FlowPET: Physics-Informed Symplectic Flow Matching for Low-Count PET Reconstruction

2026-07-13

AI Summary

This work addresses the challenge of weak lesion signal loss in low-count PET reconstruction, a common failure mode of existing generative models due to phase-space contraction. The authors propose the first integration of symplectic geometry and Hamiltonian dynamics into PET image reconstruction, modeling posterior evolution as a volume-preserving transport process in symplectic phase space. By parameterizing divergence-free vector fields through separable Hamiltonian systems and imposing conjugate boundary conditions derived from the range–nullspace decomposition of the PET operator, randomness is injected exclusively into the nullspace to preserve data consistency. This geometric approach inherently prevents probability mass collapse of weak signals, offering structural guarantees for highly ill-posed medical inverse problems. Experiments on BrainWeb, clinical pediatric, and UDPET datasets demonstrate superior SSIM and PSNR over state-of-the-art methods, with notably enhanced recovery of low-contrast lesions.

Abstract

Low-count Positron Emission Tomography (PET) reconstruction is severely hindered by the dissipative nature of prevailing generative models, where the inherent phase-space contraction leads to the numerical extinction (``wash-out'') of weak but diagnostically critical lesion signals. To overcome this geometric limitation, we propose \textbf{FlowPET}, a physics-informed framework that reformulates reconstruction as volume-preserving transport in a symplectic phase space. By parameterizing the posterior dynamics via a Separable Hamiltonian System, our approach guarantees a divergence-free vector field by construction, theoretically immunizing weak signals against probability mass collapse. To steer this conservative flow, we introduce conjugate boundary conditions based on the Range-Null space decomposition of the PET operator; this strictly enforces data consistency in the range space while confining stochastic uncertainty injection to the unobserved null space. We train the model via symplectic flow matching and perform inference using a symplectic leapfrog integrator. Extensive experiments on BrainWeb, clinical pediatric, and UDPET datasets demonstrate that \textbf{FlowPET} not only surpasses state-of-the-art deterministic and stochastic baselines in SSIM and PSNR but, more crucially, exhibits superior recovery of low-contrast lesions. The results confirm that imposing Hamiltonian structural constraints offers a robust geometric safeguard for medical inverse problems in high-noise regimes.

Problem

• Low-count PET reconstruction
• signal wash-out
• weak lesion recovery
• medical inverse problems
• phase-space contraction

Innovation

• Symplectic Flow Matching
• Hamiltonian System
• Volume-Preserving Transport
• Range-Null Space Decomposition
• Low-Count PET Reconstruction