Rethinking Cross-Dose PET Denoising: Mitigating Averaging Effects via Residual Noise Learning

1. Problem Statement

Positron Emission Tomography (PET) is critical for oncology and disease monitoring, but reducing radiation dose leads to severe Poisson noise, contrast degradation, and quantification bias. While deep learning (DL) has shown promise in denoising low-dose PET (LDPET), a significant challenge remains: generalization across varying dose levels.

• The "One-Size-Fits-All" Limitation: Models trained on a single noise level fail to generalize to other dose conditions due to domain shifts.
• The Averaging Effect: Attempts to train a unified model on multi-dose data (cross-dose training) often result in suboptimal performance. The paper argues that standard supervised training formulations implicitly optimize an expectation loss over heterogeneous noise distributions. Consequently, the model learns an "averaged" representation of the noise, leading to over-smoothing, loss of fine structural details, and degraded quantitative accuracy (e.g., SUV bias).
• Existing Solutions' Drawbacks: Previous approaches like the Unified Noise-Aware Network (UNN) require training and storing multiple specialized sub-networks (computationally expensive), while Domain Generalization (CADG) approaches may inadvertently degrade performance by enforcing domain invariance where dose and label are entangled.

2. Methodology

The authors propose a Unified Residual Noise Learning Framework that shifts the learning objective from predicting the full-dose image to estimating the residual noise directly.

A. Theoretical Formulation

• Standard Approach (Flawed): In a standard "one-size-fits-all" model, the network $f_\theta$ predicts the full-dose image $y$ from low-dose input $x$. The optimization minimizes the expected loss over all dose levels $j$:
  $$\theta^* = \arg \min_\theta E_{x}[E_{y|x}[L(f_\theta(x), y)]]$$
  This forces the model to learn an average of the noise distributions, causing the "averaging effect."
• Proposed Approach: The model is redefined to predict the noise component $n = y - x$ rather than $y$. The formulation becomes:
  $$\theta^* = \arg \min_\theta E_{x}[E_{n|x}[L(f_\theta(x), n)]]$$
  By conditioning the noise expectation strictly on the specific low-dose image, the model learns the specific noise distribution for that input without averaging across different dose levels.

B. Network Architecture & Activation Functions

• Base Architecture: The authors utilize a U-Net architecture.
• Handling Negative Noise: A critical technical innovation addresses the fact that residual noise ($y - x$) can contain negative values. Standard ReLU activations (used in DnCNN) suppress negative values, leading to information loss.
  • Training: The authors replace ReLU with LeakyReLU (slope 0.01) throughout the network to allow negative noise components to be learned.
  • Output Constraint: To ensure the final denoised image ($x + \text{predicted noise}$) remains non-negative (physically valid for PET), a LeakyReLU with a shallow slope is applied to the final output during training.
  • Inference: Standard ReLU is used for the final output to strictly enforce non-negativity.
• Loss Function: The training objective combines Mean Absolute Error (MAE) and Structural Similarity Index (SSIM) loss:
  $$L = L_{MAE} + \lambda L_{SSIM}$$
  where $\lambda = 0.5$. This ensures both pixel-level accuracy and structural preservation.

3. Key Contributions

1. Theoretical Insight: The paper explicitly identifies and mathematically demonstrates that cross-dose training in standard formulations leads to an "averaging effect" where the model minimizes an expectation over heterogeneous noise distributions, degrading performance.
2. Residual Noise Learning Framework: Proposes a unified model that learns noise directly rather than the full image, effectively decoupling the model from the need for dose-specific metadata during inference.
3. Activation Function Adaptation: Introduces the use of LeakyReLU to handle negative residual noise values, a critical modification for PET denoising that standard CNNs often overlook.
4. Efficiency: Unlike UNN or ST-UNN, which require multiple sub-networks or complex weighting mechanisms, this approach uses a single, streamlined U-Net with the same parameter count as individual dose-specific models.

4. Experimental Results

The method was validated on two large-scale, multi-dose clinical datasets:

• Dataset 1: University of Bern (Siemens Biograph Vision Quadra), 209 subjects.
• Dataset 2: Shanghai Ruijin Hospital (United Imaging uExplorer), 320 subjects.
• Conditions: Simulated dose levels ranging from 1% to 50% of the full dose.

Quantitative Performance (PSNR, SSIM, RMSE):

• Superiority: The proposed method consistently outperformed "one-size-fits-all" models, individually trained U-Nets, and UNN-based approaches across almost all dose levels.
• Averaging Effect Validation: The "one-size-fits-all" baseline showed a severe drop in SSIM (e.g., ~0.50 vs. ~0.96 for the proposed method), confirming the averaging hypothesis.
• Robustness: The proposed method showed lower standard deviation in metrics across patient cohorts compared to individually trained models, which exhibited instability at intermediate dose levels (10-25%).
• No Metadata Required: Unlike dose-embedded models, the proposed method achieved superior results without requiring dose level information as an input during inference.

Qualitative Performance:

• Visual comparisons showed that the proposed method preserved sharp edges and fine anatomical details better than individually trained U-Nets (which tended to "over-smooth" and create a "waxy" appearance).
• Difference maps indicated fewer extreme intensity artifacts (dark/bright spots) compared to UNN, particularly at very low doses (2%).

5. Significance

This work fundamentally rethinks the strategy for cross-dose PET denoising. By shifting the learning target from image reconstruction to residual noise estimation, the authors successfully mitigate the "averaging effect" that plagues unified deep learning models.

• Clinical Impact: The method enables robust denoising across a wide range of radiation doses without requiring prior knowledge of the specific dose level or complex multi-network architectures. This is crucial for longitudinal studies and vulnerable populations where dose variability is common.
• Generalizability: The approach offers a more efficient and robust alternative to existing domain generalization techniques, proving that a single, well-formulated residual learning model can outperform ensembles of specialized networks.