## 9 Multiplexed PET and Variable-Density MRI Sampling with Joint Hankel-Based Reconstruction
The sparse sampling pattern \(M_{\text{MRI}}\) is designed to balance acquisition speed with preservation of essential image information, enabling high-quality reconstruction through compressed sensing and deep learning-based methods. The joint design of \(C_{\text{PET}}\) and \(M_{\text{MRI}}\) is optimized to exploit the shared anatomical structures and complementary functional information across modalities [27], which manifest as common sparse components in an appropriately chosen transform domain. This joint sparsity is captured and enforced through structured Hankel matrix constraints [27; 31], which promote low-rankness in the lifted domain:  
\[\min_{x^{\text{MRI}},x^{\text{PET}}}\|\mathcal{H}(x^{\text{MRI}})-\mathcal{H }(x^{\text{PET}})\|_{1}+\lambda_{\text{MRI}}\|y^{\text{MRI}}-M_{\text{MRI}}F (x^{\text{MRI}})\|_{2}^{2}+\lambda_{\text{PET}}\|y^{\text{PET}}-C_{\text{ PET}}s^{\text{PET}}(x^{\text{PET}})\|_{2}^{2}, \tag{17}\]  
where \(\mathcal{H}(\cdot)\) denotes the block Hankel operator enforcing low-rank and joint sparse structure [31], and \(s^{\text{PET}}(x^{\text{PET}})\) is the mapping from the PET image domain to detector signal space. In this joint optimization problem, the goal is to reconstruct high-quality MRI and PET images \(x^{\text{MRI}}\) and \(x^{\text{PET}}\) from sparsely sampled and noisy measurements, while enforcing shared structural priors. The first term, \(\|\mathcal{H}(x^{\text{MRI}})-\mathcal{H}(x^{\text{PET}})\|_{1}\), enforces joint sparsity across the modalities by penalizing differences between their corresponding block Hankel matrix embeddings \(\mathcal{H}(\cdot)\). The \(\ell_{1}\) norm promotes sparsity in the residual Hankel domain, encouraging the MRI and PET reconstructions to share common anatomical structure while allowing for modality-specific variations. The second term, \(\lambda_{\text{MRI}}\|y^{\text{MRI}}-M_{\text{MRI}}F(x^{\text{MRI}})\|_{2}^{2}\), is a data fidelity loss for the MRI image. Here, \(y^{\text{MRI}}\) represents the acquired sparse k-space measurements, \(F(x^{\text{MRI}})\) is the Fourier transform of the image, and \(M_{\text{MRI}}\) is the binary k-space sampling mask. The squared \(\ell_{2}\) norm penalizes discrepancies between the forward model of the reconstructed MRI image and the acquired data. The weighting factor \(\lambda_{\text{MRI}}\) controls the relative importance of this fidelity term. The third term, \(\lambda_{\text{PET}}\|y^{\text{PET}}-C_{\text{PET}}s^{\text{PET}}(x^{\text{PET }})\|_{2}^{2}\), similarly enforces data consistency for the PET reconstruction. Here, \(y^{\text{PET}}\) denotes the multiplexed PET detector measurements, \(s^{\text{PET}}(x^{\text{PET}})\) represents the forward model mapping the image to the detector signal space, and \(C_{\text{PET}}\) is the structured sensing matrix applied during acquisition. Again, \(\lambda_{\text{PET}}\) balances this fidelity term relative to the other objectives. Together, this formulation enables simultaneous reconstruction of MRI and PET images that are both data-consistent and structurally aligned, leveraging shared anatomical priors via Hankel-based joint sparsity while respecting modality-specific noise characteristics and acquisition models. The hyperparameters \(\lambda_{\text{MRI}}\) and \(\lambda_{\text{PET}}\) play a critical role in tuning the tradeoff between strict data fidelity and the enforcement of joint structural priors. Higher values of \(\lambda\) promote closer agreement with the raw measurements, while lower values allow more flexibility in enforcing cross-modal alignment through the Hankel-based regularization. Optimal values are typically determined empirically based on validation performance and the noise characteristics of each modality.