## 1 Introduction  
Magnetic resonance (MR) imaging is a revolutionary medical imaging technology that plays a crucial role in clinical diagnosis and biomedical research due to its excellent soft tissue contrast, radiation-free safety, and high-resolution imaging of living structures [1]. However, due to the physical nature, the scanning process of MR takes a long time [2]. This may not only cause discomfort to the patient but may also produce motion artifacts due to the patient's small movements, thus hindering its further promotion and application [2]. To address this problem, several solutions for accelerated MR imaging have been established [3, 4, 5, 6]. The mainstream is to restore high-quality images from undersampled images with aliasing artifacts. This accelerates MR imaging through undersampling k-space data [7, 8, 9, 10]. In this way, data acquisition time can be significantly reduced. However, undersampling causes aliasing artifacts in the image (as shown in Figure 1), affecting image quality. To address this problem, previous studies have proposed compressed sensing (CS) algorithms to recover high-quality MR images from undersampled k-space data. The CS [3, 11] methods exploit the sparsity of images and reconstruct high-quality images by imposing sparsity constraints on undersampled data through optimization algorithms. However, the CS reconstruction process usually involves complex optimization algorithms and has high computational costs [12]. In addition, the performance of CS depends on the selection of multiple hyperparameters, and optimizing these parameters requires experience and extensive experiments [13]. More recently, deep learning [12; 14; 16; 17; 18; 19; 20; 21; 22; 23; 24], has excelled in reducing the computational burden and solving nonlinear problems, thus handling inverse problems more efficiently. Typically, in deep learning-based methods [12], the image reconstruction problem is formulated as an end-to-end learning task, directly mapping undersampled images to fully sampled images. In addition, these methods [16; 25] can also restore k-space data directly. Most scanners can generate multi-modal images using different acquisition parameters. The complementary information between these multi-modal images is very beneficial to MR imaging. As shown in Figure 1a, the T1-weighted images (T1 WI) and T2-weighted images (T2 WI) are different modalities from the same subject. These modalities usually have structural similarities and modality-specific information. However, due to the characteristics of MR imaging, T1 WI acquisition has a shorter repetition time (TR) and echo time (TE). Therefore, in many studies, images with shorter acquisition times are used as auxiliary modalities (prior information) to guide the accelerated imaging of the target modality with longer acquisition times [14; 26; 27]. Joint reconstruction explores complementary information among multi-modal images in varying degrees [12; 28; 29]. In addition, super-resolution reconstruction [30; 31; 32], can enhance image resolution and introduce high-resolution (HR) modalities to guide the restoration of low-resolution (LR) modalities. These methods provide new ideas for joint learning of multi-modal images. However, early studies [18; 33; 34; 33] usually spliced images of different modalities into the network or directly input images of different modalities. Although this improved the image quality to a certain extent, it has not fully utilized the potential complementarity and interaction between different modalities. In this study, to explore the complementarity between different modalities, multi-hierarchical modality correlations are investigated and a multi-scale approach is adopted to fuse the target modality (e.g., proton density-weighted images--PDs) and auxiliary modality (e.g., fat-suppressed proton density-weighted images--PDFSs) to achieve synchronous acceleration of multi-modal imaging. Specifically, this study proposed a novel multi-hierarchical complementary feature interaction network (MHCFIN).