SPINR: SELF-SUPERVISED PET
RECONSTRUCTION USING IMPLICIT NEURAL
REPRESENTATIONS

Younes Moussaoui, Diana Mateus, Lukmán El Hadi, Saïd Moussaoui, Simon Stute, Nasrin Taheri

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Younes Moussaoui, Diana Mateus, Lukmán El Hadi, Saïd Moussaoui, Simon Stute, et al.. SPINR: SELF-SUPERVISED PET RECONSTRUCTION USING IMPLICIT NEURAL REPRESENTATIONS. IEEE International Symposium on Biomedical Imaging, Apr 2026, London, United Kingdom. hal-05491824

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# SPINR: SELF-SUPERVISED PET RECONSTRUCTION USING IMPLICIT NEURAL REPRESENTATIONS

Younès Moussaoui<fnref n="1" />,<fnref n="2" />, Diana Mateus<fnref n="1" />, Lukmán El Hadi<fnref n="2" />, Saïd Moussaoui<fnref n="1" />, Simon Stute<fnref n="2" />,<fnref n="3" />, Nasrin Taheri<fnref n="2" />,<fnref n="3" />

## ABSTRACT

Implicit neural representations (INRs) have recently emerged as a continuous and flexible approach for PET image reconstruction, mitigating artifacts produced by conventional algorithms. In this work, we introduce SPINR, an enhanced implicit neural representation framework for self-supervised PET reconstruction. SPINR features a refined coordinate normalization strategy that enables spatial encoding to depend on the field of view rather than fixed image coordinates, thereby improving robustness across diverse reconstruction settings. Additionally, we conduct a spectral and performance analysis of multiple periodic and non-periodic activation functions to better understand the frequency-encoding behavior of INR architectures in PET imaging. Experimental results show that SPINR delivers good performance compared to both traditional penalized-likelihood reconstructions and deep image prior (DIP)-based approaches, providing improved contrast recovery, reduced bias, and enhanced overall image quality. Collectively, these findings establish SPINR as a data-efficient and robust framework for PET image reconstruction.

**Index Terms**— Implicit neural representations, PET imaging, image reconstruction, spectral analysis

## 1. INTRODUCTION

Positron Emission Tomography (PET) image reconstruction consists of recovering the underlying radiotracer distribution from noisy and incomplete sinogram measurements, a problem that is inherently ill-posed. Conventional iterative algorithms, such as Maximum Likelihood Expectation Maximization (MLEM) and its regularized variants, Penalized Maximum Likelihood (PML) methods, aim to minimize the discrepancy between measured and forward-projected data [1, 2]. Among them, the Block Sequential Regularized Expectation Maximization (BSREM) algorithm is widely adopted

in clinical and research settings for its effective noise control through spatial regularization. In fact, regularization schemes and post-processing filters [3] can mitigate noise amplification, but lead to additional hyperparameter tuning and loss of spatial resolution.

Deep learning has recently offered new perspectives for PET reconstruction. Fully supervised approaches such as AU-TOMAP [4] learn end-to-end mappings from sinograms to reconstructed images, producing high-quality results but requiring large, paired training datasets that limit generalization. Self-supervised strategies such as the Deep Image Prior (DIP) [5] mitigate this dependency by exploiting the implicit bias of network architectures. In PET, DIP-based methods [6, 7] have demonstrated effective noise suppression and structural preservation, benefiting from the spectral bias of convolutional networks that favor smooth structures over noise-dominated high-frequency components.

Implicit Neural Representations (INRs)—also known as coordinate-based neural fields [8]—have recently emerged as a powerful alternative for image reconstruction and analysis tasks. Similar to DIP, INRs differentiability enables their seamless integration with the PET forward model for direct optimization of the reconstructed image. However, INRs offer greater model compactness, leverage continuous representations, and exhibit enhanced spectral behavior due to advanced activation functions. For instance, the sinusoidal representation network (SIREN) [9] introduced periodic activations that enable INRs to model fine structural details and high-frequency content. Alternative activation formulations, such as Gaussian [10] and FINER [11], have since been proposed to improve representation locality, spectral smoothness, and convergence stability. These variants provide different trade-offs between frequency expressiveness and spatial regularity as shown by an spectral analysis.

Our earlier work [12] demonstrated that PET images can be reconstructed directly from sinograms using a SIREN-based INR in a self-supervised framework. Very recent studies have extended this line of research, but opting for ReLU activations and focusing on hash-based [13] or Fourier-based [14] positional encoding schemes. These developments

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

<footnote n="2">²Dépt. de Médecine Nucléaire, Centre Hospitalier Universitaire de Nantes, F-44000 Nantes, France</footnote>

<footnote n="3">³CRCI2NA, INSERM, CNRS, Université d'Angers, Université de Nantes, F-44000 Nantes, France</footnote>

<footnote>© 2026 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.</footnote>
highlight the growing interest in INR-based PET reconstruction and further validate the relevance and timeliness of our research direction.

In this study, we introduce **SPINR** (Self-supervised PET reconstruction using Implicit Neural Representations). It advances the INR PET reconstruction framework through three main contributions: (i) a coordinate normalization scheme ensuring that the input representation depends on the field of view rather than fixed image coordinates, improving generalization across reconstruction setups; (ii) a systematic evaluation of activation functions—SIREN, GAUSS, and FINER—to analyze their effect on reconstruction fidelity; and (iii) a spectral analysis offering new insights into the frequency encoding of INR-based PET reconstruction, a perspective not previously explored in PET. Together, these developments make SPINR a data-efficient framework for PET image reconstruction.

## 2. METHODS

### 2.1. Problem statement

PET image reconstruction aims to estimate the radiotracer distribution $\lambda \in \mathbb{R}^N$ from noisy sinogram measurements $y \in \mathbb{R}^M$, where $N$ is the number of image voxels and $M$ the number of sinogram bins. Each measurement $y_i$ corresponds to the $i$-th line of response (LOR) or sinogram bin. The reconstruction problem is formulated as:

$$ \min_{\theta} \{-\log p(y | \mathcal{P}(f_N(\theta | v)))\}, \quad (1) $$

where $\lambda = f_N(\theta | v)$, $\mathcal{P}(\cdot)$ denotes the forward projection operator, and $f_N(\theta | v)$ is the INR parameterized by $\theta$ and spatial coordinates $v$. Assuming Poisson-distributed data, the data fidelity term (negative log-likelihood) is expressed as:

$$ \mathcal{L}_{\text{Poisson}} = \sum_{i=1}^{M} [\mathcal{P}(f_N(\theta | v))_i - y_i \log(\mathcal{P}(f_N(\theta | v))_i)], \quad (2) $$

with the forward operator defined as $\mathcal{P}(f_N(\theta | v)) = Af_N(\theta | v) + b$, where $A$ is the system matrix (modeling projection geometry, attenuation, and detector sensitivity) and $b$ represents the additive background contribution, including scattered and random events [15]. The optimization iteratively updates the parameters $\theta_k$ at iteration $k$ via backpropagation and a gradient-based optimization strategy until convergence.

### 2.2. INRs for PET reconstruction

INRs represent images as continuous functions rather than discrete grids. In 2D PET, the network learns a mapping

$$ f_N : \mathbb{R}^2 \to \mathbb{R}, \quad (z_1, z_2) \mapsto \lambda(z_1, z_2) = f_N(\theta | (z_1, z_2)). $$

Each layer applies an affine transformation followed by a non-linear activation $\phi(\cdot)$, which can be selected to control the spectral representation of the reconstructed image.

**SIREN** uses sinusoidal activations to capture fine image details, **GAUSS** activations promote local smoothness and stability, while **FINER** combines frequency modulation and locality for improved convergence. The corresponding activation functions are defined as:

$$ \begin{aligned} \text{SIREN: } & \phi_{\text{SIREN}}(x) = \sin(\omega_0 x), \\ \text{GAUSS: } & \phi_{\text{GAUSS}}(x) = e^{-(s_0 x)^2}, \\ \text{FINER: } & \phi_{\text{FINER}}(x) = \sin(\omega_0(|x| + 1)x), \end{aligned} \quad (3) $$

where $\omega_0$ and $s_0$ are hyperparameters controlling the frequency scaling of the activation. A SoftPlus activation is used in the output layer, enforcing non-negativity while maintaining differentiability, as demonstrated in [12]. Integrated into (??), the INR directly learns a self-supervised mapping that maximizes the data likelihood.

### 2.3. Scanner-based input normalization

To achieve scanner-independent modeling, the spatial coordinates $z = (z_1, z_2)$ are normalized using the scanner's Cartesian reference frame. For an $n \times n$ image grid, the discrete indices $z_1, z_2 \in [-\frac{n}{2}, \frac{n}{2} - 1]$ are first converted to physical units (millimeters) using the voxel sizes $p_{z_1}$ and $p_{z_2}$, and then scaled by the scanner field of view (FOV) to obtain normalized coordinates: $(z_{\text{norm},1}, z_{\text{norm},2}) \in [-1, 1]^2$. This normalization is considered as the input of the model and ensures that the INR input domain remains consistent across different image resolutions and scanner configurations, providing a unified spatial encoding that improves generalization and stability during reconstruction.

## 3. EXPERIMENTS

### 3.1. Data and PET projection

A 2D BrainWeb phantom [16] was used to evaluate the proposed framework. A piecewise uniform ground-truth (GT) activity map approximating a typical FDG brain distribution was generated to provide realistic tissue contrasts. Two spatial resolutions were considered to test reconstruction robustness across different fields of view (FOVs): 160 × 160 (zero-padded) and 112 × 112 pixels. For each size, 20 independent noise realizations were produced, yielding 40 phantoms in total. Synthetic PET sinograms were obtained using a 2D analytical forward projector [15] modeling the Siemens Biograph mMR geometry. The simulations included attenuation, sensitivity, and background events, with 35% randoms, 30% scatter, and a total of 3.5 million prompts per acquisition.
<figure><img src="image_3.png" /><figcaption>Fig. 1. Overview of the proposed pipeline for PET reconstruction using SIREN. The model maps the image pixel by pixel to construct the full image. The loss is then computed in the measurement domain between the estimated sinogram obtained by projecting the estimated image and the measured sinogram.</figcaption></figure>

## 3.2. Image Quality Metrics

The peak signal-to-noise ratio (PSNR) and the mean structural similarity index measure (MSSIM) are used to assess the reconstructed image quality within the brain. Additionally, the activity recovery (AR) of a hot tumor region is defined as $AR_h = \hat{\lambda}_h / \bar{\lambda}_h$, where $\hat{\lambda}_h$ and $\bar{\lambda}_h$ represent the mean activity of the hot region in the reconstructed and GT images, respectively. The relative bias (RB) assesses cold region recovery, defined by ventricles filled with cerebrospinal fluid as $RB_c = (\hat{\lambda}_c - \bar{\lambda}_c) / \bar{\lambda}_c$ where $\hat{\lambda}_c$ and $\bar{\lambda}_c$ represent the mean activity of the cold region in the reconstructed and GT images, respectively. Finally, the image roughness (IR), computed in the white matter of the brain, measures the standard deviation of voxel values in the phantom background from their mean value: $IR = \frac{\sqrt{\frac{1}{L-1} \sum_{i \in ROI} (\hat{\lambda}_{r,i} - \bar{\lambda}_{r,ROI})^2}}{\bar{\lambda}_r}$ where $\hat{\lambda}_{r,i}$ is the activity of white matter, $\hat{\lambda}_r$ is its mean value, and $L$ is the number of voxels in the region of interest. In addition, a spectral bias analysis was performed following the method proposed by Shi et al. [17]. This analysis evaluates how the reconstructed image preserves spatial certain frequencies by comparing the ratio of the Fourier magnitudes between the reconstructed image and the reference image:

$$M(\zeta_1, \zeta_2) = \frac{|\mathcal{F}(\hat{\lambda})(\zeta_1, \zeta_2)|}{|\mathcal{F}(\lambda_{gt})(\zeta_1, \zeta_2)| + \epsilon}, \quad (4)$$

where $\mathcal{F}$ denotes the 2D Fourier transform, with $(\zeta_1, \zeta_2)$ its frequency coordinates. The constant $\epsilon$ prevents division by zero. The mean value of $M(\zeta_1, \zeta_2)$ is then computed within concentric circular frequency bands to quantify the relative preservation of low- and high-frequency components in the

reconstructed images, providing complementary information on their spectral fidelity.

## 3.3. Experimental Settings

In addition to SPINR, a DIP model was also evaluated for comparison, following a 2D U-Net architecture adapted from [7] with random noise as input, and applied to PET reconstruction in [12]. All models were optimized using the L-BFGS algorithm, which offered stable and efficient convergence. SPINR models were trained for 80 iterations with a learning rate of 1, while DIP used 650 iterations with a learning rate of 1.5. SPINR architectures comprised three hidden layers with 256 units (132k parameters), trained under three activation variants: SIREN ($\omega_0 = 35, \omega_0^{hidden} = 30$), FINER ($\omega_0 = 15, \omega_0^{hidden} = 10$), and GAUSS (scale = 11). The DIP contained approximately 382k parameters. For reference, the BSREM algorithm was implemented in CASTOR [18] with a quadratic penalty and individually tuned regularization per image to maximize PSNR. All hyperparameters were determined via grid search for optimal reconstruction quality.

# 4. RESULTS & DISCUSSION

Table 1-left demonstrates that the Cartesian normalization of the input improves the stability and resolution invariance of SPINR-SIREN reconstructions. In addition to moderate gains in PSNR, MSSIM, and $AR_h$, a substantial improvement is observed in IR, reflecting enhanced noise suppression. Overall, normalization ensures consistent performance across image sizes by expressing network inputs in the scanner's Cartesian frame. This alignment enables to keep the same activation hyper parameters, stabilizes gradients during
<table><thead><tr><th>Input Normalization</th><th>PSNR ↑</th><th>SSIM ↑</th><th>AR<sub>h</sub> ↑</th><th>IR ↓</th></tr></thead><tbody><tr><td>w/ Cartesian</td><td>18.151 ± 0.063<br>18.208 ± 0.119</td><td>0.538 ± 0.006<br>0.545 ± 0.008</td><td>96.221 ± 4.400<br>96.761 ± 3.256</td><td>76.400 ± 20.750<br>39.200 ± 15.270</td></tr><tr><td>w/o Cartesian</td><td>17.983 ± 0.106<br>18.249 ± 0.070</td><td>0.512 ± 0.010<br>0.539 ± 0.010</td><td>98.409 ± 3.205<br>98.764 ± 3.320</td><td>120.500 ± 5.635<br>102.100 ± 19.834</td></tr></tbody></table>

<table><caption>**Table 1.** (left) Quantitative performance (mean ± standard deviation) of SPINR-SIREN with and without Cartesian input normalization. Blue rows correspond to images of size 112×112, and red rows to 160×160. Cartesian normalization reduces performance gaps across resolutions and improves reconstruction stability. (right) Performance of SPINR across activation functions in terms of image quality metrics (mean ± std), compared with BSREM and DIP.</caption><thead><tr><th>Metrics</th><th>BSREM</th><th>DIP</th><th>SPINR-SIREN</th><th>SPINR-FINER</th><th>SPINR-GAUSS</th></tr></thead><tbody><tr><td>PSNR ↑</td><td>18.59 ± 0.22</td><td>17.808 ± 0.256</td><td>18.179 ± 0.091</td><td>18.264 ± 0.062</td><td>18.226 ± <strong>0.052</strong></td></tr><tr><td>MSSIM ↑</td><td>0.548 ± 0.015</td><td>0.512 ± 0.020</td><td>0.542 ± 0.007</td><td>0.546 ± 0.007</td><td>0.526 ± <strong>0.005</strong></td></tr><tr><td>AR<sub>h</sub> ↑</td><td>93.401 ± 2.393</td><td>102.79 ± 5.466</td><td>96.491 ± 3.828</td><td>99.325 ± 2.761</td><td>97.956 ± <strong>2.510</strong></td></tr><tr><td>IR ↓</td><td>20.779 ± 2.1472</td><td>28.392 ± 4.254</td><td>57.800 ± 18.010</td><td>45.300 ± 12.229</td><td>33.625 ± <strong>4.545</strong></td></tr></tbody></table>

optimization, and maintains geometric consistency with the projection model, yielding uniform reconstruction behavior independent of resolution.

Fig. 2 and Table 1-right report the comparative performance of SPINR variants, DIP, and BSREM with quadratic penalty. Among all methods, **SPINR-GAUSS** delivers the most balanced and robust results, achieving mean PSNR and MSSIM values close to BSREM while outperforming DIP. It also exhibits the lowest variability indicating enhanced stability and reduced sensitivity to initialization or data diversity. The GAUSS activation acts as an intrinsic low-pass regularizer, maintaining high AR<sub>h</sub> and substantially reducing IR, thus producing smooth yet accurate reconstructions.

The oracle performances are set by BSREM with quadratic penalty, by manually tuning regularization parameters for each dataset. SPINR and DIP, in contrast, are self-supervised frameworks with fixed hyperparameters, eliminating per-case adjustment. SPINR-SIREN and SPINR-FINER exhibit higher IR due to their greater high-frequency sensitivity, whereas DIP reconstructions remain smoother but blurrer, with reduced structural fidelity. Visual results in Fig. 2 confirm that SPINR-GAUSS produces cleaner, more contrasted images, closely approaching the quality of BSREM while maintaining a self-supervised framework, a continuous image representation, and avoiding per-case tuning.

The spectral bias analysis further elucidates these differences by examining the preservation of spatial frequencies during reconstruction. Circular statistics averaged over five concentric frequency rings (Fig. 3) quantify the recovery of

<table><caption>**Table 2.** Mean (± std) circular statistics across five concentric frequency rings for each reconstruction method.</caption><thead><tr><th>Method</th><th>Ring 1 (%)</th><th>Ring 2 (%)</th><th>Ring 3 (%)</th><th>Ring 4 (%)</th><th>Ring 5 (%)</th></tr></thead><tbody><tr><td>BSREM</td><td>92.4±0.9</td><td>74.3±2.2</td><td>53.8±4.3</td><td>33.2±3.3</td><td>18.1±1.9</td></tr><tr><td>DIP</td><td>89.2±1.0</td><td>64.8±5.6</td><td>26.6±6.2</td><td>11.2±4.2</td><td>7.4±3.2</td></tr><tr><td>FINER</td><td>87.5±1.0</td><td>68.6±2.9</td><td>39.0±3.8</td><td>17.7±2.7</td><td>9.0±1.9</td></tr><tr><td>SIREN</td><td>86.6±1.3</td><td>68.7±4.6</td><td>39.6±7.0</td><td>20.0±5.9</td><td>11.4±4.6</td></tr><tr><td>GAUSS</td><td>87.0±1.1</td><td>67.1±1.9</td><td>43.2±2.6</td><td>24.3±1.9</td><td>15.1±1.3</td></tr></tbody></table>

low- (rings 1–2) and high- (rings 4–5) frequency bands. As reported in Table 4, all methods accurately reconstruct low-frequency structures, while high-frequency recovery diminishes due to regularization. **DIP** shows a pronounced drop across rings, confirming its limited ability to capture fine details. **BSREM** maintains strong low-frequency fidelity but exhibits greater variability linked to its regularization parameter $\beta$. Conversely, SPINR variants—particularly **FINER** and **GAUSS**—achieve a more consistent balance between spectral fidelity and stability, with narrower confidence bands around their optimal iterations. Overall, **SPINR-GAUSS** demonstrates the most stable and spectrally balanced behavior, effectively recovering fine details while preserving smoothness and convergence consistency across datasets.

## 5. CONCLUSION

We proposed **SPINR**, a self-supervised PET reconstruction framework based on implicit neural representations. SPINR achieves stable, resolution-invariant performance without ex-

<figure><img src="image_4.png" /><figcaption>**Fig. 2.** Reconstructed images using BSREM with quadratic penalty, DIP, SPINR with SIREN, FINER, GAUSS.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Fig. 3.** Spectral bias (mean ± std) across five concentric frequency bands (y-axis), plotted as a function of iteration or regularization strength for each method (x-axis). The red dashed line marks the mean best-PSNR location across cases; the faint red band shows ±1 std.</figcaption></figure>
ternal supervision. Among all variants, **SPINR-GAUSS** offers the best compromise between accuracy and smoothness, matching BSREM quality while remaining self-supervised. In addition, the spectral analysis introduced in this work provides new insights into the frequency encoding behavior of INR-based PET reconstruction. Future work will extend SPINR to 3D reconstruction.

# 6. ACKNOWLEDGMENTS

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, as well as by Siemens Healthineers, the industrial partner of the NExT research industrial chair IMRAM.

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