Discrete iterative algorithms for scatter-to-attenuation reconstruction in PET

Abstract

Recently, several groups have proposed the use of scattered coincidences in positron emission tomography (PET), aiming at improved attenuation correction using the PET emission data, e.g., in PET-MRI. In this work, we analyzed the behavior of several algorithms, including reconstruction by two-branch scatter-to-attenuation back-projection (BP) and maximum likelihood expectation maximization with a one-step-late update of the system matrix (MLEM-OSL). A maximum-likelihood gradient-ascent (MLGA) approach, as previously proposed by us, was tested with four step sizes and several stabilization and acceleration techniques (Armijo step size rule, conjugate gradients, Nesterov acceleration, and subsets). The convergence speed of all algorithms was compared using phantom simulations in fields of view (FOVs) ranging from rat-sized to human-sized. For MLEM-OSL, based on a numerical criterion distinguishing low- and high-attenuation surfaces of response (SOR), the most useful (low-attenuation) SORs were isolated in order to improve convergence speed. We found that the Armijo step size rule improved convergence speed and enabled the use of conjugate gradients, further improving convergence rates. Alternatively, the use of data subsets yielded near-ideal speed-up of MLGA. Even with identical geometries (up to a spatial scale factor), performance of all algorithms depends on the FOV size, suggesting a new kind of scale problem. In particular, shortcomings of MLEM-OSL prevent convergence to the true solution in large FOVs, where MLGA behaves more favorably. Convergence rates of MLEM-OSL were improved by removing high-attenuation SORs, indicating that, opposing intuition, MLEM-OSL convergence can be improved by using less data.