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

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

Problem Statement
Low-count Positron Emission Tomography (PET) reconstruction faces a critical challenge: the faithful recovery of weak, diagnostically vital lesion signals amidst high noise levels. Current deep learning paradigms struggle with a fundamental geometric limitation. Deterministic approaches (e.g., regression-based CNNs) suffer from "regression-to-the-mean," over-smoothing high-frequency textures and suppressing subtle lesions. Conversely, stochastic generative models (e.g., Diffusion Models, SDEs) rely on dissipative dynamics. These models induce a strictly negative divergence ($\nabla \cdot v < 0$) in the phase space, causing a continuous contraction of volume. In low-count regimes, this contraction is indiscriminate; it fails to distinguish between stochastic noise and sparse pathological features, leading to the numerical extinction or "wash-out" of weak lesion signals before coherent structures can form.

Methodology
The authors propose FlowPET, a physics-informed framework that reformulates PET reconstruction as a volume-preserving transport problem within a Symplectic Phase Space.

1. Symplectic Phase Space Lift:
   Instead of operating on the standard image manifold, FlowPET lifts the reconstruction to an augmented phase space $Z = X \times P$, where $X$ represents the image position and $P$ is an auxiliary momentum space. The dynamics are governed by a Separable Hamiltonian System:
   $$H_\theta(t, x, p; y) = U_\psi(t, x; y) + K_\phi(t, p; y)$$
   where $U_\psi$ is a potential term (dependent only on position) and $K_\phi$ is a kinetic term (dependent only on momentum). By Liouville's Theorem, this structure guarantees a divergence-free vector field ($\nabla \cdot v = 0$), ensuring that the phase-space volume occupied by weak signals is invariant and not artificially contracted.

2. Physics-Informed Conjugate Boundaries:
   To address the ill-posed nature of the inverse problem without relying on contraction for denoising, FlowPET introduces a Range-Null Space Decomposition strategy for the boundary conditions of the flow:

   • Range-Space Momentum ($p_0$): At the source boundary, momentum is initialized to encode the gradient of data consistency (the "restoring force"). This pulls the trajectory toward the manifold of solutions consistent with the measured sinogram.
   • Null-Space Momentum ($p_1$): At the target boundary, stochastic uncertainty is injected strictly into the orthogonal null space of the PET operator. This confines the generation of high-frequency textures to the unobserved subspace, preventing the hallucination of artifacts that violate physical constraints.

3. Symplectic Flow Matching & Inference:
   The model is trained via Symplectic Flow Matching (SFM), minimizing the regression loss between the learned vector field and the target velocity defined by the linear interpolation of the boundary states. For inference, the authors employ a Symplectic Leapfrog (Störmer–Verlet) integrator. Unlike standard solvers (e.g., Runge-Kutta) that introduce numerical dissipation, the Leapfrog integrator preserves the unit Jacobian determinant, ensuring exact volume preservation and lossless transport of the injected null-space uncertainty to the final reconstruction.

Key Contributions

• Symplectic Generative Framework: FlowPET utilizes separable Hamiltonian dynamics to ensure divergence-free transport, theoretically preventing the "signal wash-out" pervasive in dissipative diffusion models by maintaining phase-space volume invariance.
• Physics-Informed Orthogonality: The introduction of a Range-Null space decomposition strategy rigorously enforces data consistency in the range space while confining stochastic exploration to the null space, resolving the trade-off between fidelity and diversity.
• Structure-Preserving Inference: The implementation of a symplectic Leapfrog integrator demonstrates that strict geometric conservation yields superior preservation of low-contrast lesions compared to state-of-the-art baselines.

Experimental Results
FlowPET was evaluated on three datasets: BrainWeb (20% count), an In-House pediatric dataset (1% count), and UDPET Brain (1% count).

• Quantitative Performance: FlowPET achieved state-of-the-art results across all metrics (SSIM, PSNR, RMSE). Notably, it surpassed both deterministic methods and stochastic baselines (e.g., IR-SDE, DREAM, FourierPET) in SSIM and PSNR while maintaining lower RMSE, indicating superior texture recovery without noise artifacts.
• Qualitative Analysis: Visual comparisons on pediatric scans revealed that dissipative baselines exhibited "signal wash-out" in weak lesion regions. FlowPET successfully recovered these low-contrast structures, showing structural fidelity closest to the full-count reference.
• Signal Recovery Analysis: A trajectory analysis tracking Standard Uptake Value (SUV) ratios demonstrated that while dissipative solvers caused premature signal saturation and collapse, FlowPET exhibited a sustained, monotonic recovery toward the ground truth, confirming the protection of lesion signal mass against numerical extinction.

Significance
The paper claims that FlowPET offers a robust geometric safeguard for medical inverse problems in high-noise regimes. By shifting from dissipative contraction to conservative symplectic transport, the method addresses the specific failure mode of current generative models where weak but critical diagnostic signals are lost. The work establishes that imposing Hamiltonian structural constraints is not merely a tool for physical simulation but a necessary mechanism for information conservation in ill-posed inverse problems, ensuring that probability mass is transported rather than extinguished.