## 3 Joint Sparse Sampling and Learned Acquisition for MR-PET Imaging
These approaches leverage the power of Generative Adversarial Networks (GANs) [17] to learn complex data distributions and restore textural details that are typically lost in conventional reconstructions. By integrating domain-specific priors and enforcing joint sparsity constraints between MR and PET modalities [27], the reconstruction framework not only enhances spatial resolution but also preserves functional consistency, thereby improving the reliability and clinical utility of the reconstructed images. In this work, a GAN-based framework is adopted to address the challenges of super-resolution reconstruction from sparsely sampled MR and PET data. The generator network, denoted as \(G_{\theta}\), is trained to produce high-fidelity super-resolved MR and PET images from the undersampled raw inputs, effectively recovering fine anatomical and functional details. Concurrently, a discriminator network, denoted as \(D_{\phi}\), is trained to differentiate between real high-resolution images and the synthesized outputs generated by \(G_{\theta}\). The training process is formulated as a minimax game between the generator and discriminator, leveraging the Wasserstein GAN (WGAN) formulation with gradient penalty [19]. This approach stabilizes the adversarial training, mitigates mode collapse, and encourages the generator to produce outputs that are both perceptually realistic and statistically consistent with the distribution of real data. By incorporating this adversarial learning paradigm, the model transcends traditional pixel-wise losses, enabling the recovery of high-frequency details and textures that are often lost in MSE-based reconstructions [13]. The synergy between \(G_{\theta}\) and \(D_{\phi}\) fosters enhanced anatomical fidelity and functional coherence in the reconstructed MR-PET images, which is critical for accurate clinical interpretation and quantitative analysis. The generator network \(G_{\theta}\) receives as input the sparsely sampled MR and PET images, along with the outputs of a deformable motion correction module [26]. This motion correction network is trained jointly with \(G_{\theta}\) to estimate and compensate for motion-induced artifacts across input frames. The generator output consists of super-resolved MR and PET images that are spatially aligned and corrected for motion. The discriminator network \(D_{\phi}\) receives both real and generated image pairs and is trained to produce scalar scores reflecting the perceived realism of the inputs. A lower score indicates a higher probability of the image being generated, while a higher score indicates a real image. The overall training objective consists of multiple components. The generator loss \(\mathcal{L}_{G}\) is given by the following:  
\[\mathcal{L}_{G}=\lambda_{\text{adv}}\cdot\mathbb{E}_{x}\big{[}-D_{\phi}(G_{ \theta}(x))\big{]}+\lambda_{\text{perc}}\cdot\mathcal{L}_{\text{perc}}+\lambda _{\text{pix}}\cdot\mathcal{L}_{\text{pix}}+\lambda_{\text{mc}}\cdot\mathcal{ L}_{\text{mc}}.\]  
Here, \(\mathcal{L}_{\text{adv}}\) denotes the adversarial loss [19] that encourages the generator to produce perceptually realistic outputs, while \(\mathcal{L}_{\text{perc}}\) represents the perceptual loss, calculated as the \(\ell_{2}\) distance between high-level feature representations of the generated and ground truth images, typically extracted using a pretrained VGG network [36]. The term \(\mathcal{L}_{\text{pix}}\) corresponds to the pixel-wise reconstruction loss, commonly implemented as the MSE between the generated and reference images. Additionally, \(\mathcal{L}_{\text{mc}}\) is a motion consistency loss [26] that promotes temporal coherence and spatial alignment across sequential frames or modalities, thereby enhancing the robustness of the reconstruction in dynamic imaging contexts.