# Boosting of Denoising Effect with Fusion Strategy
## 3 Combination of the NCSR and the DnCNN
### Fusion of Images Denoised by NCSR and DnCNN  
The proposed denoising effect boosting method is a linear combination of NCSR and DnCNN. That is, we apply two denoisers \(D_{1}\) and \(D_{2}\) to yield two denoised images \(\hat{x}_{1}=D_{1}(y)\) and \(\hat{x}_{2}=D_{2}(y)\). We compute the desired image \(\hat{x}\) by retaining only the \"optimal\" parts in images \(\hat{x}_{1}\) and \(\hat{x}_{2}\). This process is guided by the relative pixel intensity and the global gradient, which are consolidated into a scalar-valued weight map. The final image \(\hat{x}\) is obtained by fusing \(\hat{x}_{1}\) and \(\hat{x}_{2}\) using weighted blending. The processes involved in the proposed method are shown in Figure 3.  
To optimally fuse the initial denoised images, we compute a weight map for the \(n\)-th input image as  
\[W_{n}(i,j)=\frac{W_{1,n}(i,j)^{p_{1}}\times W_{2,n}(i,j)^{p_{2}}}{\sum_{i=1}^{ 2}W_{1,n}(i,j)^{p_{1}}\times W_{2,n}(i,j)^{p_{2}}+\epsilon} \tag{9}\]  
Figure 3: Overall framework of the proposed method.  
Figure 2: The architecture of DnCNN network.  
where \((i,j)\) represents the image pixels, and \(\epsilon\) denotes a very small positive value (e.g., \(10^{-2}\)) to avoid the denominator being zero. The parameters \(p_{1},p_{2}>0\) are set to determine the extent to which each weight should be emphasized. The number 2 indicates that there are two input images that are denoised by NCSR and DnCNN, respectively. \(W_{1,n}(i,j)\) and \(W_{2,n}(i,j)\) are two adaptive weights designed to reflect the relative pixel intensity and the global gradient of an input image. A detailed introduction to the two weights will be given in the following subsections.  
Using the weight obtained in Equation (9), the resulting denoised image \(\hat{x}\) can be obtained via a weighted sum of the initial denoised images:  
\[\hat{x}(i,j)=\sum_{n=1}^{2}W_{n}(i,j)\hat{x}_{n}(i,j), \tag{10}\]  
where \(\hat{x}_{n}\) is the input image denoised by NCSR or DnCNN and \(\hat{x}_{n}(i,j)\) is the image pixel intensity. In this study, the pixel intensity is normalized to the range of \([0,1]\). Unfortunately, applying only Equation (10) will yield an image with several artifacts. This is because the values of the weights are usually noisy and discontinuous. Therefore, we apply Equation (10) in multiple resolutions using a pyramidal image decomposition, described in [39], to avoid sharp weight map transitions. The fusion is carried out in each pyramid separately. Specifically, we set the decomposition level \(l\) to 7 based on [39]. For level \(l\), \(L\left\{\hat{x}_{n}(i,j)\right\}^{l}\) is the Laplacian pyramid of image \(\hat{x}_{n}(i,j)\) and \(G\left\{W_{n}(i,j)\right\}^{l}\) is the Gaussian pyramid of the weight map \(W_{n}(i,j)\). Note that the value of \(\hat{x}_{n}(i,j)\) determines the value of \(W_{n}(i,j)\). Then, we blend the pixel intensities in different pyramid levels in Equation (11):  
\[L\left\{\hat{x}(i,j)\right\}^{l}=\sum_{n=1}^{2}\left[L\left\{\hat{x}_{n}(i,j) \right\}^{l}G\left\{W_{n}(i,j)\right\}^{l}\right], \tag{11}\]  
The fused pyramid \(L\left\{\hat{x}_{n}(i,j)\right\}^{l}\) is collapsed to obtain the resulting denoised image \(\hat{x}\). The pyramid approach can weaken the local unnatural transition by dispersing the gray-level mutations of the whole image, which are caused by the differences in the denoising effects.