# Residual Neural Networks for the Prediction of the Regularization Parameters in PET Reconstruction

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

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Younes Moussaoui, Diana Mateus, Saïd Moussaoui, Thomas Carlier, Simon Stute. Residual Neural Networks for the Prediction of the Regularization Parameters in PET Reconstruction. 2024 46th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), Jul 2024, Orlando, France. pp.1-4, 10.1109/EMBC53108.2024.10782195. hal-04996373

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# Residual Neural Networks for the Prediction of the Regularization Parameters in PET Reconstruction

Younes Moussaoui<fnref n="1" />,<fnref n="2" />, Diana Mateus<fnref n="1" />, Saïd Moussaoui<fnref n="1" />, Thomas Carlier<fnref n="2" />,<fnref n="3" />, Simon Stute<fnref n="2" />,<fnref n="3" />

**Abstract**— Positron Emission Tomography (PET) is a medical imaging modality relying on numerical methods that integrate the statistical properties of the measurements and prior assumptions about the images. In order to maximize the computed image quality, PET reconstruction algorithms require the setting of hyperparameters that balance data fidelity with regularization. However, their optimal tuning depends on the statistical properties of the raw data and on the clinical objectives. To address this issue, we propose a supervised deep learning strategy based on a residual neural network that takes the raw measured data (sinogram) as input and automatically predicts the optimal value of the regularization parameter of the modified block Sequential Regularized Expectation Maximization (BSREM) algorithm. The proposed strategy is trained on a synthetic dataset consisting of 2D sinograms and their corresponding optimal regularization parameters. Our results demonstrate the feasibility of the approach leading to improved image reconstruction compared to classical manual tuning methods.

## I. INTRODUCTION

Positron Emission Tomography (PET) has emerged as a powerful sensing modality in the field of molecular imaging, providing valuable insights into various physiological and pathological processes at the molecular level. PET facilitates the non-invasive visualization and quantification of metabolic and molecular activities in the body and plays a pivotal role in both clinical diagnostics and research endeavors [1].

Reconstruction of PET images from raw data, usually in the form of sinograms [2], is an ill-posed inverse problem [3]. Its resolution requires the development of dedicated computational imaging methods. The classical statistical reconstruction method, based on maximum likelihood (ML) estimator combined with expectation maximization (EM) algorithm, gained prominence by incorporating the physical model and Poisson noise statistics to obtain accurate results [4]. To avoid noise amplification during MLEM iterations, instead of using early stopping, penalized ML methods (PML) offer the capability to explicitly account for prior assumption on the images (roughness, regions, edges, etc.) through the addition of a regularization term to the likelihood criterion. Among penalized reconstruction algorithms, the Block Sequential Regularized Expectation Maximization algorithm (BSREM) [5], incorporating Nuyts's Relative Difference Prior (RDP), efficiently addresses activity-dependent noise and preserves edges, demonstrating notable improvements

in quantitative evaluation [6], with adaptability for use in a medical context [7].

However, the major issue of the PML approaches is related to the choice of the appropriate values of the regularization parameters, which determine the best trade-off between data fidelity and regularization. These parameters play a crucial role in getting accurate and clinically relevant images [8].

Actually, the choice of regularization parameters is not universal and must be adapted to each imaging scenario and clinical purpose. High regularization parameters can potentially blur important details and reduce the detectability of small lesions. Conversely, low regularization parameters may produce noisy images of limited clinical utility [9].

Classical approaches to the setting of regularization parameters are based on the optimization of an additional criterion balancing data fitting and regularization level. For instance, one can apply methods such as the L-curve [10], the U-curve [11] and the Generalized Cross Validation (GCV) [12]. However, these methods are highly dependent on the specific characteristics of the data and may require several evaluations of the reconstruction problem, which can be very time consuming in the context of PET imaging routines.

The aim of this work is to predict the optimal regularization parameter from the measured data by using deep learning (DL) techniques in order to automatically adapt the regularization parameters to the specific characteristics of each patient's acquisition data. This approach differs from those incorporating deep learning for PET image reconstruction, such as unrolling methods [13], where deep learning is coupled to the reconstruction algorithm at each iteration, the plug-and-play approach [14], which uses deep learning to denoise the reconstructed image at each iteration, or the fully deep learning model [15], where the reconstruction is performed using a fully supervised model, relating sinograms to outputs without a physics-based model.

Actually, the aforementioned methods result in less interpretable reconstruction algorithms, generally require a reconstructed image as input and, in some cases, involve updating the regularization parameter at each iteration. In contrast, our approach uses deep learning exclusively to estimate the regularization parameters from the raw data before proceeding to the reconstruction stage using a state of the art method.

One of the first works on regularization parameter setting by DL techniques on raw data was applied to simulated CT scans of the Shepp-Logan phantom [16]. The reconstruction problem was formulated as the minimization of a quadratic

<footnote n="1">¹Nantes Université, École Centrale Nantes, LS2N, CNRS, UMR 6004, F-44300 Nantes, France</footnote>

<footnote n="2">²CHU Nantes, Service de Médecine Nucléaire, F-44000 Nantes, France</footnote>

<footnote n="3">³Nantes Université, CRCI2NA, F-44000 Nantes, France</footnote>
data fitting criterion with Tikhonov regularization. However, the proposed pipeline does not account for any clinical constraint. In a similar approach, Lee et al [17] proposed a DL based model in the context of PET reconstruction. However, the data generation process and the associated optimal regularization parameters were computed empirically and hence were not adapted to the clinical purpose.

The originality of our proposal lies in the development of a data-driven strategy using a residual neural network trained to account clinical constraints in terms of data generation and evaluation metrics. The proposed approach is based on realistic PET measurements simulation and the determination of optimal regularization parameters. Our model is designed to be patient-dependent and task-dependent only, with the task focused on maximizing the peak signal-to-noise ratio (PSNR) between the ground truth and the reconstructed image. The resulting model, termed SINORES, allows a fast tuning of the regularization parameters, while maintaining a sufficient level of reconstructed image quality.

## II. MATERIALS & METHODS

### A. Methods

1) Image Reconstruction by BSREM: The regularization term in BSREM algorithm, with a relative difference prior (RDP) is given by:

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

where $\lambda_j$ represents the reconstructed image intensity in the j-th voxel, p the number of voxels, $\mathcal{N}_j$ the neighbors of voxel j, and $\omega_{jk}$ a local penalty weight between voxels j and k, while $\gamma$ affects the degree of edge-preservation. The RDP is designed to preserve high spatial frequencies in the reconstructed images, while maintaining a balance between smoothness and convexity. For the sake of simplification, the $\gamma$ parameter is fixed to a pre-defined value. Finally, $\beta$ is the regularization weight, which controls the trade-off between data fidelity (likelihood) and the regularization term.

2) Optimal Regularization Parameter Setting: As shown in Fig. 1, the supervised setting of the optimal regularization parameters is achieved by a Successive Parabolic Interpolation (SPI) algorithm [18], which allows to find $\beta_{opt}$ that maximizes a target metric, such as the PSNR between the reconstructed image and the ground truth image. Note that, depending on the clinical purpose, other metrics may be retained to define the optimal set of regularization parameters.

3) Data-driven Prediction of the Regularization Parameters: Given a dataset $\{(X_i, \beta_{opt_i})\}_{i=1}^N$ of sinograms and their corresponding optimal regularization parameters, the goal is to train the parameters $\theta$ of a neural network $f(\theta)$ to predict $\hat{\beta}$ for a new, unseen sinogram. In this work, $f(\theta)$ is a Convolutional Neural Network with residual connections and absolute activation at the output to ensure the non-negativity of $\beta$. For the optimization of the parameters $\theta$, we rely on the $\ell_1$ loss function, defined as the mean absolute error between $\hat{\beta}$ and $\beta_{opt}$.

<figure><img src="image_2.png" /><figcaption>Fig. 1: Optimization of $\beta$ using Successive Parabolic Interpolation (SPI)</figcaption></figure>

4) Evaluation Criteria and Image Quality Metrics: The evaluation of the prediction model accuracy is typically performed using metrics such as the mean absolute error (MAE), the standard deviation (STD) and the mean absolute percentage error (MAPE). The MAPE is defined as:

$$MAPE = \frac{1}{N} \sum_{i=1}^{N} \left| \frac{\beta_i - \hat{\beta}_i}{\beta_i} \right| \times 100. \quad (2)$$

where N is the number of data points. To evaluate the reconstructed image quality, the peak signal-to-noise ratio (PSNR) and the mean structural similarity index map (MSSIM) are used. Additionally, the activity recovery of a hot region is also calculated and defined as $AR_h = \hat{\lambda}_h/\bar{\lambda}_h$, where $\hat{\lambda}_h$ represents the mean activity of the hot region in the reconstructed image, and $\bar{\lambda}_h$ is the mean activity of the hot region in the ground-truth image. Finally, the relative bias is also used to assess the recovery of the cold region:

$$RB_c = (\hat{\lambda}_c - \bar{\lambda}_c)/\bar{\lambda}_c. \quad (3)$$

Here, $\hat{\lambda}_c$ denotes the mean activity of the cold region in the reconstructed image, and $\bar{\lambda}_c$ represents the mean activity of the cold region in the ground truth image.

### B. Materials

1) Phantom Synthesis: A 2D simulated phantom, as illustrated in Fig. 4, was developed to represent various activity distributions. These distributions are defined by a uniform background of varying volume, encompassing a random number of smaller regions with random sizes, geometric configurations, orientations, positions, and intensity levels (cold or hot). Furthermore, non-rigid transformations using elastic deformation based on cubic B-splines were applied to the phantom. All these transformations were implemented on the reference phantom, generating multiple examples to facilitate comprehensive training and enhance the diversity of the dataset.

2) Sinogram Generation: The PET Siemens Biograph mMR geometry was considered for the simulation of the sinograms using an in-house analytical tool involving forward projection, addition of scattered and random coincidences, and finally the generation of Poisson noise [19]. The total count is set in the range $10^5$ to $10^7$, with the scatter fraction set at 0.35 ± 5% and the random fraction set at 0.4 ± 5%.
<figure><img src="image_3.png" /><figcaption>Fig. 2: Overview of the SINORES architecture, the model takes a raw sinogram as an input and estimates the regularization parameter $\beta$. The parameter is then used to reconstruct the image using the modified BSREM.</figcaption></figure>

### C. Experimental Settings

A total of 58,000 phantoms were generated. A random split is performed into 42,000 training samples and the validation and test sets containing 8,000 samples each. Prior to learning, z-score normalization was applied to the raw sinograms.

The Residual Network architecture consists of five residual blocks, as illustrated in Fig. 2. Each block includes convolutional layers, batch normalization, a Rectified Linear Unit (ReLU) activation function, followed by max-pooling, a fully connected layer, and finally an absolute activation function. The model contains 15 million parameters, trained with a batch size of 100 and the Adam optimizer with its default settings, a learning rate of $10^{-7}$, and 400 epochs. Input sinograms are cropped to a size of $184 \times 252$. The BSREM algorithm was run for 30 iterations, 28 subsets and original relaxation scheme [5].

## III. RESULTS & DISCUSSION

### A. Model Prediction Accuracy

Fig. 3 (left) shows the model results for 8,000 samples from the test set. It appears clearly that the predicted $\hat{\beta}$ values correlate strongly (83%) with the optimal values. The

coefficient of determination, R-squared ($R^2$) with a value of 0.69, further quantifies this good fitting quality.

Fig. 3 (right) shows the statistical distribution of the estimation error $|\beta_{opt} - \hat{\beta}|$. It mainly highlights the model's favorable accuracy. These results demonstrate the generalization ability to unseen data under the same acquisition/simulation conditions.

Table I provides a summary of the performance of SINORES on the test set, presenting measures such as MAE, MAPE and STD for the estimated parameter values. In addition, the table includes performance results when the regularization parameter is set to $\beta_{mean}$, defined as the average of $\beta_{opt}$ values over the training set, which can guarantees a certain level of quality. All performance indices consistently confirm the improved predictive ability of the regularization parameter using the trained SINORES model.

Table I also compares the reconstructed images obtained with $BSREM+\hat{\beta}$ (SINORES) and the baseline $BSREM+\beta_{mean}$ against the ideal reconstruction using the optimal $BSREM+\beta_{opt}$. Clearly, the quality of $BSREM+\hat{\beta}$ obtained with SINORES and requiring a single reconstruction, is closer in terms to the PSNR and MSSIM to the optimal setting, which requires several evaluations of the reconstruction algorithm.

### B. PET Image Quality

Fig. 4 gives an example of reconstructed images using different strategies of the regularization parameter settings

<table><caption>TABLE I: Regularisation parameter estimation accuracy and comparison of BSREM with $\beta_{mean}$ and $\hat{\beta}$ computed with SINORES.</caption><thead><tr><th rowspan="2">Method</th><th colspan="3">Model estimation</th><th colspan="2">Image quality</th></tr><tr><th>MAE β</th><th>MAPE β</th><th>STD β</th><th>MAE PSNR</th><th>MAE MSSIM</th></tr></thead><tbody><tr><td>β (ours)</td><td>0.073</td><td>5.31</td><td>0.043</td><td>0.16</td><td>0.03</td></tr><tr><td>β<sub>mean</sub></td><td>0.429</td><td>46.56</td><td>0.129</td><td>0.43</td><td>0.18</td></tr></tbody></table>

<figure><img src="image_4.png" /><figcaption>Fig. 3: (Left) Scatter plot of the predicted $\hat{\beta}$ using SINORES versus $\beta_{opt}$, the optimal regularization parameters; (Right) Distribution of absolute errors between $\hat{\beta}$ and $\beta_{opt}$ of the test set.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Fig. 4: Generated phantom and its reconstructions: (a) ground truth; (b) reconstruction with $\beta_{opt}$, PSNR = 36.04 dB; (c) reconstruction using *SINORES*, PSNR = 36.03 dB; (d) reconstruction using $\beta_{mean}$, PSNR = 33.02 dB; (e) zoom of a hot region $AR_h = 1$; (f) $AR_h = 0.99$; (g) $AR_h = 0.98$; (h) $AR_h = 0.95$; (i) a zoom of a cold region $RB_c = 0$; (j) $RB_c = 0.14$; (k) $RB_c = 0.15$; (l) $RB_c = 0.24$.</figcaption></figure>

(optimal, average and *SINORES*). It also shows the quality metrics in cold and hot regions of the phantom.

The visual comparison illustrates a better image quality when using $\hat{\beta}$ as compared to $\beta_{mean}$, and a similar quality as with $\beta_{opt}$. One can note that the use of $\beta_{mean}$ on this image produces a blurred reconstruction with ill-defined edges, as it can be seen in the Fig. 4(d),(h),(l). On the other hand, reconstructions with the estimated parameter $\hat{\beta}$ preserve important features while attenuating noise.

From a quantitative point of view, this improvement is confirmed by the higher PSNR and MSSIM using $\hat{\beta}$ compared to using $\beta_{mean}$. One can also note the very close prediction of $\hat{\beta}$, comparing to $\beta_{opt}$, with an error of $4 \times 10^{-3}$.

We also evaluate the recovery of activity in the hot region ($AR_h$) and the relative bias for recovery in the cold region ($RB_c$). Both measures reveal promising results regarding the quality of reconstructions obtained with *SINORES*. Fig. 4(e)-(l) show very close reconstructions in terms of these two measures when both $\beta_{opt}$ and $\hat{\beta}$ are used.

## IV. CONCLUSION

This study introduced a deep learning supervised strategy for prediction of the regularization parameters of the BSREM algorithm, using raw 2D sinograms as input and optimal regularization parameters as target. Our model adapts the regularization parameter to the measured data removing the need for a patient-specific tuning. Experimental results showed improved image reconstruction compared to a conventional method. Given the fidelity of the complete PET simulation scheme, the variability in the type and number of regions in the dataset, and the reliance on a medically approved reconstruction method (BSREM), this is a promising step towards

improving the quality and adaptability of PET reconstruction in the clinical setting.

## ACKNOWLEDGMENT

This work has been partly funded by the French "Programme d'Investissement d'Avenir" (ANR-16-IDEX-0007) and region "Pays de la Loire" through their support to I-Site NExT.

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