Numerical Algorithms for Scatter-to-Attenuation Reconstruction in PET: Empirical Comparison of Convergence, Acceleration, and the Effect of Subsets

The use of scattered coincidences for the attenuation correction of positron emission tomography data has recently been proposed. For practical applications, convergence speeds require further improvement, yet there exists a tradeoff between convergence speed and the risk of nonconvergence. In this respect, a maximum-likelihood gradient-ascent (MLGA) algorithm and a two-branch backprojection (2BP), which was previously proposed, were evaluated. MLGA was combined with the Armijo step-size rule, and accelerated using conjugate gradients, Nesterov's momentum method, and data subsets of different sizes. In 2BP, we varied the subset size, an important determinant of convergence speed and computational burden. We used three sets of simulation data to evaluate the impact of a spatial scale factor. The Armijo step size allowed tenfold increased step sizes compared with native MLGA. Conjugate gradients and Nesterov momentum lead to slightly faster, yet nonuniform convergence; improvements were mostly confined to later iterations, possibly due to the nonlinearity of the problem. MLGA with data subsets achieved faster, uniform, and predictable convergence, with a speedup factor equivalent to the number of subsets and no increase in computational burden. By contrast, 2BP computational burden increased linearly with the number of subsets due to repeated evaluation of the objective function, and convergence was limited to the case of many (and, therefore, small) subsets, which resulted in high computational burden. Possibilities of improving 2BP appear limited. While general-purpose acceleration methods appear insufficient for MLGA, results suggest that data subsets are a promising way of improving MLGA performance.