## 3 Methods
### Basic Super-Resolution Component  
Multiple basic super-resolution components form the main structure of P2P-SR. Each BSRC consists of multi-scale local feature extraction, global feature extraction and image reconstruction. Remote sensing images contain a rich variety and number of details of ground objects. The strong feature extraction capability is the basis for good super-resolution outputs. Both the CNN network and the transformer network were used for feature extraction in their network.  
The feature extraction in our network consists of two parts: convolutional neural networks for extracting multi-scale local features and a transformer-based network for extracting global features. The global feature extraction was designed based on the Swin Transformer backbone. Based on Swin Transformer's pyramidal backbone network, a plain backbone network was designed that was more suitable for super-resolution tasks. In parallel, the multi-scale local feature extraction is achieved by employing residual convolution blocks (RCB) and a feature-upsampling block (FUB). RCB comprises batch normalization, convolutional layers, and an activation function. Following RCB, the feature maps undergo a downsizing process, necessitating their upsampling through FUB before concatenation.  
The residual structure was added at both the layer level and the block level. The residual structure of the layer level was designed to be optional. This is because it is considered that lightweight transformer networks tend to contain more layers in the 3 rd block. The residual structure is more useful in the 3 rd block than in the other blocks. A residual structure that can be chosen flexibly according to the number of layers in the block is beneficial for balancing network performance and lightweighting.  
In the multi-scale local feature extraction part, RCB was designed to extract the multi-scale feature by the residual structure, and FUB was used to unify feature map resolution. The number of RCBs was set as 3, and a lightweight multi-scale feature map fusion structure was designed. The output feature map of the first RCB had the same size as the input image, and it was formulated as  
\[I_{LF,1}=F_{RCB1}(I_{LR}), \tag{2}\]  
where \(F_{RCB1}\) was the first RCB, \(I_{LR}\) was the input remote sensing images after image padding, and \(I_{LF,1}\) was the output of first RCB, Then, \(I_{LF,1}\) was sequentially fed into the two subsequent RCBs, and the feature map size was sequentially reduced by half. The outputs of the last 2 RCBs were fed to the FUB module for upsampling, and the results were formulated as  
\[I_{2}=F_{FUB}(I_{LF,2}), \tag{3}\]  
\[I_{3}=F_{FUB}(I_{LF,3}), \tag{4}\]  
where \(F_{FUB}\) was the FUB module. \(I_{1}\) was defined as \(I_{1}=I_{LF,1}\). \(I_{1}\), \(I_{2}\), and \(I_{3}\) shared the same size and were concatenated by channel. The output of the local feature extraction branch was formulated as  
\[I_{LF}=Concat(I_{1},I_{2},I_{3}), \tag{5}\]Parallelly, 4 Swin transformer layers (STL) were set in the global feature extraction branch. The output of the \(i\)th STL was formulated as  
\[I_{GF,i+1}=F_{STL,i}(I_{GF,i}),\ i=0,1,2,\ldots,N, \tag{6}\]  
where \(F_{STL,i}\) was the \(i\)th STL and the \(I_{GF,i}\) was the output of the \(i\)th STL. The output of the last STL was summed with \(I_{LR}\). The output of the global feature extraction branch was formulated as  
\[I_{GF}=I_{GF,A}+I_{LR}. \tag{7}\]  
Finally, \(I_{LF}\) and \(I_{GF}\) was fed into the image reconstruction module, and the super-resolution result is formulated as  
\[I_{SR}=F_{IR}(I_{LF}+L_{GF}), \tag{8}\]  
where \(F_{IR}\) is the image reconstruction module. Each BSRC magnified the image by a factor of \(\sqrt[N]{M}\). \(N\) was the number of BSRC, and \(M\) was the super-resolution multiplier of the P2P-SR network. Considering that \(\sqrt[N]{M}\) may often be non-integer, the classical but flexible output size method of convolution layers and upsampling was used.