# Learning Regularization Parameters for PET Image Reconstruction

Younes Moussaoui, Saïd Moussaoui, Diana Mateus, Thomas Carlier, Simon Stute

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Younes Moussaoui, Saïd Moussaoui, Diana Mateus, Thomas Carlier, Simon Stute. Learning Regularization Parameters for PET Image Reconstruction. IEEE Nuclear Science Symposium Medical Imaging Conference and Room Temperature Semiconductor Conference, Oct 2025, Yokohama, Japan. hal-05491750

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# Learning Regularization Parameters for PET Image Reconstruction

Younes Moussaoui<fnref n="1" />,<fnref n="2" />, Saïd Moussaoui<fnref n="1" />, Diana Mateus<fnref n="1" />, Thomas Carlier<fnref n="2" />, Simon Stute<fnref n="2" />

**Abstract**—PET reconstruction transforms sinogram data into images using algorithms that incorporate data statistics and regularization penalties. The type of penalty and its strength have a strong impact on the reconstructed image quality, as they control the balance between reducing noise and preserving important details. Defining appropriate values for these parameters is difficult due to the variability of clinical data properties and usage purposes. We propose a supervised deep learning approach, based on a residual neural network, that uses directly the raw sinograms to predict the optimal regularization parameters of the relative difference prior used with the modified BSREM algorithm. This network, trained on synthetic 2D sinograms paired with optimal values of regularization parameters, learns to map raw data to optimal settings. This approach allows for fast, automated parameter selection, avoiding manual tuning. It generalizes well across phantom types and consistently produces high-quality reconstructions, supporting more efficient and reliable PET imaging workflows.

**Index Terms**—PET imaging, PET reconstruction, inverse problems, regularization parameters, deep learning.

## I. INTRODUCTION

POSITRON emission tomography (PET) reconstruction, aims to recover radiotracer distributions from noisy sinogram data. Reconstruction approaches employ a likelihood function, derived from Poisson data statistics. Penalized Maximum Likelihood (PML) methods are commonly used and introduce regularization terms to stabilize the reconstruction and to reduce noise. However, selecting suitable regularization parameters is a key difficulty, especially with edge-preserving penalties that require fine-tuning to maintain clinical relevance [1].

Traditional parameter selection strategies, such as Generalized Cross-Validation (GCV) [2] can be used. While statistically grounded, these methods are computationally intensive and may not generalize well across clinical scenarios. Recent efforts adopt deep learning (DL) to automate the selection of regularization parameters. Approaches using convolutional neural networks (CNNs) [3], [4] show promise, but often lack clinical realism.

In this work, we propose a novel deep learning framework to estimate optimal regularization parameters directly from measured PET data, enabling efficient, patient-specific adaptation without embedding DL into the reconstruction loop. Our method is based on the SINORES framework [5], employing a residual neural network trained on clinical-like data to jointly predict the strength and the edge-preserving parameter of the Relative Difference Prior [6].

## II. PROPOSED APPROACH

We consider the quadratic penalty:

$$R_{\text{quad}}(\lambda) = \frac{\beta}{2} \sum_{j=1}^{p} \sum_{k \in \mathcal{N}_j} \omega_{jk} (\lambda_j - \lambda_k)^2. \quad (1)$$

Here $\lambda$ is the reconstructed image, $\mathcal{N}_j$ the neighborhood of pixel j, $\omega_{jk}$ are predefined weights, $\beta$ is the regularization strength.

We also consider the Relative Difference Prior (RDP):

$$R_{\text{RDP}}(\lambda) = \frac{\beta}{2} \sum_{j=1}^{p} \sum_{k \in \mathcal{N}_j} \omega_{jk} \frac{(\lambda_j - \lambda_k)^2}{\lambda_j + \lambda_k + \gamma |\lambda_j - \lambda_k|}, \quad (2)$$

with $\gamma$ the edge-preserving parameter.

We propose a deep learning method that predicts the regularization parameters $\beta$ and $\gamma$ directly from PET sinograms $y$ using a convolutional residual network (ResNet):

$$(\hat{\beta}, \hat{\gamma}) = f(\theta | y) \quad (3)$$

where $f(\theta)$ is a ResNet with non-negativity constraints on its outputs. The network parameters $\theta$ are learned by minimizing the beta divergence between the predicted parameters $(\hat{\beta}, \hat{\gamma})$ and the optimal regularization parameters $(\beta_{\text{opt}}, \gamma_{\text{opt}})$, which are computed using the *Successive Parabolic Interpolation (SPI)* algorithm [7]. This algorithm efficiently identifies the values of $\beta_{\text{opt}}$ and $\gamma_{\text{opt}}$ that maximize the peak signal-to-noise ratio (PSNR) between the reconstructed and ground-truth images.

An overview of the framework is shown in Fig. 1, where the predicted parameters are used in a reconstruction with the BSREM algorithm with standard relaxation parameters [8].

## III. EXPERIMENTS DETAILS

To train and evaluate the proposed method, we synthesized a 2D phantom dataset simulating FDG PET scans (Fig. 2). The phantoms have a uniform background and contain randomly placed regions with different shapes, sizes, positions and intensity levels representing different metabolic activities. Total counts ranged from $10^5$ to $10^7$, with scatter and random fractions set between 35% to 40%.

Two ResNets of 15 million parameters were trained: one to predict the regularization strength $\beta$ for a quadratic penalty and the other to jointly estimate $\beta$ and $\gamma$ for a RDP. Three 2D brain phantoms were used to assess generalization. An FDG scan was simulated using BrainWeb, and an FDG and a PE2I scan were simulated using the Zubal phantom, which provided anatomical and functional diversity. A range of prompt levels from $5 \times 10^5$ to $10^7$ was considered.

<footnote n="1">¹Nantes Université, École Centrale Nantes, F-44300 Nantes, France</footnote>

<footnote n="2">²CHU Nantes, Service de Médecine Nucléaire, F-44000 Nantes, France</footnote>
<figure><img src="image_2.png" /><figcaption>Fig. 1: Overview of the SINORES architecture; the model takes a raw sinogram as an input and estimates the regularization parameters $(\beta, \gamma)$. These parameters are then used to reconstruct the image with the BSREM algorithm.</figcaption></figure>

The relevance of the predicted regularization parameters was assessed using NRMSE. Reconstructed image quality was evaluated using PSNR, MSSIM, activity recovery (AR) in hot regions, relative bias (RB) in cold regions, and image roughness (IR) in the background.

## IV. RESULTS & DISCUSSION

Fig. 2 shows model predictions for a test phantom using BSREM with a quadratic penalty (columns 2, 4, 5), with regularization parameters estimated via GCV (column 4) and SINORES (column 5). Quantitatively, $\hat{\beta}_{SINORES}$ produced predictions close to $\hat{\beta}_{GCV}$, with reconstructions using $\hat{\beta}_{SINORES}$ producing slightly better quality in PSNR, MSSIM, AR and RB. The comparison of SINORES with GCV shows that the proposed method efficiently provides relevant regularization parameters. Computing $\hat{\beta}_{GCV}$ is costly, often requiring several hours due to iterative minimization of the GCV function. In contrast, SINORES produces optimal parameters in seconds, making it practical for clinical use.

Fig. 2 also shows a reconstruction using RDP with parameters predicted by SINORES (column 6), as well as the optimal reconstruction based on PSNR (column 3). Visually, the SINORES-based reconstruction closely matches the optimal result.

Table I reports the performance of the model in the test and generalization sets. The results confirm that SINORES yields high-quality reconstructions on the test set, as measured by PSNR and MSSIM, and illustrate a good generalization capability.

<table><caption>TABLE I: Model's performance on test and generalization sets in terms of regression error and reconstruction quality (measured by NRMSE).</caption><thead><tr><th>Data</th><th>εβ</th><th>εγ</th><th>εPSNR</th><th>εMSSIM</th></tr></thead><tbody><tr><td>Test set</td><td>0.84</td><td>0.09</td><td>0.06</td><td>0.06</td></tr><tr><td>Generalization</td><td>0.66</td><td>0.53</td><td>0.16</td><td>0.02</td></tr></tbody></table>

<figure><img src="image_3.png" /><figcaption>Fig. 2: Reconstructed images with zoomed-in hot and cold regions. From left to right: Ground Truth, optimal reconstructions using quadratic and RDP penalties, quadratic penalty with GCV-based regularization, quadratic penalty with SINORES, and RDP with SINORES.</figcaption></figure>

## V. CONCLUSION

We demonstrate the feasibility of predicting optimal regularization parameters for PML reconstruction directly from raw PET data using a supervised deep learning approach. It allows a fast, efficient and automatic patient-specific tuning that leads to an improved image quality without performing several reconstructions.

## REFERENCES

[1] M. Millardet et al., IEEE TRPMS, pp. 629–640 (2021).

[2] P. Hansen, SIAM (1998).

[3] J. Lee et al., Medical Imaging, pp. 506–511 (2023).

[4] B. Afkham et al., Inverse Problems, 105017 (2021).

[5] Y. Moussaoui et al., IEEE EMBC (2024).

[6] J. Nuyts et al., IEEE TNS, pp. 56–60 (2002).

[7] R. Brent, Courier Corporation (2013).

[8] S. Ahn et al., IEEE TMI, pp. 613–626 (2003).

[9] S. Stute et al., 2015 IEEE NSS/MIC, CA, USA, pp. 1–3.