## 3 Joint Sparse Sampling and Learned Acquisition for MR-PET Imaging  
The process of sparse sampling in hybrid MR-PET systems presents a unique opportunity to move beyond traditional acquisition protocols and toward optimized joint acquisition and reconstruction [27,29]. In conventional workflows, MR and PET data are acquired using separate hardware and software chains, with sampling patterns designed independently for each modality. However, recent advances in both physics-driven and learning-based methods have demonstrated that jointly optimized sparse sampling strategies can noticeably enhance image quality while drastically reducing acquisition time [8,27]. By exploiting the complementary nature of MR and PET signals, sparsity can be enforced not only within each modality but also across them, enabling cross-modality information sharing during reconstruction [29,31]. This joint sparsity paradigm leverages shared anatomical structures and functional correlations, enabling more accurate recovery from undersampled data. Modern formulations incorporate structured sparse representations, including block Hankel and Toeplitz matrix models [27,31], which capture the inherent low-rank and sparse patterns in MR and PET data more effectively than unstructured approaches. Moreover, integrating deep learning frameworks, particularly generative adversarial networks (GANs) trained to respect joint sparsity constraints [11,17], provides a powerful means to reconstruct high-fidelity images from heavily undersampled raw data. These approaches dynamically learn the manifold of plausible MR-PET images, reducing noise and artifacts while preserving fine anatomical and metabolic details critical for diagnosis. Additionally, adaptive sampling schemes guided by uncertainty quantification [33] and task-specific criteria enable further acceleration by selectively acquiring the most informative k-space and sinogram samples. This strategy ensures efficient utilization of scanner time and patient comfort without compromising diagnostic accuracy. In summary, the synergy between advanced sparse sampling designs, structured joint sparsity models, and learning-based reconstruction methods heralds a new era of fast, reliable, and high-resolution MR-PET imaging that holds immense promise for clinical applications and research. In the present work, a unified approach to sparse sampling has been adopted, in which the underlying goal is to maximize anatomical and functional consistency across modalities while respecting hardware and physiological constraints. Specifically, the PET acquisition  
Figure 2: Pipeline for high-resolution MR-PET image reconstruction. Sparse MR k-space and PET sinogram data are reconstructed into initial low-resolution images \(x^{\text{MRLLR}}\) and \(x^{\text{PETLR}}\), corrected for motion via deformation fields \(T_{\theta}\), and passed through a shared generator network \(G_{\theta}\) to produce high-resolution images. Structured joint sparsity \(\|\mathcal{H}(x^{\text{MRI}})-\mathcal{H}(x^{\text{PET}})\|\) is enforced during training. The discriminator \(D_{\phi}\) supervises adversarial training to improve image fidelity. has been performed using a non-uniform angular sampling based on the PROPELLER scheme [34], in which individual projection subsets (blades) are adaptively selected to prioritize regions of high anatomical relevance. The blade selection has been optimized through an iterative process that balances coverage and redundancy, ensuring sufficient sampling density in critical areas while minimizing total acquisition time. Concurrently, the MR acquisition employs a hybrid sparse sampling strategy combining Poisson-disc random sampling [35] with Partial Fourier coverage and conjugate symmetry exploitation [8]. This design ensures full sampling of the low-frequency k-space components critical for anatomical delineation, while higher frequency components are acquired sparsely to allow acceleration.