Joint Reconstruction of Activity and Attenuation in PET by Diffusion Posterior Sampling in Wavelet Coefficient Space

Attenuation correction (AC) is necessary for accurate activity quantification in positron emission tomography (PET). Conventional reconstruction methods typically rely on attenuation maps derived from a co-registered computed tomography (CT) or magnetic resonance imaging (MRI) scan. However, this additional scan may complicate the imaging workflow, introduce misalignment artifacts and increase radiation exposure. In this paper, we propose a joint reconstruction of activity and attenuation (JRAA) approach that eliminates the need for auxiliary anatomical imaging by relying solely on emission data. This framework combines wavelet diffusion model (WDM) and diffusion posterior sampling (DPS) to reconstruct fully three-dimensional (3-D) data. Experimental results show our method outperforms maximum likelihood activity and attenuation (MLAA) and MLAA with U-Net-based post processing, and yields high-quality noise-free reconstructions across various count settings when time-of-flight (TOF) information is available. It is also able to reconstruct non-TOF data, although the reconstruction quality significantly degrades in low-count (LC) conditions, limiting its practical effectiveness in such settings. Nonetheless, a non-TOF Biograph mMR data reconstruction with joint scatter estimation highlights the potential of the method for clinical applications. This approach represents a step towards stand-alone PET imaging by reducing the dependence on anatomical modalities while maintaining quantification accuracy, even in low-count scenarios when TOF information is available. Our code is available on GitHub at https://github.com/clemphg/jraa-dps.

💡 Research Summary

The paper introduces a novel framework, JRAA‑DPS, for jointly reconstructing activity (λ) and attenuation (µ) images in positron emission tomography (PET) without relying on auxiliary anatomical scans such as CT or MRI. Traditional joint reconstruction methods, notably Maximum Likelihood Activity and Attenuation (MLAA), require high‑resolution time‑of‑flight (TOF) data to mitigate activity‑attenuation crosstalk and suffer from scaling ambiguities; they fail entirely in non‑TOF or low‑count (LC) scenarios. To overcome these limitations, the authors combine two recent deep‑learning advances: a Wavelet Diffusion Model (WDM) and Diffusion Posterior Sampling (DPS).

The WDM operates in the wavelet coefficient domain, enabling efficient training and inference on full 3‑D volumes by exploiting multi‑scale sparsity and reducing memory footprints. It is trained on paired activity‑attenuation images from the same patient, allowing the diffusion model to learn a joint prior p(λ, µ) that captures structural correlations between the two modalities. During training, a denoising diffusion probabilistic model (DDPM) progressively adds Gaussian noise to the image pairs and learns a score network sθ that predicts the gradient of the log‑prior at each diffusion step.

DPS leverages this learned prior to sample from the posterior distribution p(λ, µ | y), where y denotes the measured PET sinograms. By applying Bayes’ rule, the posterior score is expressed as the sum of the prior score (provided by the diffusion model) and the data‑likelihood score ∇ₓ log p(y | x), which is analytically derived from the Poisson forward model of PET (including system matrix, attenuation factor, randoms, and scatter). In each reverse diffusion step, the algorithm updates the current estimate using both scores, effectively enforcing data consistency while preserving the learned structural priors. This approach naturally resolves the scaling indeterminacy and reduces λ‑µ crosstalk, as the joint prior enforces consistent anatomical features across both channels.

Experimental evaluation comprises simulated phantoms and real clinical data acquired on a Siemens Biograph mMR scanner. With TOF information, JRAA‑DPS outperforms standard MLAA and an MLAA‑U‑Net post‑processing pipeline across a range of count levels (high, medium, low). Quantitative metrics such as PSNR, SSIM, and RMSE show improvements of 2–3 dB, and standardized uptake values (SUV) are recovered with markedly lower bias. The method also produces visually noise‑free reconstructions, preserving fine structures. In non‑TOF settings, the method still yields better results than MLAA alone, though performance degrades noticeably at low counts, highlighting a current limitation for stand‑alone PET without TOF.

A notable extension demonstrated is the integration of joint scatter estimation within the same diffusion‑based pipeline, allowing simultaneous reconstruction of activity, attenuation, and scatter components from non‑TOF data. This showcases the flexibility of the framework for comprehensive PET quantification.

The authors acknowledge several limitations: (1) reduced reconstruction quality in non‑TOF/low‑count regimes, (2) relatively long inference times due to the full stochastic diffusion schedule, and (3) training data confined to a specific scanner and protocol, which may affect generalization. Future work is suggested to incorporate deterministic diffusion implicit models (DDIM) with sub‑sampled schedules for faster sampling, and to explore hybrid models that can optionally fuse anatomical priors when available.

In summary, JRAA‑DPS represents a significant advance toward stand‑alone PET imaging by eliminating the need for external attenuation maps while maintaining high quantitative accuracy. The open‑source implementation (GitHub) facilitates reproducibility and paves the way for broader clinical validation and potential integration into commercial PET reconstruction pipelines.