## 3 Methods
### Efficient Structure Extraction Module  
Based on the mechanism of the human visual system, the human eyes are highly sensitive to edge information in images [28]. Consequently, edge information holds significant importance in visual perception [29], and the visual quality of an image is closely related to its edge information [30]. In the context of image super-resolution, the inherent structural information of low-resolution face images, which can be considered as a form of edge information, plays a crucial role. Building upon the inspiration from [31], we have developed an efficient structural information extraction module that extracts edge features and structural information from a multi-scale perspective. This approach allows us to fully leverage the structural feature correlations present in images and effectively recover more precise and distinct structural details in facial images.  
Given the high sensitivity of edges and structures to scale variations, the input features \(F_{in}\) are divided into four segments with varying channel counts using three Cross-Convolution Blocks (CCBs) to extract features at different scales. The architecture of the CCBs is illustrated in Figure 3a. To progressively weigh the features in the channel dimension, we employ Cross Convolution [31] and Efficient Channel Attention (ECA) [32]. This weighting strategy enables the network to prioritize the restoration of facial structure and enhance the preservation of contour details. Subsequently, the features extracted by the CCB across different channels are fused, and the hierarchical features of the channels are consolidated through two convolutional layers with PRelu activation functions and ECA. This process yields the final output feature \(F_{out}\), which can be expressed as follows:  
\[F_{out}=H_{fuse}\Big{(}\left[HC_{3}^{0},HC_{2}^{1},HC_{1}^{2},HC_{0}^{3}\right] \Big{)} \tag{5}\]  
where \(HC_{j}^{k}\) denotes the \(k\)th group after the \(j\)th CCB and \(H_{fuse}\) represents the fusion operation.  
In contrast to conventional convolutions, cross-convolution employs two distinct vertical filters \(k_{1\times m}\) and \(k_{m\times 1}\), for asymmetric filtering of the features. By leveraging gradient information in both the vertical and horizontal directions, these filters emphasize the structural characteristics of edge contours. The edge information is subsequently fused and reinforced to obtain the final output feature \(F_{out}^{Cross}\). The process can be expressed as follows, where \(b\) represents the bias term:  
\[F_{out}^{Cross}=Conv(k_{1\times m^{\prime}},F_{in}^{CCB}\Big{)}+Conv\Big{(}k_{ m\times 1},F_{in}^{CCB}\Big{)}+b \tag{6}\]  
Figure 2: The architecture of MSRB.  
The structure of ECA, as depicted in Figure 3b, offers a more efficient and lightweight alternative to SE-Net [21]. It overcomes the challenge of diminished learning caused by dimensionality reduction. ECA starts by performing channel-wise averaging of the features. The weight for each channel is then computed by jointly considering the aggregated features of that channel and its neighboring channels. This is accomplished using a one-dimensional convolution with a kernel size of \(k\), which facilitates inter-channel feature interactions. The calculation can be expressed as follows:  
\[w=Sigmoid(Conv_{1\times k}(y)) \tag{7}\]  
For a given channel dimension \(C\), the size \(k\) of the convolution kernel can be calculated by Equation (8).  
\[k=\Big{|}\frac{log_{2}(C)+1}{2}\Big{|}_{odd} \tag{8}\]  
where \(\big{|}t\big{|}_{odd}\) indicates the nearest odd number. Finally, the information of each channel is fused to obtain the output features, which are used to enhance the information exchange between channels at different levels by capturing the local cross-channel information.