noise. The ML-EM algorithm finds the solution for Poisson noise as the fixed point of iterating a forward projection and a non-linear back projection. In tomography we have several hundred million equations and unknowns. The elements of the huge matrix are high-dimensional integrals, which cannot be stored, but must be re-computed with Monte Carlo (MC) quadrature when needed. In this paper we address the problems of how the quadrature error affects the accuracy of the reconstruction, whether it is possible to modify the back projection to speed up convergence without compromising the accuracy, and whether we should always take the same MC estimate or modify it in every projection.