## 3 Blur Kernel Estimation
### Gradient Norm Ratio Priori  
The essence of image degradation is the loss and aliasing of pixel information. This process will result in the loss of high-frequency information of the image, which is reflected in the intuitive visual effect, that is, the edge texture of the low-resolution image is not clear and the contrast is low. Therefore, it is necessary to find an index as a constraint. When the reconstructed image tends to blur latent image, the index increases, and when it tends to clear latent image, the index decreases. As a result, the restored image in the iterative process is close to the latent HR image, achieving an accurate estimation of blur kernel. In optimization problems, the most common constraints are \(L_{1}\) norm, \(L_{2}\) norm and \(L_{p}\) norm. Generally, \(0<p<1\). Taking \(L_{1}\) norm as an example, this is mostly used to punish the high frequency band in image processing. When the image contains noise, the noise can be removed by minimizing the \(L_{1}\) norm. When the image is degraded, the high-frequency band itself will decay, and minimizing the \(L_{1}\) norm of the high-frequency component of the image will lead to further blur of the reconstructed image. For the other norms, because its mechanism is similar to the \(L_{1}\) norm, if the norm constraint is directly introduced as the regular term, it cannot improve the quality of image reconstruction. However, it is found that although the cost decreases with the decrease of the resolution of the input image, that is, the blur of the reconstructed image, the cost reduction speed of different norms is different, and the cost reduction speed of \(L_{2}\) norm is significantly higher than that of \(L_{1}\) norm. Therefore, the introduction of high-frequency information norm ratio as a priori information can ensure that the penalty cost increases with the reduction of reconstructed image quality. In other words, the purpose of improving reconstructed image quality can be achieved by reducing the cost. The image high-frequency information can be obtained in many ways. In this paper, the transverse and longitudinal gradient is selected as an example to constrain the reconstructed image in the form of norm ratio. Figure 1 is a schematic diagram of the change of the high-frequency information cost of reconstructed images when different constraints are adopted at twice the down-sampling ratio. High-frequency information is obtained by convoluting the reconstructed image with the transverse difference operator \(f_{1}=\left[1-1\right]\) and the longitudinal difference operator \(f_{2}=\left[1-1\right]^{T}\), respectively. The reconstructed image is a series of images obtained from the infrared image of typical power equipment, which is reconstructed back to the original size through cubic interpolation after Gaussian kernel action with different width and 2 times down sampling. It can be clearly seen from Figure 1 that with the increase of ambiguity, the cost of high-frequency information under each norm constraint is decreasing, while the cost of the ratio of \(L_{1}\) norm to \(L_{2}\) norm is increasing. Define \(\text{GNR}(x)=\parallel x_{1}\parallel/\parallel x_{2}\parallel\text{GNR}(x)\) is introduced into the objective function as a constraint term. During the process of minimizing an objective function, the closeness of the intermediate latent image to the clear image can help estimate blur kernels more accurately. The estimation model of the blur kernel is:  
\[ \begin{split}\left(\hat{X},\hat{h}\right)=\underset{X,h}{ \text{argmin}}\parallel Y-S_{l}(h\otimes X)S_{r}\parallel_{2}^{2}+\lambda(GNR (f_{1}\otimes X)+GNR(f_{2}\otimes X))\\ +\gamma\parallel h\parallel_{2}^{2}+\delta\parallel\Psi^{T}X \parallel_{1}\end{split}  \tag{2}\]  
where \(\lambda,\delta\), and \(\gamma\) are weighting coefficients; \(S_{l}\) is a row sampling matrix; \(S_{r}\) is a column sampling matrix; \(X\) and \(Y\) is the two-dimensional form corresponding to \(x\) and \(y\) respectively; \(\Psi\) is sparse basis.