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Manifold-Constrained PET Reconstruction with Learned Flow-Matching Priors

Hengjia Ran, Jie Luo, Yutao Zhu, Rui Hu, Huafeng Liu, Bo Zhao

arXiv:2608.29158v1eess.IV

TL;DR

PET reconstruction is an ill-posed Poisson inverse problem prone to noise amplification and artifacts, particularly in low-count regimes. The paper uses a flow-matching generator as a learned manifold constraint within an ADMM-based regularized Poisson optimization, combining EM data-consistency updates with latent-space projections. Across simulated and real PET experiments, it reports improved noise suppression, structural fidelity, quantitative accuracy, and robustness versus conventional and deep-learning baselines, while identifying nonconvexity and noisy training targets as limitations.

  • Problem

    PET reconstruction is an ill-posed Poisson inverse problem that can suffer from noise amplification and artifacts, especially in low-dose or short-acquisition settings.

  • Method

    A flow-matching generator learned from high-quality PET images is imposed as a differentiable manifold prior within regularized Poisson likelihood reconstruction, solved using ADMM with EM and latent-space updates.

  • Results

    The method consistently outperforms conventional, supervised, and unsupervised deep-learning baselines in noise suppression, structural fidelity, quantitative accuracy, and low-count robustness.

  • Takeaways & Limitations

    Flow matching can serve as an effective learned manifold prior that balances noise suppression, structural preservation, and quantitative accuracy in low-dose PET reconstruction.

  • Takeaways & Limitations

    The optimization is nonconvex without a complete theoretical convergence guarantee, and training on noisy high-quality reconstructions limits performance relative to true activity maps.

Abstract

from arXiv · show

Image reconstruction for positron emission tomography (PET) is an ill-posed Poisson inverse problem that often suffers from severe noise amplification and artifacts. 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 flow-matching model on high-quality PET images to learn a deterministic ordinary differential equation transport from a Gaussian latent distribution to the empirical PET image distribution, yielding a differentiable generator of anatomically plausible images. We incorporate this generator as an explicit manifold constraint into a regularized Poisson likelihood formulation. We solve the resulting optimization problem using an alternating direction method of multipliers algorithm, in which an expectation-maximization-type surrogate update enforces data consistency and a gradient-based latent-space projection enforces manifold proximity. We evaluate the proposed method on both simulated and real PET datasets, assessing dose-level robustness, lesion-insertion generalization, and cross-scanner transfer. Compared with conventional reconstruction methods and state-of-the-art deep learning baselines, the proposed method provides superior noise suppression, structural preservation, and quantitative accuracy, while maintaining high computational efficiency.

I. INTRODUCTION

PET reconstruction is challenged by an ill-posed, Poisson-noisy inverse problem, especially in low-dose settings. The paper addresses this with a flow-matching generator used as a differentiable learned manifold prior within a Poisson reconstruction framework.

  • Low-dose and short-acquisition PET protocols intensify the challenge of reconstructing high-quality images from noisy measurements.
  • Classical maximum-likelihood reconstruction can amplify noise and streak artifacts as iterations proceed, motivating stronger prior information.
  • Flow matching transports Gaussian samples to the empirical image distribution through a deterministic ODE, enabling differentiable generation with fewer evaluations than score-based diffusion.
  • The proposed framework constrains reconstruction to a flow-matching manifold while retaining Poisson data fidelity, using ADMM with EM-style and latent-space updates.
  • The method is designed to combine realistic anatomical priors with physical consistency for improved reconstruction accuracy, visual fidelity, and low-count robustness.

B. Solution Algorithm

The reconstruction is solved with an ADMM scheme that alternates Poisson data-consistency optimization and latent-space manifold projection. EM-style updates preserve nonnegativity, while L-BFGS handles the smooth nonlinear latent subproblem.

  • ADMM decomposes the constrained problem into an image-domain data-consistency subproblem, a latent projection subproblem, and a multiplier update.
  • The image-domain subproblem uses an EM-style majorization approach for the Poisson term while keeping the image near the current manifold prediction.
  • The EM-type closed-form update is applied element-wise and inherently preserves nonnegativity, allowing efficient inexact ADMM iterations.
  • The latent subproblem is a smooth nonlinear projection optimized with L-BFGS using gradients computed by automatic differentiation.
  • Warm starts improve convergence efficiency, while multi-start latent initialization can be used because the overall formulation is nonconvex.

A. Experimental Setting

The simulation study uses BrainWeb digital phantoms to evaluate reconstruction under synthetic FDG PET conditions, with lesions withheld from training for generalization testing.

  • The simulation uses 20 subject-realistic BrainWeb digital phantoms, split into 18 training cases, one validation case, and one independent test case.
  • Each volume receives three random deformation realizations to augment the simulated FDG PET data.
  • Tumor lesions are excluded from training and introduced only in the test case to assess generalization to unseen pathological uptake patterns.

1) Simulation Dataset:

The clinical dataset comprises full-dose FDG PET scans from 372 patients acquired on a Siemens Biograph Vision Quadra scanner, with standardized 360-second acquisitions.

  • The Siemens Biograph Vision Quadra dataset contains FDG PET scans from 372 patients acquired at the University of Bern.
  • Scans were acquired for 360 s, with mean administered activity of 219.2±50.9 MBq and reconstructed volumes of 440 × 440 × 645 voxels.
  • The reconstructed voxel dimensions are 1.65×1.65×5 mm3.

2) Clinical Datasets:

Clinical data comprise PET scans from two total-body scanner systems, with cross-scanner testing designed to assess robustness under distribution shift.

  • The uEXPLORER dataset includes 301 patients scanned for 360 seconds, with 360 × 360 × 673 images and 1.6667 × 1.6667 × 2.8860 mm3 voxels.
  • The Siemens Biograph Vision Quadra and uEXPLORER datasets differ in spatial resolution and reconstruction parameters, creating a clear distribution shift.
  • Axial head slices were used, and cross-scanner robustness was evaluated by training on the full uEXPLORER dataset and testing on Siemens data.
  • PET slices were centrally cropped to 128×128 to reduce GPU memory consumption and improve computational efficiency.
  • Projection-domain measurements used a Siemens Biograph Vision PET forward model implemented with parallelproj and matched the clinical list-mode reconstruction geometry.

3) PET Forward Model:

The PET forward model predicts expected projection data from activity, attenuation, normalization, and additive contamination terms, with low-dose data generated by event downsampling.

  • The expected projection data are modeled using a ParallelProj projector, CT-based attenuation, normalization factors, activity, and additive random and scatter contributions.
  • Low-dose sinograms were generated by downsampling events from full-dose coincidence measurements at preset fractions.
  • BrainWeb simulations used a corresponding 2D forward model that independently projects each axial slice.
  • Reconstruction performance was evaluated with PSNR, SSIM, NRMSE, CRC, and background STD.

4) Evaluation Metrics:

The evaluation combines image-quality, quantitative-accuracy, and noise metrics across reconstruction comparisons and dose levels.

  • SSIM measures similarity between reconstructed and ground-truth images through luminance, contrast, and structure.
  • NRMSE provides a normalized measure of reconstruction error magnitude.
  • CRC evaluates quantitative accuracy in defined regions of interest using target and background values relative to ground truth.
  • STD quantifies noise in background regions using mean values from background ROIs across realizations.
  • Figure 1 compares ML, TV, FBSEM, DIP, DPS, and the proposed method on 50% dose simulations using slices, relative-error maps, and NRMSE/PSNR/SSIM.
  • Figure 2 analyzes tumor CRC versus white-matter STD through convergence trajectories averaged over 10 independent realizations.

5) Comparison method:

The implementation uses a residual U-Net velocity field and evaluates convergence and parameter sensitivity under a computational PET reconstruction setup.

  • The velocity field uses a residual U-Net with a five-slice axial stack as input, four encoder stages, four decoder stages, and skip connections.
  • The network uses residual blocks with 3 × 3 convolution-BN-ReLU layers and 1 × 1 shortcut projections when channel dimensions change.
  • Figure 3 examines NRMSE sensitivity to ADMM and latent regularization parameters and convergence under varying EM-type and L-BFGS update counts.

B. Simulation study

The proposed method outperformed conventional and learning-based baselines across simulated and real PET evaluations, while remaining robust to unseen lesions, scanner mismatch, initialization, and solver settings. Its flow-matching ODE also enabled efficient reconstruction with few Euler steps.

  • Reconstruction quality: The proposed method achieved the best overall performance at 50% simulated dose, improving structure preservation and quantitative metrics over FBSEM and DPS.ML showed noise and streaking, TV oversmoothed fine anatomy, while learning-based baselines preserved more structure than traditional methods.
  • Generalization: The method achieved the highest CRC and lowest STD in lesion-insertion tests, demonstrating robustness to unseen lesions.Lesions were absent from training, and each method was evaluated over 10 realizations.
  • Optimization robustness: Stable final NRMSE values across ADMM, latent-regularization, EM-update, and L-BFGS settings indicate limited sensitivity to these optimization parameters.With EM=15 and L-BFGS=12, the algorithm reached a stable solution within 10 ADMM iterations.
  • Initialization: All initialization strategies reached the same converged likelihood values, indicating that the optimization was not strongly dependent on the starting point.The compared strategies included zero, Gaussian random, uniform random, and MLEM-based initialization.
  • Computational efficiency: NRMSE dropped from 16.5% to 7.49% with 10 Euler steps and remained essentially unchanged thereafter.Diffusion-based methods such as DPS typically require about 1000 steps for comparable performance.
  • Patient study: On real patient data, the method consistently achieved the lowest NRMSE, PSNR above 34 dB, and SSIM above 0.96 across three orthogonal views.It outperformed all baselines using a multi-slice training strategy with five consecutive slices and full 3D reconstruction.
  • Cross-scanner transfer: Under scanner mismatch, the method showed only a small degradation while outperforming DPS across slices in NRMSE, PSNR, and SSIM.The comparison used United Imaging training data and Siemens test data for the mismatch setting.

IV. DISCUSSION

The framework combines a learned flow-matching manifold prior with Poisson data fidelity through an optimization-based ADMM reconstruction scheme. Experiments indicate effective noise suppression, structural preservation, robustness, and computational practicality, while broader validation and theoretical limitations remain.

  • Method: The method balances noise suppression, structural preservation, and quantitative accuracy by constraining reconstructions to a learned PET manifold while enforcing Poisson data fidelity.The prior is learned from high-quality PET images rather than specified through a hand-crafted regularizer.
  • Method: Flow matching provides a differentiable, optimization-friendly generator that enables efficient latent-space manifold projection within ADMM.Its ODE-based transport supports deterministic generation, while L-BFGS performs manifold projection.
  • Robustness: The algorithm is relatively insensitive to EM updates, L-BFGS steps, ADMM penalty choices, and initialization, while lesion insertion suggests generalization to unseen local abnormalities.Lesion experiments also report low background noise despite abnormal uptake patterns.
  • Limitations: The current approach uses neighboring-slice information rather than a fully 3D flow-matching model, which could better capture volumetric correlations but would increase memory and computation.The neural generator also makes the optimization nonconvex, and complete theoretical convergence guarantees are outside the work’s scope.
  • Limitations: The optimization-based formulation produces a point estimate, whereas sampling-based Bayesian integration could provide uncertainty quantification at substantially longer computation times.This marks a trade-off between computational efficiency and uncertainty information.
  • Robustness: The proposed method consistently outperforms DPS across matched and cross-scanner settings, with smaller degradation under scanner mismatch.The cross-scanner evaluation compares Siemens-to-Siemens testing with United Imaging training followed by Siemens testing.
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