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 the attenuation factors, causing the diagonal of the Hessian to span over 5 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 that is 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 evaluated our method on the PET Rapid Image reconstruction Challenge dataset that includes a wide range of phantoms, scanner models, and count levels. We also varied the regularization strengths. Our preconditioner was implemented in a conjugate gradient descent algorithm without subsets or stochastic acceleration.

We show that the proposed preconditioner consistently 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.

These results demonstrate that decomposing the Hessian into diagonal and circulant components is an effective strategy for accelerating MAP PET reconstruction. The proposed preconditioner significantly improves convergence speed in challenging, ill-conditioned Poisson PET inverse problems.