# An Unsupervised Weight Map Generative Network for Pixel-Level Combination of Image Denoisers
## 2 Related Work
### Structural Patch Decomposition Approach  
In addition to the image-level combination strategy, image combination can also be performed at the patch level. The patch-level combination strategy improves the granularity of image processing and retains more available details in images compared to the image-level combination strategy. In recent years, various researchers have proposed their patch-level fusion schemes, among which [PERSON] et al. proposed SPD for multi-exposure image fusion in [32]. Although the SPD was not originally used in image denoising, we found that applying it to denoised image combination yielded higher-quality results compared to typical methods, such as CsNet. Therefore, in this section, we present a discussion on structural patch decomposition. Initially, the SPD approach uses a moving window with a fixed stride to extract \(K\) overlapping image patch sequences \(\{\mathbf{p}_{k}\}=\{\mathbf{p}_{k}|\leq k\leq K\}\) from the entire image. They decompose a given patch \(\mathbf{p}_{k}\) into three parts, such as signal strength, signal structure and mean intensity. Its decomposition formula is  
\[ \begin{split}\mathbf{p}_{k}&=\left\|\mathbf{p}_{k} -\mu_{\mathbf{p}_{k}}\right\|\cdot\frac{\mathbf{p}_{k}-\mu_{\mathbf{p}_{k}}}{ \left\|\mathbf{p}_{k}-\mu_{\mathbf{p}_{k}}\right\|}+\mu_{\mathbf{p}_{k}}\\ &=\left\|\tilde{\mathbf{p}}_{k}\right\|\cdot\frac{\tilde{\mathbf{ p}}_{k}}{\left\|\tilde{\mathbf{p}}_{k}\right\|}+\mu_{\mathbf{p}_{k}}\\ &=c_{k}\cdot\mathbf{s}_{k}+l_{k}\end{split}  \tag{6}\]  
where \(\|\cdot\|\) is the \(l_{2}\) norm of a vector, \(\mu_{\mathbf{p}_{k}}\) is the mean value of the patch, and \(\tilde{\mathbf{p}}_{k}=\mathbf{p}_{k}-\mu_{\mathbf{p}_{k}}\) is a mean-removed patch. As a result, the scalar \(c=\left\|\tilde{\mathbf{p}}_{k}\right\|\), \(l=\mu_{\mathbf{p}_{k}}\) and vector \(s=\frac{\mathbf{p}_{k}}{\left\|\mathbf{p}_{k}\right\|}\) have physical significances that correspond to the strength component, mean intensity component and structure component of \(\mathbf{p}_{k}\), respectively. First, the strength component is analyzed. Here, the contrast of the combined patch is determined by the highest contrast. It is expressed as  
\[\hat{c}=\max_{1\leq k\leq K}c_{k}=\max_{1\leq k\leq K}\left\|\tilde{\mathbf{ p}}_{k}\right\| \tag{7}\]  
Regarding the structure component, to make the combined image patch structure reflect all source image patch structures, a basic implementation of this relationship is provided  
\[\mathbf{\hat{s}}=\frac{\mathbf{\overline{s}}}{\left\|\mathbf{\overline{s}} \right\|}\quad\text{ and }\quad\mathbf{\overline{s}}=\frac{\sum_{k=1}^{K}S(\tilde{\mathbf{p}}_{k}) \mathbf{s}_{k}}{\sum_{k=1}^{K}S(\tilde{\mathbf{p}}_{k})} \tag{8}\]  
where \(S(\cdot)\) is a weighting function given by \(S(\tilde{\mathbf{p}}_{k})=\left\|\tilde{\mathbf{p}}_{k}\right\|^{p}\) that defines the contribution factor for each source image patch. For the mean intensity component, it is denoted as  
\[\hat{l}=\frac{\sum_{k=1}^{K}L(\mu_{k},l_{k})l_{k}}{\sum_{k=1}^{K}L(\mu_{k},l_{ k})} \tag{9}\]  
where \(L(\cdot)\) is a weighting function. It contains two parameters, including the local mean value of the current patch and the global mean value \(\mu_{k}\) of the image, as inputs. \(L(\cdot)\) measures the exposures of \(\mathbf{p}_{k}\) in \(x_{k}\), imposing a high penalty when the image and/or the current patch are under/over-exposed. After determining the \(\hat{c}\), \(\hat{\mathfrak{s}}\) and \(\hat{l}\) components using Equations (7)-(9), respectively, the reconstructed patch can be expressed as  
\[\hat{x}=\hat{c}\cdot\hat{\mathfrak{s}}+\hat{l} \tag{10}\]  
After the overlapping image patches are reconstructed, they are placed back into the entire image and the pixels in the overlapping patches are averaged to produce a great overall result. In our study, we apply SPD to image denoising. For the co-located initial denoised patches produced by stand-alone methods, SPD decomposes them into signal intensity, signal structure and average intensity.