## 2 Joint Sparseness on MR/PET  
Although the modalities employed exhibit distinct physical principles, reconstruction is frequently conducted independently for each modality despite their shared spatial geometry. This traditional approach overlooks the opportunity to leverage the complementary anatomical and functional information embedded in the multimodal data [29]. By jointly reconstructing MR and PET images while explicitly modeling their inherent structural similarities, the reconstruction process can be substantially improved. This integrated framework enables enhanced suppression of motion-induced artifacts, augments spatial resolution, and promotes more accurate cross-modality alignment [27]. Consequently, the synergistic fusion of MR and PET data not only facilitates superior image quality but also supports more reliable quantitative analyses, thereby advancing diagnostic precision and clinical decision-making [30]. To exploit this potential, a joint sparsity framework has been adopted [27; 31]. Rather than reconstructing the modalities independently, a joint sparsity algorithm has been applied, which simultaneously leverages structural similarities and enforces shared sparse representations. This is particularly effective in reducing the impact of involuntary patient motion during image acquisition and in enhancing the overall resolution by combining information from both sparse datasets. The approach builds on compressed sensing (CS) principles [8], using Partial Fourier sampling and exploiting conjugate symmetry to induce sparsity in the MRI and PET datasets. Furthermore, it enhances the classical voxel-wise sparsity model by introducing structured sparsity through matrix lifting [27]. Specifically, the data are embedded into block Hankel or Toeplitz matrix forms, which are known to exhibit low-rank structure when the underlying images contain spatial redundancy and self-similarity [31]. The choice of block Hankel matrices for representing structured sparsity in this work is motivated by several key advantages over traditional voxel-wise sparsifying transforms. First, Hankel embeddings are well suited to capture both local and global redundancies present in anatomical and functional images. Anatomical structures such as tissue boundaries, organ contours, and smooth textures naturally induce strong correlations between overlapping local patches in the image domain [31]. When such overlapping patches are reorganized into Hankel matrices, the resulting structures exhibit low-rank properties that can be effectively exploited for denoising and reconstruction. Second, Hankel matrix representations inherently encode shift-invariant features, allowing the model to capture repetitive and self-similar patterns across the image, which are often missed by voxel-wise sparse models. This is particularly advantageous in multimodal MR-PET imaging where shared anatomical features manifest with varying intensities and contrast, yet maintain consistent spatial patterns. By enforcing low-rank constraints on the joint Hankel embeddings of MR and PET images, the reconstruction framework encourages coherent structural recovery across modalities, thereby enhancing the quality and anatomical fidelity of the resulting images [32]. Third, the structured sparsity induced by Hankel matrices facilitates robust motion compensation and artifact suppression. The low-rank structure is resilient to localized distortions and noise [31], enabling more accurate separation of true anatomical signals from motion-induced inconsistencies. This property improves the stability and convergence of the reconstruction algorithm, especially under aggressive undersampling conditions characteristic of accelerated MR-PET acquisitions. Moreover, the low-rank structure of Hankel matrices has been demonstrated to be highly effective in compressed sensing MRI and CT applications [27], enabling superior reconstruction quality under aggressive undersampling.