## 3 Joint Sparse Sampling and Learned Acquisition for MR-PET Imaging
Together, these complementary sampling strategies are synchronized to leverage the joint sparsity and structural correlations between PET and MR data, enhancing the quality and consistency of the reconstructed images. Partial Fourier sampling exploits Hermitian symmetry in k-space [8], effectively halving the acquisition time by reconstructing missing data from conjugate symmetric counterparts. The Poisson-disc random sampling pattern, characterized by a minimum distance between samples [35], reduces coherent aliasing artifacts and improves reconstruction robustness by spreading undersampling artifacts incoherently. Moreover, conjugate symmetry enables efficient k-space completion algorithms [8], which utilize the inherent redundancy in MR data to fill in missing samples. This approach complements the compressed sensing framework by providing additional constraints that enhance image fidelity, especially in regions with fine anatomical details. Together, these strategies harmonize to accelerate MR acquisition without sacrificing diagnostic quality, thereby reducing patient discomfort and increasing throughput. When combined with PET's inherently sparse sinogram data acquisition [29], this joint sparse sampling methodology unlocks new possibilities for simultaneous MR-PET imaging, enabling faster scans with improved spatial and temporal resolution. Mathematically, the PET acquisition can be modeled as the application of a sparse system matrix \(A_{\text{PET}}^{\text{sparse}}\), constructed from the full system matrix \(A_{\text{PET}}\) by selecting a subset of projection angles and detector bins. Let \(M_{\text{PET}}\) denote the corresponding binary sampling mask, with ones indicating active detector bins and angles. The observed PET sinogram is then given by the following:  
\[y^{\text{PET}}=M_{\text{PET}}\cdot A_{\text{PET}}(x^{\text{PET}})+\eta^{ \text{PET}},\]  
where \(\eta^{\text{PET}}\) denotes acquisition noise. The acquisition masks for both MR and PET are not static but learned through a gradient-based refinement process, which adaptively prioritizes anatomically informative regions. The scheme synchronizes sampling density across modalities, leveraging the shared anatomical priors to enhance structural fidelity while minimizing redundancy. In parallel, the MR acquisition is modeled as the application of a variable density Fourier sampling operator \(F_{\text{MRI}}^{\text{sparse}}\). The corresponding binary sampling mask \(M_{\text{MRI}}\) indicates which k-space samples are acquired. The observed k-space data are thus expressed as follows:  
\[y^{\text{MRI}}=M_{\text{MRI}}\cdot F(x^{\text{MRI}})+\eta^{\text{MRI}},\]  
where \(F\) denotes the full Fourier transform and \(\eta^{\text{MRI}}\) accounts for thermal and system noise. In this formulation, \(y^{\text{PET}}\) represents the observed PET sinogram data after sparse sampling and acquisition. The matrix \(M_{\text{PET}}\) is a binary sampling mask that selects a subset of projection angles and detector bins used during acquisition, effectively modeling the sparse sampling pattern applied to the PET system. The operator \(A_{\text{PET}}(\cdot)\) denotes the forward projection operator, which maps the PET image \(x^{\text{PET}}\) from the image domain to the sinogram domain by simulating the physical process of photon detection along projection paths (i.e., the PET system matrix). The term \(\eta^{\text{PET}}\) accounts for acquisition noise in the PET data, typically modeled as a combination of Poisson noise (arising from the quantum nature of radioactive decay) and Gaussian noise (introduced by detector electronics and signal processing). The resulting equation describes how the sparse and noisy PET measurement \(y^{\text{PET}}\) is generated from the underlying PET image \(x^{\text{PET}}\) under the current acquisition protocol. Importantly, the design of the sampling masks \(M_{\text{PET}}\) and \(M_{\text{MRI}}\) is not performed independently.