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PC-UNet: An Enforcing Poisson Statistics U-Net for Positron Emission Tomography Denoising

Yang Shi, Jingchao Wang, Liangsi Lu, Mingxuan Huang, Ruixin He, Yifeng Xie, Hanqian Liu, Minzhe Guo, Yangyang Liang, Weipeng Zhang, Zimeng Li, Xuhang Chen

TL;DR

PET denoising at low doses suffers from Poisson noise that degrades lesion detectability. The authors propose PC-UNet, a Poisson statistics–enforcing U-shaped network that couples a standard L1 fidelity term with PVMC-Loss, a physics-informed constraint derived from Poisson count statistics. PVMC-Loss uses patches to align the residual variance with a Poisson slope k via the ratio pi_p = Var_p(r) / ( k * Mean_p(y_hat) + epsilon ), and k can be learned for a given scanning protocol. Theoretical results establish asymptotic unbiasedness, gradient adaptivity, and a Generalized Method of Moments interpretation, while experiments on UDPET data show improved PSNR and SSIM and robust k convergence, highlighting improved physical consistency and image fidelity under low-dose constraints.

Abstract

Positron Emission Tomography (PET) is crucial in medicine, but its clinical use is limited due to high signal-to-noise ratio doses increasing radiation exposure. Lowering doses increases Poisson noise, which current denoising methods fail to handle, causing distortions and artifacts. We propose a Poisson Consistent U-Net (PC-UNet) model with a new Poisson Variance and Mean Consistency Loss (PVMC-Loss) that incorporates physical data to improve image fidelity. PVMC-Loss is statistically unbiased in variance and gradient adaptation, acting as a Generalized Method of Moments implementation, offering robustness to minor data mismatches. Tests on PET datasets show PC-UNet improves physical consistency and image fidelity, proving its ability to integrate physical information effectively.

PC-UNet: An Enforcing Poisson Statistics U-Net for Positron Emission Tomography Denoising

TL;DR

PET denoising at low doses suffers from Poisson noise that degrades lesion detectability. The authors propose PC-UNet, a Poisson statistics–enforcing U-shaped network that couples a standard L1 fidelity term with PVMC-Loss, a physics-informed constraint derived from Poisson count statistics. PVMC-Loss uses patches to align the residual variance with a Poisson slope k via the ratio pi_p = Var_p(r) / ( k * Mean_p(y_hat) + epsilon ), and k can be learned for a given scanning protocol. Theoretical results establish asymptotic unbiasedness, gradient adaptivity, and a Generalized Method of Moments interpretation, while experiments on UDPET data show improved PSNR and SSIM and robust k convergence, highlighting improved physical consistency and image fidelity under low-dose constraints.

Abstract

Positron Emission Tomography (PET) is crucial in medicine, but its clinical use is limited due to high signal-to-noise ratio doses increasing radiation exposure. Lowering doses increases Poisson noise, which current denoising methods fail to handle, causing distortions and artifacts. We propose a Poisson Consistent U-Net (PC-UNet) model with a new Poisson Variance and Mean Consistency Loss (PVMC-Loss) that incorporates physical data to improve image fidelity. PVMC-Loss is statistically unbiased in variance and gradient adaptation, acting as a Generalized Method of Moments implementation, offering robustness to minor data mismatches. Tests on PET datasets show PC-UNet improves physical consistency and image fidelity, proving its ability to integrate physical information effectively.
Paper Structure (18 sections, 3 theorems, 30 equations, 4 figures, 1 table)

This paper contains 18 sections, 3 theorems, 30 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Method
  3. PVMC-Loss
  4. Theoretical Analysis of PVMC-Loss
  5. Asymptotic Unbiasedness with Bounded Bias
  6. Gradient Structure and Adaptive Learning
  7. Interpretation as GMM
  8. PC-UNet
  9. Experiments
  10. Dataset
  11. Comparision Experiments
  12. Setting
  13. Evaluation Metrics
  14. Results
  15. Analysis of parameter $k$
  16. ...and 3 more sections

Key Result

Theorem 1

When $\mathcal{L}_{\mathrm{PVMC}} \to 0$, the expectation of the network output $\hat{y}$ satisfies: where $\mathbb{E}_{p \sim D}(\cdot)$ is defined as first calculating the variance within each patch and then taking the expectation of the variance values of all patches globally.

Figures (4)

  • Figure 1: (a) is a low-dose PET. For the small number of photons, the signal of (a) is completely overwhelmed by noise. (b) is a full-dose PET, we consider it as a clean image. (c) is obtained by denoising (a) through U-Net. The red rectangular area marked out is a region with relatively low photon count. This characteristic is manifested as lower brightness but clear structure in (b), while (c) appears as blurred structure and severe noise artifacts.
  • Figure 2: Framework of PC-UNet. We use the denoised image and the patch of the residual image, calculating the mean and variance to obtain our proposed PVMC-Loss.
  • Figure 3: The denoising results of different methods.
  • Figure 4: (a) $k$ converges as epochs increase; (b) ablation over 9 values of $\lambda$; (c) ablation over 6 patch sizes.

Theorems & Definitions (6)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof