## 3 Methods
### The Scale Factor Division Method  
For the super-resolution process with a certain scale factor, a method was needed to divide the overall scale factor into each BSRC. As the number of BSRCs increases, the number of supervisions increases, and the super-resolution ill-posedness is reduced. At the same time, the total number of network layers increases, and the number of computations increases. The sum of super-resolution network performance and computational complexity can be formulated as  
\[Cost=\min\left[\sum_{n}P_{n}+\sum_{n}Loss_{n}\left(I_{n}^{SR},GT_{n}\right) \right], \tag{9}\]  
where \(P_{n}\) is the high-resolution of the parameters of the \(n\)th BSRC, and \(I_{n}^{SR}\) and \(GT_{n}\) are the super-resolution output and the ground truth of the \(n\)th BSRC, respectively. \(P_{n}\) and \(Loss_{n}\) are vastly different in order of magnitude and need to be normalized before summing. As seen from the above equation, the correlation between \(Cost\) and the number of BSRCs \(n\) is difficult to predict. It may be assumed that the number of BSRCs increases from 1. Initially, the number of BSRCs is small, and with the network being sufficiently trained, the increase in the number of BSRCs benefits the increase in the number of supervisions in the network, which in turn helps to reduce the super-resolution ill-posedness. The reduction in super-resolution ill-posedness helps to reduce the probability of detail loss and pseudo-detail in the output image, which leads to a reduction in the \(Loss_{n}\) of the \(n\)th BSRC. In this case, the increase in the number of BSRCs also leads to an increase in the number of network parameters. If the network is properly designed to be lightweight so that the increase of \(P_{n}\) is smaller than the decrease of \(Loss_{n}\), then \(Cost\) has the possibility to be reduced. In the comparison of experimental results, the performance improvement of our method on 4-times super-resolution results is more significant than 2 times, although 4-times super resolution requires more BSRCs. As the number of BSRCs continues to increase, the number of BSRCs may be saturated for a super-resolution task of certain scale factor. More BSRCs bring limited performance gains, but significantly increase the number of network parameters. Therefore, finding the appropriate number of BSRCs is important for our multi-level supervision-based super-resolution network. After determining the number of BSRCs, the scale factor of each BSRC is also worth discussing. Let the number of BSRCs in the network be \(N\) and the \(i\)th BSRC super-resolutionmultiplier be \(m_{i}\), then the relationship between the number of BBRCs, the scale factor of each BSRC, and the overall scale factor of our super-resolution network is as follows:  
\[M=\prod_{i=1}^{N}m_{i},m_{i}>1. \tag{10}\]  
From the above equation, it can be seen that when \(N\) increases, the scale factor of each component decreases. When \(N\) tends to infinity, \(m_{i}\) tends to 1. This means that each BSRC only magnifies the input image at a very small factor. Each BSRC needs only a very small number of parameters to perform its own super-resolution task. Therefore, when \(N\) tends to infinity, the trend of increasing or decreasing the total number of parameters is difficult to predict. Therefore, it is necessary to optimize the BSRC combination and find out the relationship between super-resolution network performance, the number of parameters, and the scale-factor division method. In the experiments on the various scale-factor division ways, it was proposed to start from two aspects, namely the number of BBRCs in the network and the super-resolution scale factor of each BSRC. In terms of the number of BBRCs, the higher the number of BBRCs, the more training constraints were introduced into the network, but the greater the computational complexity. The optimal number of BBRCs was obtained by comparing the super-resolution network performance and the computational complexity of different numbers of BBRCs.