## 3 Joint Sparse Sampling and Learned Acquisition for MR-PET Imaging
In this framework, the generator loss \(\mathcal{L}_{G}\) combines several complementary terms:  
* \(\lambda_{\text{adv}}\cdot\mathbb{E}_{x}\big{[}-D_{\phi}(G_{\theta}(x))\big{]}\) is the adversarial loss term, which encourages the generator \(G_{\theta}\) to produce images that are realistic enough to fool the discriminator \(D_{\phi}\). * \(\lambda_{\text{perc}}\cdot\mathcal{L}_{\text{perc}}\) is the perceptual loss, which ensures that the high-level semantic features of the generated images match those of the ground-truth images, helping to preserve important anatomical details. * \(\lambda_{\text{pix}}\cdot\mathcal{L}_{\text{pix}}\) is the pixel-wise loss, typically computed as mean squared error (MSE), which penalizes differences at the pixel level to promote accurate image reconstruction. * \(\lambda_{\text{mc}}\cdot\mathcal{L}_{\text{mc}}\) is the motion consistency loss, which enforces spatial and temporal alignment across sequential frames or modalities, thereby reducing motion artifacts. The hyperparameters \(\lambda_{\text{adv}}\), \(\lambda_{\text{perc}}\), \(\lambda_{\text{pix}}\), and \(\lambda_{\text{mc}}\) serve as weighting factors that balance the contributions of each term to the total loss \(\mathcal{L}_{G}\). Their relative values are tuned empirically to achieve optimal trade-offs between realism, anatomical accuracy, and motion robustness in the reconstructed images. The discriminator loss \(\mathcal{L}_{D}\) follows the WGAN-GP formulation and is defined as follows:  
\[\mathcal{L}_{D}=\mathbb{E}_{x_{\text{real}}}\big{[}D_{\phi}(x_{\text{real}}) \big{]}-\mathbb{E}_{x_{\text{gen}}}\big{[}D_{\phi}(G_{\theta}(x_{\text{gen}})) \big{]}+\lambda_{\text{gp}}\cdot\mathbb{E}_{\hat{x}}\Big{[}\big{(}\|\
abla_{ \hat{x}}D_{\phi}(\hat{x})\|_{2}-1\big{)}^{2}\Big{]},\]  
where \(\hat{x}\) are interpolated samples between real and generated images, and \(\lambda_{\text{gp}}\) is the gradient penalty coefficient ensuring Lipschitz continuity of \(D_{\phi}\)[19]. The discriminator loss \(\mathcal{L}_{D}\) follows the Wasserstein GAN with Gradient Penalty (WGAN-GP) formulation and consists of three key terms:  
* \(\mathbb{E}_{x_{\text{real}}}\big{[}D_{\phi}(x_{\text{real}})\big{]}\) encourages the discriminator \(D_{\phi}\) to assign higher scores to real high-resolution images \(x_{\text{real}}\). * \(-\mathbb{E}_{x_{\text{gen}}}\big{[}D_{\phi}(G_{\theta}(x_{\text{gen}}))\big{]}\) penalizes the discriminator for assigning high scores to generated (fake) images produced by the generator \(G_{\theta}\), thus encouraging discrimination between real and generated samples. * \(\lambda_{\text{gp}}\cdot\mathbb{E}_{\hat{x}}\Big{[}\big{(}\|\
abla_{\hat{x}} D_{\phi}(\hat{x})\|_{2}-1\big{)}^{2}\Big{]}\) is the gradient penalty term, which enforces the Lipschitz continuity constraint required for stable WGAN training. Here, \(\hat{x}\) are random interpolations between real and generated samples, and \(\lambda_{\text{gp}}\) controls the strength of the penalty. Together, these components guide the discriminator to robustly distinguish between real and generated images while ensuring stable adversarial training. The generator architecture \(G_{\theta}\) is based on a deep residual network with sub-pixel convolution layers [37] for efficient upsampling. The network consists of an initial feature extraction block, multiple residual blocks with skip connections, and a final upsampling block. The residual blocks enable effective learning of high-frequency details, while the sub-pixel convolutions avoid artifacts commonly introduced by naive upsampling layers. The deformable motion correction module is implemented as a separate convolutional network [26] that estimates spatial deformation fields between input frames and a canonical reference frame. These deformation fields are applied to align the input data prior to super-resolution reconstruction. The motion correction loss \(\mathcal{L}_{\text{mc}}\) is computed as the temporal consistency error between aligned frames and the reference.