Bayesian PET Reconstruction with Learned Normalizing Flow Priors

Image reconstruction for positron emission tomography (PET) is an ill-posed Poisson inverse problem. While traditional methods like MLEM and MAP-TV suffer from noise or artifacts, deep learning approaches can be data-hungry and lack interpretability. We propose a novel Bayesian framework using a pre-trained Normalizing Flow (NF) as a powerful image prior. We reformulate the problem as posterior sampling in the NF's simple, Gaussian latent space, using the Metropolis-Adjusted Langevin Algorithm for efficient inference. This latent-space approach combines an expressive learned prior with rigorous Bayesian inference, enabling robust uncertainty quantification. By decoupling the prior from measurement physics, our unsupervised method adapts to varying noise levels and forward models without retraining. Experiments show our method outperforms baselines, yielding superior image quality and detail preservation, especially in low-count scenarios.