arXiv:2608.29158v2 Announce Type: replace Abstract: Image reconstruction for positron emission tomography (PET) is an ill-posed Poisson inverse problem. In this work, we introduce an unsupervised, optimization-based reconstruction framework that employs a flow-matching generative model as a learned manifold prior. We train the generative model on high-quality PET images to learn a differentiable ordinary differential equation (ODE)-based transport from a Gaussian latent distribution to the PET image distribution. We incorporate this generator as an explicit manifold constraint into a regularized Poisson likelihood formulation. To solve the resulting problem, we develop an alternating direction method of multipliers (ADMM) algorithm that integrates an expectation-maximization (EM)-type surrogate update for Poisson data fidelity with gradient-based latent-space optimization. Numerical simulations and in vivo PET experiments demonstrate improved reconstruction accuracy compared with conventional reconstruction methods and recent deep learning-based approaches, particularly in low-count settings, with competitive computational efficiency.