## 5 The Methods Utilized for the Reconstruction of High-Resolution MR-PET Images
It includes the following components:  
* \(\lambda_{\text{adv}}\cdot\mathbb{E}_{x}\big{[}-D_{\phi}(G_{\theta}(x))\big{]}\) is the adversarial loss term that encourages the generator \(G_{\theta}\) to produce images that can fool the discriminator \(D_{\phi}\). The weight \(\lambda_{\text{adv}}\) controls the contribution of this term. * \(\lambda_{\text{perc}}\cdot\mathcal{L}_{\text{perc}}\) is the perceptual loss, where \(\mathcal{L}_{\text{perc}}\) measures the feature-space difference between the generated and ground truth images using a pre-trained deep neural network (typically VGG). This promotes the preservation of high-level semantic and structural features. \(\lambda_{\text{perc}}\) controls the influence of this term. * \(\lambda_{\text{pix}}\cdot\mathcal{L}_{\text{pix}}\) is the pixel-wise loss, typically implemented as an \(L_{1}\) or \(L_{2}\) distance (e.g., MSE) between the generated and reference images. It enforces low-level fidelity. The corresponding weight is \(\lambda_{\text{pix}}\). * \(\lambda_{\text{mc}}\cdot\mathcal{L}_{\text{mc}}\) is the motion consistency loss, which penalizes discrepancies between temporally or spatially aligned image frames after motion correction. This ensures coherent reconstruction across sequences. The parameter \(\lambda_{\text{mc}}\) tunes the importance of this term. * \(\lambda_{\text{sparse}}\cdot\mathcal{L}_{\text{sparse}}\) is the structured sparsity regularization loss, designed to enforce joint sparsity and low-rank structure in the Hankel matrix representations of MR and PET images. It promotes anatomical consistency across modalities. The hyperparameter \(\lambda_{\text{sparse}}\) controls this term's contribution. By jointly optimizing all these terms, the generator is trained to produce high-quality, anatomically accurate, and perceptually realistic MR-PET images that are robust to noise, motion artifacts, and sparse sampling. Here, \(\mathcal{L}_{\text{adv}}\) denotes the adversarial loss, encouraging the generator to produce perceptually realistic outputs [17; 18; 19]. The perceptual loss \(\mathcal{L}_{\text{perc}}\) is computed as the \(\ell_{2}\) distance between feature representations extracted from intermediate layers of a pretrained VGG network [13], ensuring that the generated images preserve semantic and structural fidelity. The pixel-wise reconstruction loss \(\mathcal{L}_{\text{pix}}\), implemented as mean squared error (MSE), penalizes deviations in intensity values and supports the accurate recovery of fine image details. The motion consistency loss \(\mathcal{L}_{\text{mc}}\) enforces temporal coherence by aligning the generated frames in dynamic imaging scenarios, thereby reducing artifacts caused by patient movement or physiological motion during acquisition [26]. The structured sparsity regularization term \(\mathcal{L}_{\text{sparse}}\) incorporates domain-specific anatomical priors by promoting joint sparsity across the MR and PET modalities [27], improving cross-modality alignment and reducing modality-specific artifacts. Together, these loss components synergize to produce super-resolved images that are both quantitatively accurate and visually plausible, facilitating improved diagnostic utility. The motion consistency loss \(\mathcal{L}_{\text{mc}}\) is computed as follows:  
\[\mathcal{L}_{\text{mc}}=\mathbb{E}_{t}\Big{[}\|G_{\theta}(x_{t})-T_{t\to 0}G_{ \theta}(x_{0})\|_{2}^{2}\Big{]},\]  
where \(T_{t\to 0}\) is the estimated deformation field mapping frame \(t\) to the reference frame 0 [26]. The discriminator loss \(\mathcal{L}_{D}\) follows the WGAN-GP formulation [19]:  
\[\mathcal{L}_{D}=\mathbb{E}_{x_{\text{real}}}\big{[}D_{\phi}(x_{\text{real}}) \big{]}-\mathbb{E}_{\text{sgn}}\big{[}D_{\phi}(G_{\theta}(x_{\text{gen}})) \big{]}+\lambda_{\text{gp}}\cdot\mathbb{E}_{\hat{x}}\Big{[}\big{(}\|\
abla_{ \hat{x}}D_{\phi}(\hat{x})\|_{2}-1\big{)}^{2}\Big{]},\]  
where \(\hat{x}\) represents interpolated samples between real and generated images, and \(\lambda_{\text{gp}}\) is the gradient penalty coefficient.