# Multi-Scale Factor Image Super-Resolution Algorithm with Information Distillation Network
## 2 Materials and Methods
### Multi-Factor Upsampling Module  
The upsampling module mainly includes position projection, weight prediction, and feature mapping. The Location Projection projects pixels onto the LR image. The Weight Prediction Module predicts the weights of the filter for each pixel on the SR image. Finally, the Feature Mapping function maps the feature on the LR image with the predicted weights back to the SR image to calculate the value of the pixel. After \(I_{LR}\) extracts image features through the information distillation module, the output feature map is \(F_{LR^{\prime}}\), and the network finally outputs \(I_{SR}\). According to the principle that a pixel on the HR image can be back-projected to the \(I_{LR}\), pixel \((i,j)\) on the \(I_{SR}\) can be determined by a pixel \((i^{\prime},j^{\prime})\) on the LR image and the filter weight. Therefore, the upsampling module needs a specific filter to match \((i^{\prime},j^{\prime})\) and \((i,j)\). The formula is shown in Formula (7). \(\Phi(\bullet)\) is the mapping function from \(I_{LR}\) to \(I_{HR}\). \(F_{LR^{\prime}}(i^{\prime},j^{\prime})\) represents the pixel on the \(I_{LR}\), and \(I_{SR}(i,j)\) represents the pixel on the \(I_{SR}\). \[I_{SR}(i,j)=\Phi\big{(}F_{LR^{\prime}}\big{(}i^{\prime},j^{\prime}\big{)},W(i, j)\big{)} \tag{7}\]  
(1) Position projection  
Position projection is to back-project \(I_{SR}\) onto \(F_{LR^{\prime}}\), as shown in Figure 5. The value of pixel \((i,j)\) on \(I_{SR}\) is determined by the point \((i^{\prime},j^{\prime})\) on \(F_{LR^{\prime}}\).the relationship between these two pixels is expressed by Formula (8). \[\big{(}i^{\prime},j^{\prime}\big{)}=T(i,j)=\left(\left|\frac{i}{r^{\prime}}, \frac{j}{r}\right|\right) \tag{8}\]  
Among them, \(T(\bullet)\) is the conversion function, which converts the point \((i,j)\) into \((i^{\prime},j^{\prime})\). \(\left|\frac{i}{r},\frac{j}{r}\right|\) is floor function, and \(r\) is scale-factor. It can be seen that adding a scale factor to calculate the relationship between two pixels is suitable for SISR with any scale factor. The Location Projection can upscale the feature maps with arbitrary scale factor. The scale factor \(r\) is divided into two types: integer and non-integer. When \(r\) is an integer, for example, when \(r\) is 2, one pixel in the LR image can determine two pixels in the HR image, as shown in Figure 6a. When the scale factor is a non-integer, for example, \(r\) is 1.5, one pixel in the LR image determines one or two pixels in the HR image, as shown in Figure 6b. No matter whether \(r\) is an integer or a non-integer, there is always a unique point on the LR image corresponding to a point on the SR image, and these two pixels are called the most relevant pixel pair. Figure 5: Location projection schematic diagram. Different from the typical upscale module, we use a network to predict the filter weights. This process is called weight prediction, expressed by Formula (9):  
\[W(i,j)=\varphi\big{(}I_{i,j};\theta\big{)} \tag{9}\]  
\(\varphi\big{(}I_{i,j};\theta\big{)}\) represents the weight prediction process, \(I_{i,j}\) is the input of the weight prediction network, \(\theta\) is the parameter of the weight prediction network, and \(W(i,j)\) is the weight at the pixel \((i,j)\). At the pixel \((i,j)\), the input \(I_{ij}\) of \(\varphi(\bullet)\) can be expanded to the relative offset of \((i^{\prime},j^{\prime})\), which is expressed as followed by Formula (10):  
\[I_{ij}=\left(\frac{i}{r}-\left|\frac{i}{r}\right|\frac{i}{r}-\left|\frac{j}{r} \right|\right) \tag{10}\]  
To train multiple scale factors for a network, we add scale factor \(r\) to the expression of \(I_{ij}\). Assuming that the image is upscaled by 2 and 4, then \(I_{SR_{2}}\) and \(I_{SR_{4}}\) are obtained. Arbitrary pixels \((i,j)\) on \(I_{SR_{2}}\) will have the same filter weights and position projection coordinates as \((2i,2j)\) pixels on \(I_{SR_{4}}\).