## 2 Related Work
### The Ill-Posedness of Super Resolution  
The development of super-resolution methods has always been accompanied by the discussion of the ill-posedness of super resolution. In [17], the smaller receptive fields of convolutional networks were thought to worsen the ill-posedness of super resolution. The convolutional neural networks used for super resolution at that time had fewer layers and smaller network receptive fields. Local information contained by small patches was insufficient to recover image detail. The VDSR proposed in [7] used a deeper convolutional network to increase the perceptual field, using the information contained in the larger patches for image-detail recovery. The performance improvement was significant compared to SRCNN, which has fewer layers. With the rapid development of deep networks, the network depth and number of parameters increased dramatically. The Transformer for vision tasks expanded the field of perception even further to the entire image. Therefore, it is no longer appropriate to reduce the ill-posedness of super resolution by deepening the network. As the first application of GAN to super resolution, reducing the ill-posedness of super resolution was the main motivation of this work [36]. In [36], the researchers considered that multiple different super-resolution images could be reconstructed from the same low-resolution image. Pixel-wise loss functions such as MSE, which were commonly used at the time, were considered to be averaging over all super-resolution images. This resulted in over-smoothing of the image and loss of image detail. Therefore, a GAN-based super-resolution network was proposed to drive the super-resolution results towards a perceptually more convincing solution. Both of these methods were effective in improving the performance of super-resolution networks, but unfortunately the relationship between ill-posedness and network constraints was not discussed.  
Multiple different high-resolution images can be downsampled to the same low-resolution image. In turn, multiple different super-resolution images can be reconstructed from the same low-resolution image. The task of super resolution is to select the closest one of these super-resolution images to the ground truth. In SRFlow [18], a super-resolution method based on normalized flow was proposed to learn the conditional distribution of the output. SRFlow therefore directly accounted for the ill-posedness of the problem and learned to predict diverse photo-realistic high-resolution images. In [10], the ill-posedness of super resolution was reduced from another perspective. Super resolution was considered to be the process of fitting a mapping function from a low-resolution image to a high-resolution image. Due to the ill-posedness of super resolution, the space of all possible mapping functions is huge. This makes it difficult to train the network. In order to address this issue, a dual regression network (DRN) [20] was proposed. During the training process of the DRN, the super-resolution image was reconstructed from the low-resolution input image, and the loss between it and the ground truth was calculated. The super-resolution image was then downsampled to the same size as the input image and compared to the input image, and another loss was calculated. The essence of DRN is the increase in the number of network supervisions. The increase in supervision benefits the introduction of more ground truth and imposes more constraints on the network. The improved performance of super-resolution networks due to DRN illustrated the effectiveness of increased supervision in reducing the ill-posedness of super resolution.