The importance of preconditioners in fast Poisson-based iterative reconstruction algorithms for SPECT

We study the effects of different preconditioners on Poisson-based iterative reconstruction algorithms. Preconditioners are linear transformations that map the image solution space for the reconstruction problem into a space where the likelihood function can be more efficiently optimized. We apply preconditioners to conjugate gradient (CG) algorithms seeking to optimize the Poisson log likelihood function for SPECT. We show that, without a preconditioner, such algorithms may converge more slowly than the ML-EM algorithm. Previous research has applied a preconditioner that depends on the current iteration's image estimate. However, these algorithms do not generate conjugate step directions and do not obtain the full benefit of the CG algorithm's speed. We propose a preconditioner that depends only on the measured projection data and remains constant with each iteration, thus generating nearly conjugate step directions. We show that our method optimizes the log likelihood function more efficiently than the previously proposed methods. We also show that, if the measured projection data contains few zero or near-zero projection bins, the Poisson CG algorithms have convergence rates comparable with those from weighted least-squares (WLS-CG) algorithms. We conclude that the performance of Poisson CG algorithms depends heavily on the preconditioner chosen, and that they can be made competitive with WLS-CG by manipulation of the preconditioners.