# An Unsupervised Weight Map Generative Network for Pixel-Level Combination of Image Denoisers
## 3 Methodology
### Loss Function  
The UWMGN utilizes the loss function as the training target to assess the quality of the combined image without the ground truth. We discuss our loss function is this section.  
Information entropy (IE) is a method for evaluating the information content of images. The higher the information entropy, the more details there are in the image and the better the quality of the image. As a result, information entropy loss is used to suppress noise interference. The expression of the information entropy loss function is as follows:  
\[L_{\text{IE}}=-\sum_{g=0}^{d}p(g)\log_{2}p(g) \tag{13}\]  
where \(p(g)\) denotes the probability of the occurrence of the intensity value \(g\) in the combined image \(\hat{x}\), and \(d\) represents the range of intensity values in the combined image, which usually ranges from 0 to 1 or 0 to 255. We set the information entropy to a negative number such that the \(L_{\text{IE}}\) amount of the combined image will continue decreasing and more details will be retained, further improving the quality of the final combined image.  
Total variation (TV) calculates a value based on the input signal. It was observed by [PERSON] et al. that the TV of a noisy image is much higher than that of a noise-free image. Therefore, image denoising can be performed by minimizing the TV value [35]. Hence, we utilized TV to regularize the loss function, which will be minimized by CNN to reduce noise in the image. The loss function is given as  
\[L_{\text{TV}}=\sum_{i=1}^{H}\sum_{j=1}^{W}\left[(\hat{x}(i,j+1)-\hat{x}(i,j))^ {2}+(\hat{x}(i+1,j)-\hat{x}(i,j))^{2}\right]^{\frac{1}{2}}, \tag{14}\]  
where \(W\) and \(H\) are the width and height of the combined image, respectively, and \(\hat{x}(i,j)\) is the \((i,j)\)-th pixel in the combined image.  
Figure 4: Structure of our backbone network. \(Z\) refers to the initial random noise. \(O_{1},O_{2},\dots,O_{K}\) are the generated original weight maps. \(W_{1},W_{2},\dots,W_{K}\) are the weight maps after normalization.  
As the PSNR [36] is widely used to assess denoising performance, we added an MSE to the loss function to adjust our mapping parameters. MSE is a measure that reflects the difference between the estimated and actual values. In this study, we used MSE loss to measure the difference between the initial noisy image \(y\) and the output image \(\hat{x}\). Then, the loss function is given as \(L_{\text{MSE}}\) in Equation (15).  
\[L_{\text{MSE}}=\frac{1}{WH}\sum_{i=1}^{H}\sum_{j=1}^{W}|y(i,j)-\hat{x}(i,j)|, \tag{15}\]  
where \(y(i,j)\) is the \((i,j)\)-th pixel in the noisy image. In comparison to the standard MSE, Equation (15) can significantly improve the smoothness and contrast of the uniform area of the image.  
We obtained the total loss function by combining all the aforementioned loss functions. The total loss is indicated by \(L\).  
\[L=\lambda_{\text{MSE}}L_{\text{MSE}}+\lambda_{\text{IE}}L_{\text{IE}}+\lambda_ {\text{TV}}L_{\text{TV}}, \tag{16}\]  
where the parameters of \(\lambda_{\text{MSE}}\), \(\lambda_{\text{IE}}\) and \(\lambda_{\text{TV}}\) are set to 1000, 1 and 10, respectively, in the experiments. We emphasize that, based on the design of the aforementioned loss function, the proposed weight convolutional neural network can be trained in an unsupervised manner. It combines the advantages of \(L_{\text{IE}}\), \(L_{\text{TV}}\) and \(L_{\text{MSE}}\). Here, \(L_{\text{IE}}\) and \(L_{\text{TV}}\) ensure that the obtained output weight maps can reduce noise interference to obtain the optimal quality of combined image, and \(L_{\text{MSE}}\) effectively reduces image artifacts to guarantee that the fidelity term makes the output image close to the ground truth.