# Boosting of Denoising Effect with Fusion Strategy
## 2 Related Work  
### Nonlocally Centralized Sparse Representation (NCSR) for Image Denoising  
The NCSR algorithm involves decomposing a noisy image into a set of overlapping patches, learning the sub-dictionaries to sparsely code the image patches, making an estimate of the sparse coding vectors, and combining the estimated patches to form the denoised image. It can be equivalently introduced as below.  
Using the notation employed in [9], for an image \(x\in\mathbb{R}^{N}\), we denote \(x_{i}=R_{i}x\) as an image patch of size \(\sqrt{n}\times\sqrt{n}\) at pixel \(i\); furthermore, \(R_{i}\) represents a matrix for extracting the patch \(x_{i}\) from \(x\). For a given dictionary \(D\in\mathbb{R}^{n\times M}\), \(n\leq M\), \(x_{i}\) can be sparsely coded as \(x_{i}\approx D\alpha_{x,i}\) by solving an \(l_{1}\)-minimization problem written as:  
\[\min_{\alpha_{i}}\sum_{i=1}^{P}(\|x_{i}-D\alpha_{i}\|_{2}^{2}+\lambda\|\alpha_ {i}\|_{1}), \tag{2}\]  
where \(\alpha_{i}\) represents the sparse coding coefficient of \(x_{i}\), \(P\) is the sum of sparse codes for image \(x\), and \(\lambda\) is the regularization parameter. The redundant patch-based representation is obtained by overlapping the image patches. This aims at suppressing the boundary artifacts. The entire image \(x\) is denoted by a set of sparse codes \(\{a_{x,i}\}\). A straightforward least-square solution to reconstruct \(x\) from \(\{a_{x,i}\}\) is  
\[x\approx\left(\sum_{i=1}^{N}R_{i}^{T}R_{i}\right)^{-1}\sum_{i=1}^{N}(R_{i}^{T }D\alpha_{x,i}). \tag{3}\]  
For the convenience of expression, let  
\[x\approx\Phi\circ\alpha_{x}\ =\ \ \left(\sum_{i=1}^{N}R_{i}^{T}R_{i}\right)^{-1} \sum_{i=1}^{N}(R_{i}^{T}D\alpha_{x,i}), \tag{4}\]here, \(\alpha_{x}\) is the concatenation of all the sparse codes. As mentioned in the model of the noisy image in Equation (1), the sparsity coding denoising model to recover \(x\) from \(y\) is obtained by solving a minimization problem:  
\[\alpha_{y}\ =\ arg\min_{\alpha}(\|x_{i}-D\circ\alpha\|_{2}^{2}+\lambda\|\alpha\| _{1}). \tag{5}\]  
Then, the image \(x\) is estimated as \(\hat{x}=D\circ\alpha_{y}\). \(\hat{x}\) is an estimate obtained by averaging each of the reconstructed patches in \(x_{i}\). The reconstruction of \(x\) from \(y\) in NCSR algorithm is defined as the following minimization problem:  
\[\alpha_{y}\ =\ arg\min_{\alpha}\Big{(}\|x_{i}-D\circ\alpha\|_{2}^{2}\Big{)}+ \lambda\sum_{i}\|\alpha_{i}-\beta_{i}\|_{p}, \tag{6}\]  
where the regularization parameter \(\lambda\) balances the centralized sparsity and the fidelity terms for better performance. This should be adaptively determined. \(\beta_{i}\) represents the nonlocal estimate of the unknown sparse code \(\alpha_{i}\). \(\|\alpha_{i}-\beta_{i}\|_{p}\) is the only regularization term in the aforementioned model. In the case of \(p=1\), the estimated \(\beta_{i}\) can be computed from the nonlocal redundancy of natural images, and this is why the model is called the nonlocally centralized sparse representation (NCSR).  
An iterative shrinkage strategy is employed to calculate \(\beta_{i}\) in Equation (6). Let \(\Omega_{i}\) be denote a set of patches similar to patch \(x_{i}\) and \(\alpha_{i,q}\) be the sparse codes of patch \(x_{i,q}\) within set \(\Omega_{i}\). Thereafter, \(\beta_{i}\) can be computed as:  
\[\beta_{i}=\sum_{q\in\Omega_{i}}w_{i,q}\alpha_{i,q}, \tag{7}\]  
where \(w_{i,q}\) is the corresponding weight and it is set inversely proportional to the distance between patches \(x_{i}\) and \(x_{i,q}\):  
\[w_{i,q}=\frac{1}{W}exp(-(\|x_{i}-x_{i,q}\|_{2}^{2})/h), \tag{8}\]  
where \(x_{i}=D\alpha_{i}\) and \(x_{i}=D\alpha_{i,q}\), respectively. \(h\) is a pre-determined scalar, and \(W\) is a normalization factor. Specifically, with the nonlocal estimate \(\beta_{i}\) taking full advantage of the nonlocal redundancy of images, the NCSR algorithm naturally integrates the nonlocal self-similarity prior into the sparse representation framework and shows a promising performance in terms of denoising natural images with many repetitive structures.