## 3 Joint Sparse Sampling and Learned Acquisition for MR-PET Imaging
Both are constructed with awareness of the expected joint sparsity structure of the target images, as described in Section 3[27]. In particular, anatomical regions that are expected to exhibit correlated structure in MR and PET domains are given higher priority in the sampling design. For PET, this translates to denser sampling of angular sectors corresponding to such regions; for MR, it translates to targeted k-space coverage emphasizing spatial frequencies representing those anatomical features. This coordinated sampling strategy maximizes the mutual information between the two modalities and enhances the efficacy of joint reconstruction algorithms. By incorporating prior knowledge about the common anatomical support and exploiting structured sparsity, the sampling patterns reduce redundant acquisition of uninformative data [33], thereby optimizing total acquisition time. This joint approach facilitates synergistic reconstruction, allowing the use of coupled sparsity constraints and structured low-rank priors, which improve noise resilience and artifact suppression in the fused image space. Furthermore, this co-optimized sparse sampling framework enables adaptive allocation of sampling density based on clinical priorities or pathology-specific information [33], potentially allowing dynamic scan protocols tailored to individual patients or diagnostic tasks. The resulting acceleration in data acquisition directly benefits clinical workflow by shortening scan times without compromising image quality, as demonstrated by improved quantitative metrics and enhanced visual fidelity in subsequent experimental results. To formalize this, a joint acquisition operator can be defined as follows:  
\[\mathcal{A}_{\text{joint}}(x^{\text{MRI}},x^{\text{PET}})=\Big{(}M_{\text{MRI }}\cdot F(x^{\text{MRI}}),\ M_{\text{PET}}\cdot A_{\text{PET}}(x^{\text{PET}}) \Big{)}.\]  
In this expression, \(\mathcal{A}_{\text{joint}}(x^{\text{MRI}},x^{\text{PET}})\) denotes the joint acquisition operator, which models the combined data acquisition process for both MRI and PET modalities. Specifically, the term \(M_{\text{MRI}}\cdot F(x^{\text{MRI}})\) represents the sparsely sampled k-space data in the MRI domain. Here, \(M_{\text{MRI}}\) is a binary sampling mask that selects a subset of k-space samples to be acquired, \(F(\cdot)\) denotes the full Fourier transform operator, and \(x^{\text{MRI}}\) is the underlying MRI image to be reconstructed. Similarly, \(M_{\text{PET}}\cdot A_{\text{PET}}(x^{\text{PET}})\) represents the sparsely sampled PET sinogram data. In this term, \(M_{\text{PET}}\) is the PET sampling mask selecting a subset of projection angles and detector bins, \(A_{\text{PET}}(\cdot)\) is the PET system matrix (forward projection operator), and \(x^{\text{PET}}\) is the PET image volume being reconstructed. This joint modeling captures the fact that both modalities undergo different acquisition processes but are coupled through shared anatomical structures. By formulating the acquisition operator jointly, the reconstruction framework can exploit cross-modality correlations during image recovery, enabling improved consistency and quality in the reconstructed MR-PET images. The corresponding data fidelity term used in the reconstruction is then:  
\[\mathcal{L}_{\text{data}}=\|M_{\text{MRI}}\cdot F(x^{\text{MRI}})-y^{\text{MRI }}\|_{2}^{2}+\|M_{\text{PET}}\cdot A_{\text{PET}}(x^{\text{PET}})-y^{\text{ PET}}\|_{2}^{2}.\]In this formulation, \(\mathcal{L}_{\text{data}}\) represents the total data fidelity loss, which ensures that the reconstructed images for both MRI and PET modalities remain consistent with the actually acquired raw data. The first term, \(\|M_{\text{MRI}}\cdot F(x^{\text{MRI}})-y^{\text{MRI}}\|_{2}^{2}\), measures the squared \(\ell_{2}\) norm (Euclidean distance) between the sparsely sampled k-space data predicted from the reconstructed MRI image \(x^{\text{MRI}}\) and the acquired MRI k-space data \(y^{\text{MRI}}\).