## 3 Methods  
### Multi-Level Supervision Super-Resolution Network  
As mentioned earlier, super resolution is a typically ill-posed problem. Its ill-posedness mainly comes from the huge space of all possible functions for mapping from low-resolution images to high-resolution images. Increasing the number of supervisions in the network can increase the constraints on the network training, gradually reduce the space of possible functions, and reduce the ill-posedness of super resolution. Most of the current super-resolution methods adopted a one-step supervision structure, which was only supervised once at the output of the super-resolution network, resulting in insufficient constraints in the super-resolution process. The unsuppressed ill-posedness of super resolution is visualized in the network output by the loss of image details and the appearance of pseudo-detail. In order to reduce the ill-posedness of the super-resolution process and improve the super-resolution network performance, a progressive super-resolution network was proposed for remote sensing images based on multi-level supervision. As shown in Figure 1, the P2P-SR network consists of several basic super-resolution components and multi-level supervision. The overall super-resolution scalar factor of the P2P-SR network can be set flexibly. The BSRCs can be increased or decreased like building blocks, with fewer BSRCs for a smaller overall super-resolution scale factor and more BSRCs for a larger overall super-resolution scale factor. The scale factor of each BSRC is determined by the overall scale factor and the number of BSRCs. The BSRC consists of a multi-scale feature extraction module, a global feature extraction module, and an image reconstruction module. Considering that the scale factor of each BSRC should have some flexibility, the network structure of each BSRC is the same except for the image reconstruction module. This helps to shorten the overall training time of the network. During the network training, the training of the first BSRC was completed first. Its trained parameters were shared to the subsequent BSRCs, and all BSRCs were tuned together simultaneously to complete the overall network training. Our network used a progressive super-resolution structure and multi-level supervision. But this is not a simple stacking of network components. The cross-level loss functions were designed to facilitate better transfer of image detail loss across the network components. The effect of the number of BSRCs and each BSRC scale factor on network performance was explored when the overall scale factor was certain. These will be explained in detail in later sections. To conveniently describe the multilevel supervision network, the overall scale factor was set to \(M\), and the number of BSRCs was set to \(N\). The P2P-SR net used \(N\) BSRC modules to accomplish \(M\)-times super resolution. At the output of each component, the super-resolution result was compared with the ground truth downsampled to the same size. The loss function was calculated between the result and the ground truth. The progressive super-resolution process of low-resolution remote sensing images can be expressed as  
\[\left\|I\right\|_{SR}=\text{min}\Bigg{(}\lambda_{1}Loss_{N}+\lambda_{2}\sum_{n =1}^{N-1}Loss_{n}\Bigg{)}, \tag{1}\]  
where \(I\) is the input low-resolution image, \(\left\|\cdot\right\|_{SR}\) is the process of super resolution, \(Loss_{n}\) is the loss function of the \(n\)th level, and \(\lambda_{1}\) and \(\lambda_{2}\) are adjustable weights. From Equation (1), it is clear that the multi-level supervision network aims to reduce the loss of each BSRC and the overall network loss at the same time. The loss of the \(N\)th BSRC component is given a separate adjustable weight \(\lambda_{1}\). This is because the \(N\)th component is the last component of the network, and its output is also the output of the overall network. The output of the \(N\)th component is evaluated not only by the loss function, but also by the visual perception of the human eye.