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

This paper proposes a joint reconstruction framework for PET activity and attenuation that utilizes wavelet diffusion models and diffusion posterior sampling to eliminate the need for auxiliary anatomical scans, demonstrating superior performance over existing methods in simulated 3D TOF data while showing promising potential for clinical non-TOF applications.

Original authors: Clémentine Phung-Ngoc, Alexandre Bousse, Antoine De Paepe, Thibaut Merlin, Baptiste Laurent, Hong-Phuong Dang, Olivier Saut, Catherine Cheze-Le-Rest, Dimitris Visvikis

Published 2026-07-14

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Clémentine Phung-Ngoc, Alexandre Bousse, Antoine De Paepe, Thibaut Merlin, Baptiste Laurent, Hong-Phuong Dang, Olivier Saut, Catherine Cheze-Le-Rest, Dimitris Visvikis

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to solve a giant, 3D jigsaw puzzle of the inside of a human body. Usually, to see the picture clearly, you need two things: a flashlight to show you where the glowing pieces are (the "activity" from a radioactive tracer) and a map of the foggy glass the light has to pass through (the "attenuation" map, usually taken from a CT scan).

For decades, doctors have relied on that extra CT map to fix the picture. But getting a CT scan means extra radiation, extra time, and extra cost. What if you could solve the puzzle using only the flashlight data? That's the big question this paper tackles.

The authors propose a new method called JRAA-DPS. Think of it as a super-smart AI detective that has studied thousands of solved puzzles. It knows that in the human body, the "glowing" parts (like a tumor) and the "foggy" parts (like bone or muscle) usually fit together in specific, predictable ways.

Here is how the magic happens:

The Training: The AI was trained on pairs of images (glow + fog) from 360 real patients. It learned the "rules" of how these two images look together, kind of like how a chef learns that a perfect cake always has a specific ratio of flour to sugar.
The Wavelet Trick: To handle the massive 3D size of a human body without the computer crashing, the team didn't feed the AI the raw images. Instead, they broke the images down into "wavelet coefficients." Imagine taking a high-resolution photo and turning it into a set of musical notes that describe the shapes and textures. The AI learns to compose these notes.
The Sampling (The "DPS" part): When the AI gets a new, noisy puzzle (just the flashlight data), it doesn't just guess. It starts with a cloud of random static (white noise) and slowly, step-by-step, refines it. At every step, it asks two questions: "Does this look like a realistic human body?" (based on its training) and "Does this match the flashlight data we actually measured?" It tweaks the image until both answers are "Yes."

What the paper actually found (and what it didn't):
In their simulations (computer-generated puzzles), this new method was a star player. When the data was "High Count" (lots of light, clear signal), the JRAA-DPS method produced activity images that were sharper and less noisy than the old standard methods (called MLAA and MLAA-UNet). It even managed to work when the "Time-of-Flight" (TOF) information was missing, though the quality dropped a bit.

However, the paper is very honest about the limits. When the data was "Low Count" (very dim light, like a faint whisper), the method struggled. The reconstructed "fog" map (attenuation) got a bit wobbly and distorted, though the glowing picture (activity) still looked okay.

They also tested this on real patient data from a Siemens Biograph mMR scanner. Here, the method successfully reconstructed a picture of a patient without using any CT or MR map for the attenuation. It found tumors and body structures. But, there was a catch: the method tended to slightly overestimate the brightness in quiet areas (like the liver) because its estimate of the "scattered" light was a bit too low.

The "Hallucination" Warning
Because this AI is a generative model (it creates things from scratch based on patterns), it can sometimes "hallucinate." The authors noticed that in the reconstructed fog maps, small details like tiny air pockets in the lungs or specific bone edges sometimes looked a bit off or were missing. The paper suggests these errors didn't ruin the main glowing picture, but they are a real limitation. It's like an artist who can paint a perfect landscape but might accidentally forget to draw a specific small rock.

The Speed Bump
There is one more thing to keep in mind: speed. While the old methods took about 2 to 3 minutes to solve the puzzle, this new AI detective took about 43 minutes for simulated data and 77 minutes for real clinical data. The authors admit this is slow, but they suggest that with some mathematical tweaks (like using a different sampling schedule called DDIM), this time could be cut down significantly in the future.

The Bottom Line
This paper suggests that we might one day be able to do PET scans without the extra CT scan, saving radiation and time. The method works well in simulations and shows promise on real data, proving it can reconstruct high-quality images even without a pre-made map. But it's not a perfect, instant solution yet. It needs more work to handle low-light situations better, to stop the "hallucinations" in the fog maps, and to run faster. For now, it's a very promising step toward a simpler, safer way to see inside the body.

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

Problem Statement
Positron Emission Tomography (PET) requires attenuation correction (AC) to accurately quantify radiotracer activity. Conventional workflows rely on co-registered Computed Tomography (CT) or Magnetic Resonance (MR) scans to generate attenuation maps. However, these auxiliary scans increase radiation exposure, complicate workflows, and introduce misalignment artifacts. While Joint Reconstruction of Activity and Attenuation (JRAA) using emission data alone offers a solution, existing methods face significant limitations. Traditional approaches like Maximum Likelihood Activity and Attenuation (MLAA) suffer from activity-attenuation crosstalk when Time-of-Flight (TOF) resolution is insufficient and fail entirely without TOF information. Furthermore, deep learning (DL) methods for AC often rely on direct image-to-image mappings that lack consistency with measured emission data or indirect methods that depend on noisy intermediate reconstructions, making them sensitive to out-of-distribution cases (e.g., unusual anatomy or varying count levels).

Methodology
The authors propose JRAA-DPS, a framework that performs joint reconstruction of activity ( $\lambda$ ) and attenuation ( $\mu$ ) solely from emission data using Diffusion Posterior Sampling (DPS) guided by a Wavelet Diffusion Model (WDM).

Wavelet Diffusion Model (WDM): To handle large 3-D volumes efficiently, the authors employ a WDM trained on pairs of activity and attenuation images. Instead of operating in image space, the model learns a prior in the wavelet coefficient space. An orthogonal Discrete Wavelet Transform (DWT) decomposes the 3-D image pairs into eight-channel wavelet coefficients, reducing computational complexity while preserving structural information. The model is trained using a Denoising Diffusion Probabilistic Model (DDPM) objective to learn the joint probability distribution $p(\lambda, \mu)$ .
Diffusion Posterior Sampling (DPS): During reconstruction, the trained WDM acts as a prior rather than a direct predictor. The algorithm samples from the posterior distribution $p(\lambda, \mu | y)$ $p (λ, μ ∣ y)$ , where $y$ $y$ represents the measured emission data. At each diffusion step, the algorithm:
- Generates a noisy estimate based on the learned prior.
- Updates the activity and attenuation latent variables separately using gradients of the log-likelihood of the measured data (data consistency).
- Explicitly incorporates the forward model, including scanner geometry, TOF resolution, scan duration, and noise characteristics, ensuring the reconstruction remains consistent with the specific acquisition settings without retraining the model.
Joint Scatter Estimation: The framework integrates a joint scatter estimation strategy where scatter parameters are progressively refined during the sampling process using gradient ascent, eliminating the need for a pre-estimated attenuation map to compute scatter.

Key Contributions

Novel JRAA Framework: Introduction of a DM-based JRAA method (JRAA-DPS) that leverages a diffusion prior learned from activity-attenuation pairs to reconstruct both maps from emission data alone.
Decoupling of Prior and Acquisition: The diffusion prior is independent of the acquisition model. A single trained model can be applied across diverse settings (different scan durations, TOF resolutions, and scanner geometries) provided the forward model is adapted to the specific data, removing the need for retraining.
Scalability via Wavelets: The use of a WDM allows for the processing of large 3-D volumes, overcoming the computational barriers typically associated with 3-D diffusion models.
Joint Scatter Integration: The ability to perform joint scatter estimation within the DPS loop, further reducing dependence on external anatomical priors.

Experimental Results
The method was evaluated on both simulated data (Experiment 1) and real clinical data from a Siemens Biograph mMR scanner (Experiment 2).

Simulated Data: JRAA-DPS outperformed both MLAA and MLAA-UNet (a U-Net post-processed MLAA) in terms of Peak Signal-to-Noise Ratio (PSNR), Structural Similarity Index (SSIM), and Normalized Root Mean Squared Error (NRMSE) for activity reconstruction in both High-Count (HC) and Low-Count (LC) settings.
- TOF vs. Non-TOF: With TOF information, JRAA-DPS produced high-quality, noise-free reconstructions. In non-TOF settings, performance degraded, particularly in LC conditions, though it still surpassed MLAA and MLAA-UNet in activity recovery.
- Attenuation Reconstruction: In LC settings, MLAA-UNet produced smoother attenuation maps with better metric scores, while JRAA-DPS introduced some erroneous details. However, these details did not significantly compromise the quality of the reconstructed activity image.
- Lesion Recovery: JRAA-DPS demonstrated robust recovery of lesion intensity and contrast, particularly in LC settings with sufficient hyperparameter tuning.
Clinical Data: The method successfully reconstructed non-TOF clinical data, recovering meaningful structures such as tumors. While MLAA-UNet achieved higher image quality metrics (partially due to favorable comparison conditions using the same OSEM algorithm and scatter estimates), JRAA-DPS reconstructed both activity and attenuation without relying on MR-derived attenuation maps.
Limitations Observed:
- Scatter Underestimation: In clinical data, the joint scatter estimation tended to underestimate scatter, leading to activity overestimation in low-uptake regions (e.g., the liver).
- Hallucinations: As with generative models, the method produced "hallucinations" in attenuation maps (e.g., missing bone details, incoherent lung structures), though these had minimal impact on the activity reconstruction.
- Computational Cost: Reconstruction is computationally expensive, taking approximately 1 hour 20 minutes for clinical data, compared to ~20 minutes for MLAA-based methods.

Significance and Claims
The paper positions JRAA-DPS as a significant step toward stand-alone PET imaging, aiming to reduce dependence on anatomical modalities (CT/MR) while maintaining quantification accuracy. The authors claim that by explicitly modeling acquisition factors in the forward model, the method offers robustness to varying count levels and the presence or absence of TOF information without requiring retraining.

The authors are modest regarding the current state of the method, acknowledging that:

Non-TOF LC limitations: Reconstruction quality in low-count, non-TOF scenarios is significantly degraded, limiting practical effectiveness in these specific settings.
Scatter Estimation: The current joint scatter estimation strategy is imperfect and requires further refinement to prevent activity overestimation.
Generative Artifacts: The potential for hallucinated features in attenuation maps remains a limitation that requires clinical validation.
Speed: The reconstruction time is currently a barrier to clinical adoption, though the authors suggest that accelerated sampling methods (e.g., DDIM) could mitigate this.

Ultimately, the work demonstrates the feasibility of using diffusion priors for CT-less PET reconstruction, offering a pathway to radiation dose reduction and improved workflow efficiency, provided that scatter estimation and sampling speed are further optimized.

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

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