## 2 Joint Sparseness on MR/PET
Recent studies have further extended these techniques to multimodal fusion scenarios, where cross-modality correlations can be expressed through joint constraints on Hankel embeddings [27]. In this context, the joint low-rankness of the MRI and PET Hankel matrices reflects shared anatomical content while allowing modality-specific variations. Furthermore, this approach mitigates artifacts arising from motion, noise, and sparse sampling by leveraging the redundancy and correlation inherent in multimodal images. The structured low-rank modeling via Hankel embeddings thus acts as a powerful regularizer that preserves salient features such as tissue boundaries and functional uptake regions, which are critical for accurate diagnosis and quantification. Importantly, the joint low-rank constraint aligns with the underlying physics and biology captured by MR and PET modalities, making it a natural and principled choice for integrated image reconstruction. The incorporation of Hankel matrix-based structured sparsity into a unified deep learning framework enables end-to-end optimization, where learned priors can further adapt to complex multimodal distributions. This hybrid strategy combines the interpretability and robustness of model-based low-rank regularization with the flexibility and representation power of neural networks [27], resulting in well-established performance in high-resolution MR-PET image reconstruction. Hankel-based models also provide a flexible and data-driven framework for structured sparsity, without requiring the selection of a fixed sparsifying basis. This is particularly advantageous in multimodal reconstruction, where the optimal representation may vary between modalities and anatomical regions. The use of block Hankel embeddings allows the model to adaptively capture the inherent redundancy in the data, resulting in more robust and consistent fusion of MRI and PET images. In this framework, the MRI and PET image volumes, denoted as \(x^{\text{MRI}}\) and \(x^{\text{PET}}\), are mapped into structured matrices using a block Hankel operator \(\mathcal{H}(\cdot)\). The joint sparsity constraint is then formulated as an optimization problem:  
\[\min_{x^{\text{MRI}},x^{\text{PET}}}\ \|\mathcal{H}(x^{\text{MRI}})\|_{*}+\| \mathcal{H}(x^{\text{PET}})\|_{*}+\lambda\|\mathcal{H}(x^{\text{MRI}})- \mathcal{H}(x^{\text{PET}})\|_{1}+\sum_{i=\text{MRLPET}}\mathcal{L}_{\text{ data}}(x^{i}),\]  
where the nuclear norm \(\|\cdot\|_{*}\) promotes low-rank structure, the \(\ell_{1}\) norm of the residuals \(\|\mathcal{H}(x^{\text{MRI}})-\mathcal{H}(x^{\text{PET}})\|_{1}\) encourages joint sparsity between modalities, and the data fidelity terms \(\mathcal{L}_{\text{data}}(x^{i})\) ensure consistency with the acquired raw data. The data fidelity terms are given by  
\[\mathcal{L}_{\text{data}}(x^{\text{MRI}})=\|A_{\text{MRI}}(x^{\text{MRI}})-y^ {\text{MRI}}\|_{2}^{2},\quad\mathcal{L}_{\text{data}}(x^{\text{PET}})=\|A_{ \text{PET}}(x^{\text{PET}})-y^{\text{PET}}\|_{2}^{2},\]  
where \(A_{\text{MRI}}\) represents Fourier-based k-space sampling and \(A_{\text{PET}}\) models the PET projection operator. The nuclear norm \(\|\cdot\|_{*}\), defined as the sum of the singular values of a matrix, serves as a convex surrogate for the non-convex matrix rank. In this framework, applying the nuclear norm to the block Hankel embeddings \(\mathcal{H}(x^{\text{MRI}})\) and \(\mathcal{H}(x^{\text{PET}})\) promotes low-rank structure, which corresponds to the inherent spatial redundancy and self-similarity in anatomical and functional images. This regularization enhances the denoising capability and supports accurate reconstruction from highly undersampled raw data. The use of the nuclear norm is thus critical for enforcing structured sparsity in the lifted matrix domain, complementing the \(\ell_{1}\) term that enforces joint sparsity between modalities.