## 3 Methods and Materials  
### Background and Problem Definition  
Due to the inherent nature of MRI, its scanning process tends to be time-consuming, which can lead to patient discomfort and potential motion artifacts caused by slight movements during the scan. As a result, undersampling the k-space has become a main approach to reduce the scanning time. Let \(x_{f}\) represent the fully sampled MR image and \(x_{u}\) represent the zero-filled image. Then the k-space undersampled process in the real scene can be expressed as:  
\[x_{u}=\mathrm{F}^{-1}\Big{(}\mathrm{M}\circ\mathrm{F}x_{f}+\epsilon\Big{)} \tag{1}\]  
where F and F\({}^{-1}\) denote the Fourier and the inverse Fourier transformer respectively, M denotes the undersampling mask, '\(\circ\)' denotes the Hadamard multiplication operation, and \(\epsilon\) denotes the noise. However, undersampling the k-space leads to aliasing artifacts. The CS leverages the inherent sparsity (in MRI, most of the image information is contained in a few key coefficients, allowing the rest to be ignored or set to zero without losing critical details. This enables data compression and reconstruction from fewer samples) of MR images and applies regularization techniques (regularization adds constraints to the reconstruction process, guiding it towards better results) to recover high-fidelity images from noisy k-space data. This approach enables sampling rates far below the Nyquist limit, significantly accelerating MR imaging. In general, the CS-MRI reconstruction model can be expressed as an optimization problem:  
\[x_{f}^{\prime} =\underset{x_{f}}{\mathrm{argmin}}\Big{\|}\mathrm{M}\circ \mathrm{F}x_{f}-k_{u}\Big{\|}_{2}^{2}+\lambda\psi\Big{(}x_{f}\Big{)} \tag{2}\]  
where \(\psi(\cdot)\) denotes a priori regularization which is used to introduce additional knowledge or assumptions about the image, \(\lambda\) denotes the hyperparameter used to balance the reconstruction quality and the regularization effect, \(k_{u}\) denotes the undersampled k-space data, and \(x_{f}^{\prime}\) denotes the reconstructed image close to the ground truth. In the deep learning framework, image reconstruction is transformed into a direct end-to-end learning process, which can learn the direct mapping from zero-filled k-space to fully sampled images. This process can be described as:  
\[x_{f}^{\prime} =\underset{\Theta}{\mathrm{argmin}}\|\mathrm{MF}(\mathcal{N}_{ \Theta}(x_{u}\left|\Theta\right.))-k_{u}\|_{2}^{2}+\lambda\|x_{f}-\mathcal{N}_ {\Theta}(x_{u}\left|\Theta\right.)\|_{2}^{2} \tag{3}\]  
where \(\mathcal{N}_{\Theta}\) denotes the network MHCFIN proposed in this paper driven by parameter \(\Theta\). Based on the above problem description, we will elaborate on the specific details of the proposed network in this section. Specifically, we first outline the overall architecture of MHCFIN, which adopts a dual-branch encoder-decoder network based on the attention mechanism. Then, we deeply explore the different hierarchical multi-modal feature interaction module (MMFIM) and the single-modal feature fusion module (SMFFM). The overall architecture of the proposed method is shown in Figure 2. The network uses a dual-branch learning structure, which can process multi-modal data simultaneously and learn the complementarity between them.