## 7 Registration of MR Scans  
Accurate motion correction through spatial registration of MR scans constitutes a critical component of the proposed joint MR-PET super-resolution framework. The registration step ensures that spatial correspondences across temporal frames and modalities are preserved, thereby enabling effective joint sparsity exploitation and consistent training of the super-resolution network [26, 40, 43, 44]. A multi-scale pyramidal registration network [26, 44] has been implemented to estimate deformation fields between MR frames at multiple resolution levels. The registration network is trained in an unsupervised manner [40, 43] to optimize a spatial transform \(\phi_{\omega}\), parameterized by the network weights \(\omega\), which maps source frames \(I_{t+i}^{\text{MRI}}\) to the target frame \(I_{t}^{\text{MRI}}\), see Figure 3. The pyramidal structure allows the network to capture both large-scale and fine-scale motion components. see Figure 3.  
Figure 3: Multi-scaleregistration and super-resolution network flow. At each training iteration, the registration network receives as input a target MR frame \(I_{t}^{\text{MRI}}\) and a set of surrounding frames \(\{I_{t-R},\ldots,I_{t+R}\}\). The network predicts deformation fields \(\phi_{\omega_{t+i}}\) for each source frame, and the transformed images \(I_{t+i}^{\text{MRI}}\circ\phi_{\omega_{t+i}}\) are compared to the target frame. The registration parameters \(\omega^{*}\) are optimized by minimizing the following objective function:  
\[\min_{\text{MBI},\text{PET}}\|\mathcal{H}(x_{\text{MRI}})\|_{*}+\|\mathcal{H}( x_{\text{PET}})\|_{*}+\mu\,\|\mathcal{H}(x_{\text{MRI}})-\mathcal{H}(x_{\text{PET}})\|_{1}+ \sum_{i\in\{\text{MRI},\text{PET}\}}L_{\text{data}}(x_{i}). \tag{1}\]  
where \(\mathcal{R}(\cdot)\) denotes a spatial smoothness regularization term, which penalizes excessive deformation gradients and enforces physiologically plausible motion. In this optimization objective, \(\omega^{*}\) represents the optimal parameters of the registration network, which are learned to align a set of neighboring MR frames \(\{I_{t-R}^{\text{MRI}},\ldots,I_{t+R}^{\text{MRI}}\}\) to a reference frame \(I_{t}^{\text{MRI}}\). The goal is to minimize the alignment error while encouraging smooth and physiologically plausible deformations. * \(I_{t}^{\text{MRI}}\) is the target MR frame at time \(t\). * \(I_{t+i}^{\text{MRI}}\) denotes neighboring MR frames at relative time offsets \(i\in[-R,R]\). * \(\phi_{\omega_{t+i}}\) is the deformation field, parameterized by \(\omega_{t+i}\), that spatially transforms frame \(I_{t+i}^{\text{MRI}}\) to align it with \(I_{t}^{\text{MRI}}\). * The notation \(I_{t+i}^{\text{MRI}}\circ\phi_{\omega_{t+i}}\) indicates the application of the deformation field to the source frame. * The first term \(\sum_{i=-R}^{R}\|I_{t}^{\text{MRI}}-I_{t+i}^{\text{MRI}}\circ\phi_{\omega_{t+i }}\|_{2}^{2}\) measures the mean squared error (MSE) between the target frame and each motion-compensated neighboring frame. * \(\lambda_{\text{reg}}\) is a hyperparameter that controls the trade-off between registration accuracy and the smoothness of the estimated deformation fields. * \(\mathcal{R}(\phi_{\omega_{t+i}})\) is a spatial regularization term that penalizes large or non-smooth gradients in the deformation fields, typically encouraging spatial smoothness and diffeomorphic properties to ensure physiologically realistic motion. A common choice for \(\mathcal{R}(\cdot)\) is the squared norm of the gradient of the deformation field, promoting gradual, non-abrupt deformations. The combined objective balances accurate frame alignment with physiologically plausible motion modeling, which is critical for preserving anatomical fidelity in the subsequent joint MR-PET reconstruction pipeline.