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

arXiv: https://arxiv.org/abs/2607.11104 · HTML: https://mlsanity.com/paper/2607.11104/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Medical imaging and segmentation, one of 14 directions in Diffusion models.
- Builds on 18 of its 41 references that are in the map.
- First posted on arXiv 13 Jul 2026.

## Placed in

- Diffusion models / [Medical imaging and segmentation](https://mlsanity.com/diffusion/medical-imaging-and-segmentation/)

## Builds on (most cited first, 18 of 18)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Image-to-Image Translation with Conditional Adversarial Networks](https://mlsanity.com/paper/1611.07004/) (2016)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [FiLM: Visual Reasoning with a General Conditioning Layer](https://mlsanity.com/paper/1709.07871/) (2017)
- [Image Super-Resolution via Iterative Refinement](https://mlsanity.com/paper/2104.07636/) (2021)
- [Image Restoration with Mean-Reverting Stochastic Differential Equations](https://mlsanity.com/paper/2301.11699/) (2023)
- [Hamiltonian Generative Networks](https://mlsanity.com/paper/1909.13789/) (2019)
- [DeepPET: A deep encoder-decoder network for directly solving the PET reconstruction inverse problem](https://mlsanity.com/paper/1804.07851/) (2018)
- [TransEM:Residual Swin-Transformer based regularized PET image reconstruction](https://mlsanity.com/paper/2205.04204/) (2022)
- [Likelihood-Scheduled Score-Based Generative Modeling for Fully 3D PET Image Reconstruction](https://mlsanity.com/paper/2412.04339/) (2024)
- [RED: Residual Estimation Diffusion for Low-Dose PET Sinogram Reconstruction](https://mlsanity.com/paper/2411.05354/) (2024)
- [MCAD: Multi-modal Conditioned Adversarial Diffusion Model for High-Quality PET Image Reconstruction](https://mlsanity.com/paper/2406.13150/) (2024)
- [Pseudo-MRI-Guided PET Image Reconstruction Method Based on a Diffusion Probabilistic Model](https://mlsanity.com/paper/2403.18139/) (2024)
- [Diffusion Transformer Meets Random Masks: An Advanced PET Reconstruction Framework](https://mlsanity.com/paper/2503.08339/) (2025)
- [Symplectic Generative Networks (SGNs): A Hamiltonian Framework for Invertible Deep Generative Modeling](https://mlsanity.com/paper/2505.22527/) (2025)
- [FourierPET: Deep Fourier-based Unrolled Network for Low-count PET Reconstruction](https://mlsanity.com/paper/2601.11680/) (2026)

## Abstract (arXiv)

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 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 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.