## 5 The Methods Utilized for the Reconstruction of High-Resolution MR-PET Images  
Reconstructing high-resolution MR-PET images from sparsely sampled, noisy, and motion-degraded raw data presents a significant challenge [8; 26; 40]. Conventional super-resolution (SR) approaches based solely on voxel-wise pixel losses tend to produce over-smoothed images that lack critical anatomical detail and perceptual realism [11; 37]. Moreover, in hybrid MR-PET imaging, the presence of motion artifacts and modality-specific noise characteristics further complicates the reconstruction task [26; 40]. To address these challenges, an integrated approach is employed that combines adversarial training [17; 18; 19] with multi-level perceptual and motion-consistency losses, enabling the recovery of fine structural details while preserving cross-modality coherence. This comprehensive loss framework promotes the generation of anatomically faithful, high-resolution images that are robust to noise and motion-induced distortions, thereby enhancing diagnostic reliability. The reconstruction framework builds upon the joint sparsity and optimized sampling strategies described in Sections 4 and 9. The pipeline consists of three main components: (i) initial low-resolution MR and PET reconstruction from sparsely sampled data; (ii) a deformable motion correction network that estimates and compensates for inter-frame motion [26; 40]; and (iii) a WGAN-based super-resolution network [18; 19] that jointly enhances spatial resolution and restores fine anatomical details. The generator network \(G_{\theta}\) takes as input the motion-corrected low-resolution MR and PET images and produces high-resolution outputs that preserve and enhance the anatomical structures shared across both modalities. The architecture employs a deep residual network design [11], incorporating sub-pixel convolution layers [37] to enable efficient and high-quality upsampling. Specifically, the generator consists of an initial feature extraction block, followed by a cascade of 16 residual blocks equipped with skip connections [41] to facilitate gradient flow and mitigate vanishing gradients during training. The final stage includes a sub-pixel convolutional layer that increases spatial resolution by rearranging feature maps [37]. This design allows for the reconstruction of fine image details while maintaining global contextual information. The network is trained end-to-end using a composite loss function that balances pixel-wise fidelity, perceptual similarity, adversarial realism, and motion consistency across frames. This encourages \(G_{\theta}\) not only to produce visually sharp images but also to maintain anatomical coherence between MR and PET reconstructions, which is critical for accurate clinical interpretation. Batch normalization and parametric ReLU activations [42] are employed throughout to improve convergence and model robustness. The generator's capacity to leverage shared anatomical priors from both modalities enhances super-resolution performance, particularly under severe undersampling and motion degradation scenarios [27]. The discriminator network \(D_{\phi}\) is trained to distinguish between real high-resolution MR-PET image pairs and those generated by \(G_{\theta}\). A patch-based discriminator design is adopted [11], where image patches are classified independently to encourage the generator to produce realistic textures at both local and global scales. The overall generator loss \(\mathcal{L}_{G}\) is defined as follows:  
\[\mathcal{L}_{G}=\lambda_{\text{adv}}\cdot\mathbb{E}_{x}\big{[}-D_{\phi}(G_{ \theta}(x))\big{]}+\lambda_{\text{perc}}\cdot\mathcal{L}_{\text{perc}}+\lambda_ {\text{pix}}\cdot\mathcal{L}_{\text{pix}}+\lambda_{\text{mc}}\cdot\mathcal{L}_ {\text{mc}}+\lambda_{\text{sparse}}\cdot\mathcal{L}_{\text{sparse}}.\]  
The generator loss \(\mathcal{L}_{G}\) is a composite objective designed to balance multiple aspects of the image reconstruction quality.