## 3 Joint Sparse Sampling and Learned Acquisition for MR-PET Imaging
The second and third terms in the objective, \(\beta\cdot\|M_{\text{PET},\theta}\|_{1}\) and \(\gamma\cdot\|M_{\text{MRI},\theta}\|_{1}\), are \(\ell_{1}\) norm penalties applied to the learned sampling masks. The \(\ell_{1}\) norm encourages sparsity in the masks by penalizing the total number of active (non-zero) sampling locations. This regularization ensures that the learned acquisition scheme remains efficient and avoids trivial solutions where all samples are acquired. * The hyperparameter \(\beta\) controls the trade-off between PET sampling sparsity and reconstruction quality. Larger values of \(\beta\) will promote sparser PET sampling, leading to shorter scan times but possibly higher reconstruction errors. * The hyperparameter \(\gamma\) similarly controls the trade-off for MRI sampling. Larger \(\gamma\) promotes more aggressive k-space undersampling in MRI, again balancing acquisition time versus image fidelity. In practice, \(\beta\) and \(\gamma\) are typically tuned empirically. Reasonable ranges are
* \(\beta\in[10^{-4},10^{-2}]\)
* \(\gamma\in[10^{-4},10^{-2}]\)  
with exact values depending on the desired target sparsity level and acceptable tradeoff in reconstruction fidelity. Overall, this formulation allows for automatic optimization of the acquisition masks in synergy with the reconstruction network, resulting in a data-driven, task-adaptive sensing strategy that can outperform conventional hand-crafted sampling schemes. This hybrid design--combining physics-inspired sampling schemes with learned optimization--has yielded significant improvements in both MR and PET reconstruction quality under extreme undersampling regimes [27; 33]. Moreover, the learned sensing matrices have been integrated with the joint structured sparsity framework and GAN-based super-resolution pipeline [11], resulting in a fully unified system. Sparse raw data from both modalities are jointly processed through a pipeline that enforces shared anatomical structure, addresses modality-specific variations, and leverages generative adversarial networks to enhance resolution and suppress artifacts. This holistic approach achieves accelerated acquisition without compromising image fidelity, thus paving the way for more efficient and clinically viable hybrid MR-PET imaging protocols. In summary, the sparse sampling strategy presented here goes beyond traditional acceleration techniques. By embracing a joint, learned, and structure-aware approach to MR and PET acquisition, it provides a foundation for highly efficient multimodal imaging that is robust to noise, motion, and hardware limitations [33]. The resulting system constitutes a well-established example of end-to-end optimized MR-PET acquisition and reconstruction. Future work will explore the development of fully co-designed joint sampling schemes for MR-PET systems, in which the acquisition patterns of both modalities are optimized together in an end-to-end fashion. This approach has the potential to further enhance joint reconstruction quality, reduce scan time, and improve robustness to motion and hardware constraints. The Application of Generative Adversarial Networks (GANs) Within the Framework of Super Resolution Image Reconstruction  
The problem of super-resolution image reconstruction from sparsely sampled and noisy MR-PET data is inherently ill-posed, posing significant challenges for accurate image recovery. Traditional pixel-wise reconstruction losses, such as mean squared error (MSE), tend to produce overly smooth and blurred outputs, which lack the critical high-frequency anatomical details necessary for precise diagnostic evaluation [13]. This limitation is especially pronounced in hybrid MR-PET imaging, where maintaining fine structural fidelity is essential for accurate cross-modality registration and quantitative PET analysis. To overcome these obstacles, advanced reconstruction strategies have been developed that incorporate perceptual [13] and adversarial losses [17; 19], enabling the recovery of sharper, more realistic images.