HDDPM: Heteroscedastic Denoising Diffusion Probabilistic Model for Quantitative Low-Count Brain PET Recovery

Technical Summary: HDDPM for Quantitative Low-Count Brain PET Recovery

Problem Statement
Positron Emission Tomography (PET) faces a critical trade-off between diagnostic image quality and patient radiation exposure. Reducing the injected radiotracer dose to minimize radiation leads to "low-count" acquisitions characterized by severe noise. Unlike standard image processing tasks where noise is often assumed to be Gaussian and stationary, low-count PET noise is inherently non-Gaussian, non-stationary, and spatially dependent. Specifically, noise variance scales directly with local radioactivity concentration (Poisson statistics) and is influenced by the reconstruction algorithm and physical corrections.

Standard Denoising Diffusion Probabilistic Models (DDPMs) fail to address these physical realities. Their forward diffusion process adds isotropic, homoscedastic (constant variance) Gaussian noise to the target image. This assumption ignores the activity-dependent nature of PET degradation, leading to a mismatch between the learned corruption model and the actual physical noise structure of the scanner. Consequently, standard deep learning denoisers often struggle to generalize across different dose levels or scanner types, exhibiting issues like over-smoothing or under-denoising when applied to data with noise characteristics different from their training distribution.

Methodology
The authors propose the Heteroscedastic Denoising Diffusion Probabilistic Model (HDDPM), a framework designed to incorporate the physical noise structure of PET directly into the diffusion process.

1. Problem Formulation: The task is framed as a paired 3D image-to-image reconstruction where the model predicts the residual ($r = x_{HD} - x_{LD}$) between a standard-dose reference ($x_{HD}$) and a low-count input ($x_{LD}$). The final output is reconstructed by adding the predicted residual back to the low-count input. This approach preserves the anatomical structure present in the input while focusing the network's capacity on learning the count-dependent degradation.
2. Heteroscedastic Forward Process: Unlike standard DDPMs that use a scalar variance schedule, HDDPM replaces the scalar variance with a voxel-wise variance-scaling map, $g(v)$.
   • This map is generated by a frozen, Poisson-based module (not learned during training) that computes local variance based on the intensity of a reference image ($v$).
   • The variance is inversely proportional to local activity: low-activity regions receive stronger noise perturbation, while high-activity regions receive weaker perturbation, mimicking the heteroscedastic nature of PET count statistics.
   • During training, the standard-dose target serves as the variance reference. During inference, the model is conditioned only on the low-count image and the dose fraction, without requiring a target-derived map.
3. Network Architecture & Conditioning: The denoising network utilizes a 3D U-Net backbone in $x_0$-prediction mode. It takes the noisy residual, the low-count conditioning image, the diffusion timestep, and the dose fraction as explicit conditioning variables. Metadata (age, weight, scanner model) is also included where available.
4. Training Strategy: The model is trained to minimize the residual reconstruction loss. To handle the clinical challenge of extreme dose reduction, severe low-count cases were upweighted during sampling.

Key Contributions

• Heteroscedastic Reformulation: The paper introduces a novel reformulation of the DDPM forward process for medical imaging, replacing isotropic Gaussian noise with an intensity-aware, spatially varying noise model derived from Poisson statistics.
• Residual Learning with Dose Conditioning: The framework combines residual learning (predicting the difference between low and standard counts) with explicit dose-fraction conditioning, allowing the model to adapt to various simulated dose levels (1% to 50%) without retraining.
• Controlled Evaluation: The study provides a strictly controlled comparison between the proposed HDDPM and an isotropic DDPM baseline, isolating the specific impact of modeling heteroscedastic noise by keeping all other architectural and training parameters identical.

Results
The HDDPM was evaluated across three different scanner types (Siemens Biograph Vision Quadra, United Imaging uExplorer, and an external Siemens Biograph mMR) using both internal and external datasets at simulated dose levels ranging from 1% to 50%.

• Overall Image Quality: HDDPM and standard isotropic DDPM demonstrated comparable global image quality (measured by PSNR, SSIM, and SUV NMAE) across most dose levels and datasets.
• Low-Dose Robustness: The primary advantage of HDDPM emerged in the lowest-dose (1%) external scans. In this challenging, out-of-distribution scenario, HDDPM significantly reduced measurement errors compared to the standard model. Specifically, in the external cohort at 1% counts, HDDPM reduced low-uptake SUVr error to 2.40% compared to 3.10% for the isotropic DDPM ($p < 0.001$).
• Regional Quantification: While internal validation showed similar performance between the two models, HDDPM showed superior reliability in preserving regional quantitative accuracy (SUVr) in heavily degraded or external data.
• Noise Texture: Noise Power Spectrum (NPS) analysis indicated that both models produced similar noise textures internally, though HDDPM preserved slightly greater high-frequency residual power in external data, suggesting better preservation of fine details in difficult scenarios.

Significance and Claims
The authors claim that HDDPM provides a physically motivated inductive bias for quantitative low-count PET recovery. By reflecting the activity-dependent noise structure of PET within the diffusion process, the model achieves greater reliability in scenarios where standard isotropic assumptions fail—specifically in extreme dose reduction and cross-scanner generalization.

The paper modestly positions HDDPM as a proof-of-concept demonstrating the feasibility of heteroscedastic noising. The authors note that architectural parameters (e.g., training duration, variance scaling) were not fully optimized, meaning the reported results represent the basic behavior and potential of the approach rather than its absolute performance ceiling. The core significance lies in proving that modeling the specific physical degradation of the imaging system (heteroscedasticity) yields tangible benefits for quantitative accuracy in low-count PET, particularly when data is scarce or comes from unseen scanner configurations.