Generalizable Preconditioning Strategies for MAP PET Reconstruction Using Poisson Likelihood

The Positron Emission Tomography (PET) problem with Poisson log-likelihood is notoriously ill-conditioned. This stems from its dependence on the inverse of the measured counts and the square of attenuation factors, causing the diagonal of the Hessian to span over five orders of magnitude. Optimization is therefore slow, motivating decades of research into acceleration techniques. In this paper, we propose a novel preconditioner tailored for Maximum a Posteriori (MAP) PET reconstruction priors designed to achieve approximately uniform spatial resolution. Our approach decomposes the Hessian into two components: one diagonal and one circulant. The diagonal term is the Hessian expectation computed in an initial solution estimate. As the circulant term we use an apodized 2D ramp filter. We evaluate our method on the PETRIC challenge dataset that includes a wide range of phantoms, scanner model, count levels. We also varied regularization strengths. Our preconditioner is implemented in a conjugate gradient descent algorithm without subsets nor stochastic acceleration. We show that it constantly achieves convergence in fewer than 10 full iterations—each consisting of just one forward and one backward projection. We also show that the circulant component, despite its crude 2D approximation, provides very meaningful acceleration beyond the diagonal-only case.