## 2 Proposed Method
### Overview  
The proposed method, illustrated in Figure 2, is a multi-scale connected symmetrical hierarchical network containing three stages: encoding, bottleneck, and decoding. The encoding stage aims to extract and promote both local facial details and global facial structures, while the bottleneck stage is designed to optimize the encoded low-level features. Finally,the decoding stage is introduced for multi-scale feature fusion and image reconstruction. For convenience of description, we denote \(\mathbf{I}_{\text{LR}}\), \(\mathbf{I}_{\text{SR}}\), and \(\mathbf{I}_{\text{HR}}\) as the low-resolution (LR) images, super-resolved (SR) images, and ground-truth high-resolution (HR) images, respectively. _(1) Encoding Stage_: The encoding stage aims to extract and promote both local facial details and global facial structures. First, a \(3\times 3\) convolution layer is applied to extract shallow features. Here, we suggest using 32 output channels for optimal performance. It is essential to keep in mind that the number of output channels should exceed the number of input channels, while an excessive number of output channels may lead to a significant increase in computation complexity and a decrease in feature information storage effectiveness [53]. Afterwards, the extracted shallow features are passed through three encoding stages. Each stage contains a carefully designed Spatial-Channel Mutual Attention-guided Transformer Module-Top (SCATM-T) and a downsampling block. The SCATM-T contains a Spatial-Channel Mutual Attention Guiding Block-Top (SCAGB-T) and a Channel-wise Multi-head Transformer Block (CMTB), which are discussed in the following subsection. The downsampling block consists of three layers: a \(3\times 3\) convolutional layer with a stride of 2, followed by a LeakyReLU activation function layer, then another \(3\times 3\) convolutional layer with a stride of 1. Please note that the first convolutional layer doubles the feature channels while simultaneously halving the feature map size. _(2) Bottleneck Stage_: The bottleneck stage contains a large number of encoded feature maps, though each map is relatively small compared to those in the encoding stage. To better utilize these feature map features in the decoding stage, we introduce the Spatial  
Figure 2: The structure of the proposed spatial-channel mutual attention-guided transformer network. It is a symmetrical hierarchical network containing three stages: encoding, bottleneck, and decoding. The encoding stage aims to extract and promote both local facial details and global facial structures, while the bottleneck stage is designed to optimize the encoded low-level features. Finally, the decoding stage is introduced for multi-scale feature fusion and image reconstruction. Channel Mutual Attention-guided Transformer Module-Bottom (SCATM-B). Unlike the SCATM-T in the encoding stage, the guiding blocks in SCATM-B aim to further enhance the low-level encoded features. With the help of SCATM-Bs, the model can focus on a greater variety of facial structures and continuously strengthen different facial features. _(3) Decoding Stage_: The decoding stage aims to reconstruct high-quality face images based on the previously extracted and refined multi-layer features. In the decoding stage, the low-level features are first fed to the upsampling block, which contains a \(6\times 6\) transposed convolutional layer with a stride of 2 followed by a LeakyReLU activation function layer and a \(3\times 3\) convolution layer with a stride of 1. The feature channels are halved and the feature map size is doubled in the first transposed convolutional layer, which is the opposite of the downsampling process in the encoding stage. Afterward, the upsampled features are combined with features from other scales by the Multi-scale Feature Fusion Module (MFFM), extending the network flexibility and resulting in better restoration performance. Then, the well-combined features are fed to the SCATM-T for further image detail enhancement.