# An Unsupervised Weight Map Generative Network for Pixel-Level Combination of Image Denoisers
## 3 Methodology
### Basic Concept  
We illustrate below, with an example, the complementary nature of the model- and learning-based image denoising methods. Figure 2 shows the model- and learning-based denoisers called BM3D and FFDNet, respectively. These images are corrupted by a medium level of noise (\(\sigma=30\)). For better visual comparison, a particular region is enlarged for images. Based on the magnified region in the Barbara images, the BM3D excels in areas with repeating textures. In contrast, in the Boat image, we can observe that FFDNet is better in generic content than BM3D because FFDNet has learned image priors from external data, which is specific to the image to be denoised. These results show that BM3D is successful at retaining the repeating textures, whereas FFDNet is capable of removing noise in the generic content. Therefore, FFDNet and BM3D are complementary to each other. To further leverage each of their advantages, we aimed to combine the two results. Moreover, we introduced a pixel-level combination strategy, which makes full use of the complementarity among different denoisers, to retain the well-denoised details of each preprocessed denoised image in finer granularity. Therefore, the problem of improving the denoising effect in this study is a pixel-level image combination problem, which typically follows a weighted summation framework that combines the \((i,j)\)-th pixel \(x_{k}(i,j)\) in image \(x_{k}\) with the corresponding weighted \(w_{k}(i,j)\) in weight map \(W_{k}\) to achieve an optimal combined result \(\hat{x}(i,j)\). Specifically, in our study, the aforementioned input images are the initial denoised images \(\hat{x}_{1},\hat{x}_{2},\ldots,\hat{x}_{K}\) generated by different denoisers \(D_{1},D_{2},\ldots,D_{K}\). Thus, for each pixel, our study can be described mathematically as  
\[\hat{x}(i,j)=\sum_{k=1}^{K}w_{k}(i,j)\hat{x}_{k}(i,j) \tag{11}\]  
Considering that these initial denoised images are complementary to each other, we set weights for all pixels of each denoised image, where the sum of the weights in the same position is 1 after normalization. The weight map of each denoised image is difficult to be set through manual designing since it is complex and different. Hence, we used the unsupervised deep learning method to find these optimal weight maps. In other words, we intended to use the unsupervised deep learning network to generate a normalized weight map for each denoised image. The normalization operation ensures that the intensity value of the combined images remains within the 0 to 1 or 0 to 255 ranges. The normalization formula is defined as:  
\[w_{k}(i,j)=\frac{o_{k}(i,j)}{\sum_{n=1}^{K}o_{n}(i,j)} \tag{12}\]  
where \(o_{k}(i,j)\) represents the original weight of the (_i,j_)-th pixel of the k-th initial denoised image generated by our unsupervised deep learning network, and \(w_{k}(i,j)\) represents the (_i,j_)-th pixel of the k-th initial denoised image after normalization.  
In summary, we used an unsupervised deep learning network to generate a weight map for each initial denoised image under the guidance of our designed loss function, where we combined these images at the pixel level to provide the optimal denoising result.