## 2 Joint Sparseness on MR/PET
The data fidelity terms employ the squared \(\ell_{2}\) norm \(\|\cdot\|_{2}^{2}\), which corresponds to the sum of squared differences between the forward model predictions and the acquired raw measurements. Specifically, \(\mathcal{L}_{\text{data}}(x^{\text{MRI}})\) measures the discrepancy between the Fourier-domain representation of the reconstructed MR image, \(A_{\text{MRI}}(x^{\text{MRI}})\), and the observed k-space samples \(y^{\text{MRI}}\). Likewise, \(\mathcal{L}_{\text{data}}(x^{\text{PET}})\) quantifies the error between the PET forward projection \(A_{\text{PET}}(x^{\text{PET}})\) and the measured sinogram data \(y^{\text{PET}}\). Minimizing these terms ensures that the reconstructed images remain consistent with the original measured data, thereby preserving physical fidelity while allowing for the incorporation of additional regularization priors such as structured sparsity. This optimization is solved using an alternating minimization scheme, in which updates to \(x^{\text{MRI}}\) and \(x^{\text{PET}}\) are performed iteratively via proximal gradient descent and low-rank matrix factorization [27]. Additionally, this joint sparsity term is incorporated as an auxiliary loss in the GAN-based super-resolution architecture described in the subsequent sections of this work. The total generator loss is expressed as follows:  
\[\mathcal{L}_{\text{GAN}}^{\text{joint}}=\mathcal{L}_{\text{GAN}}+\alpha \Big{(}\|\mathcal{H}(x^{\text{MRI}})\|_{*}+\|\mathcal{H}(x^{\text{PET}})\|_{* }+\lambda\|\mathcal{H}(x^{\text{MRI}})-\mathcal{H}(x^{\text{PET}})\|_{1} \Big{)},\]  
where \(\alpha\) balances the influence of the structured sparsity prior. In the joint loss formulation \(\mathcal{L}_{\text{GAN}}^{\text{joint}}\), the hyperparameters \(\alpha\) and \(\lambda\) play critical roles in balancing the influence of the structured sparsity prior relative to the GAN-based adversarial and perceptual components of the total loss. The parameter \(\alpha\) globally weights the entire structured sparsity regularization term within the total objective. A larger value of \(\alpha\) emphasizes the enforcement of low-rank and joint-sparse representations through the nuclear norm and \(\ell_{1}\) penalty terms, thereby pro moting cross-modality anatomical consistency and denoising effects. Conversely, a smaller \(\alpha\) shifts the optimization focus toward achieving perceptual realism through the GAN and perceptual losses. In practice, \(\alpha\) is typically selected from the range \(\alpha\in[10^{-3},10^{-1}]\), and is tuned empirically to balance anatomical fidelity with perceptual quality. The parameter \(\lambda\) governs the relative contribution of the cross-modality joint sparsity constraint \(\|\mathcal{H}(x^{\text{MRI}})-\mathcal{H}(x^{\text{PET}})\|_{1}\). Higher values of \(\lambda\) enforce stronger alignment and consistency of structural patterns between MRI and PET modalities, which is beneficial for tasks such as multi-modal fusion and anatomical correspondence preservation, see Figure 1. However, excessively large \(\lambda\) may over-constrain the optimization and suppress modality-specific features. In this work, \(\lambda\) is selected from the range \(\lambda\in[0.1,1.0]\), with optimal values determined via cross-validation on a validation subset. The joint tuning of \(\alpha\) and \(\lambda\) is critical to achieving the desired tradeoff between anatomical coherence, modality-specific contrast, and perceptual sharpness in the final super-resolved MR-PET images. Experimental validation was performed using both phantom and in vivo datasets acquired on a Biograph mMR hybrid MR-PET scanner. Comparisons were conducted between reconstructions with and without the structured joint sparsity term. Performance was evaluated using PSNR and SSIM across various raw data sparsity levels (20%, 40%, 60%, 80%, and 100%).