## 3 Proposed Method
### Overview  
The proposed method, illustrated in Figure 2, is a symmetrical hierarchical network consisting of three stages: encoding, bottleneck, and decoding. The encoding stage aims to extract and enhance both local facial details and global facial structures. Meanwhile, the bottleneck stage is intended to optimize the encoded low-level features. Finally, the decoding stage is introduced to facilitate multi-scale feature fusion and image reconstruction. To simplify the description, we use the notations \(\mathbf{I}_{LR}\), \(\mathbf{I}_{SR}\), and \(\mathbf{I}_{HR}\) to represent the low-resolution (LR) images, the super-resolved (SR) images, and the ground-truth high-resolution (HR) images, respectively. **(1) Encoding Stage**: The goal of the encoding stage is to extract and enhance both local facial details and global facial structures. To begin with, the face images traverse a \(3\times 3\) convolution layer to extract their low-level features. Since the output channel number should exceed the input ones while an excessive number of output channels would significantly increase the computational complexity [53], we suggest using 32 output channels for optimal performance. Afterward, the extracted shallow features are passed through three encoding stages. Each stage comprises an Attention-Guided Transformer Module--Top (AGTM-T) and a Pixel-Related Downsample Module (PRDM). The AGTM-T comprises an Attention Guiding Block--Top (AGB-T) and a Channel-wise Multi-head Transformer Block (CMTB). All the blocks and modules mentioned above will be discussed in the following subsection. Moreover, it is worth noting that the channel of the input feature maps doubles, and the size of the input feature maps halves after each encoding stage. **(2) Bottleneck Stage**: In the bottleneck stage, there are a large number of encoded feature maps, but each one is relatively small in size compared to those in the encoding  
Figure 2: The structure of the proposed attention-guided transformer with pixel-related deconvolution network. It is a symmetrical hierarchical network containing three stages: encoding, bottleneck, and decoding. The goal of the encoding stage is to extract and enhance both local facial details and global facial structures. Meanwhile, the bottleneck stage is intended to optimize the encoded low-level features. Finally, the decoding stage is introduced to facilitate multi-scale feature fusion and image reconstruction. Here, the red and green arrows in MFFM represent downsampling and upsampling, respectively. stage. To better use these features in the decoding stage, we introduce the Attention-Guided Transformer Module--Bottleneck (AGTM-B). Unlike AGTM-T in the encoding stage, the guiding blocks in AGTM-B aim to further enhance the low-level encoded features. By applying AGTM-Bs, the model can continuously strengthen different facial features and focus on a broader range of facial structures. **(3) Decoding Stage**: The decoding stage of the proposed method aims to reconstruct high-quality face images by utilizing previously extracted and refined multi-scale features. In this stage, low-level features are initially fed into the Pixel-Related Upsample Module (PRUM). The module halves the feature map channel and doubles the feature map size, which is the opposite of the PRDM in the encoding stage. After this, the upsampled features are combined with features from other scales using the Multi-scale Feature Fusion Module (MFFM) to enhance network flexibility and achieve better restoration performance. The well-combined features are then fed to AGTM-T for further refinement of image details. Finally, a convolutional layer with a size of \(3\times 3\) is utilized to convert the learned feature maps to the output face image \(I_{Out}\). The final SR face image output \(I_{SR}\) is obtained by adding the LR face image \(I_{LR}\), which has been upsampled to the same size as the HR image through bicubic interpolation, to the output face image \(I_{Out}\).