# Weighted Group Sparse Regularized Tensor Decomposition for Hyperspectral Image Denoising
## 3 Proposed Method  
Assuming that the noise is independent in the denoising problem, we usually consider the following degradation model:  
\[\mathcal{Y}=\mathcal{X}+\mathcal{S}+\mathcal{N}, \tag{6}\]  
where \(\mathcal{X}\in\mathbb{R}^{l_{1}\times l_{2}\times l_{3}}\) is the clean hyperspectral image; \(\mathcal{N}\in\mathbb{R}^{l_{1}\times l_{2}\times l_{3}}\) is Gaussian noise; \(\mathcal{S}\in\mathbb{R}^{l_{1}\times l_{2}\times l_{3}}\) is sparse noise; \(I_{1}\times I_{2}\) is the hyperspectral images spatial size; and \(I_{3}\) is the number of bands in the spectral analysis. Based on this degradation model, we propose LRTRDGS, which is recovered by tensor ring constraints and preserves fine details by group sparse regularization. The  
Figure 1: A graphical representation of tensor ring decomposition. framework of the proposed LRTRDGS method is depicted in Figure 2. The LRTRDGS model is represented as follows:  
\[ \begin{array}{l}\min_{\mathcal{X},\mathcal{N},\mathcal{S}}\delta\|D\mathcal{X} \|_{W,2,1}+\lambda_{1}\|\mathcal{S}\|_{1}+\lambda_{2}\|\mathcal{N}\|_{F}^{2} \\ \text{s.t.}\ \mathcal{Y}=\mathcal{X}+\mathcal{N}+\mathcal{S},\\ \mathcal{X}=\Phi([\mathcal{G}]).\end{array}  \tag{7}\]  
where \(D\) is composed of \(D_{x}\) and \(D_{y}\), which are two differential operators in two spatial dimensions. The weighted \(l_{2,1}\)-norm of \(D\mathcal{X}\) is as follows:  
\[\|D\mathcal{X}\|_{W,2,1}=\sum_{i=1}^{m}\sum_{j=1}^{n}W_{x}(i,j)\|D_{x}\mathcal{ X}(i,j,:)\|_{2}+\sum_{i=1}^{m}\sum_{j=1}^{n}W_{y}(i,j)\|D_{y}\mathcal{X}(i,j,:)\|_{2}.\]  
To solve the LRTRDGS model (7), we propose a symmetric alternating direction method of the multiplier (ADMM) that incorporates proximity point operators. In the upcoming phases, the suggested ADMM will be implemented to address the proposed model. Two auxiliary variables can be introduced to the model (7) and are expressed as follows:  
\[ \begin{array}{l}\min_{\mathcal{X},\mathcal{N},\mathcal{S}}\delta\|\mathcal{ P}\|_{W,2,1}+\lambda_{1}\|\mathcal{S}\|_{1}+\lambda_{2}\|\mathcal{N}\|_{F}^{2} \\ \text{s.t.}\ \mathcal{Y}=\mathcal{X}+\mathcal{N}+\mathcal{S},\\ \mathcal{X}=\mathcal{Q},\mathcal{P}=D\mathcal{Q},\\ \mathcal{X}=\Phi([\mathcal{G}]).\end{array}  \tag{8}\]  
Under the constraint of tensor ring decomposition \(\mathcal{X}=\Phi([\mathcal{G}])\), this problem has the following augmented Lagrangian function:  
\[ \begin{array}{l}L_{\mu}(\mathcal{X},\mathcal{S},\mathcal{N},\mathcal{P}, \mathcal{Q},\Lambda_{1},\Lambda_{2},\Lambda_{3})=\delta\|\mathcal{P}\|_{W,2,1}+\lambda_{1}\|\mathcal{S}\|_{1}+\lambda_{2}\|\mathcal{N}\|_{F}^{2}+\langle \Lambda_{1},\mathcal{Y}-\mathcal{X}-\mathcal{N}-\mathcal{S}\rangle\\ +\langle\Lambda_{2},\mathcal{X}-\mathcal{Q}\rangle+\langle\Lambda_{3},D \mathcal{Q}-\mathcal{P}\rangle+\frac{\mu}{2}\big{(}\|\mathcal{Y}-\mathcal{X}- \mathcal{N}-\mathcal{S}\|_{F}^{2}+\|\mathcal{X}-\mathcal{Q}\|_{F}^{2}+\|D \mathcal{Q}-\mathcal{P}\|_{F}^{2}\big{)},\end{array}  \tag{9}\]  
where \(\Lambda_{1}\), \(\Lambda_{2}\) and \(\Lambda_{3}\) are the Lagrange multipliers and \(\mu\) is a positive scalar. As seen from the Lagrangian function, all variables are independent. We fix other variables and choose suitable ones to optimize the incremental Lagrangian function in (9). To optimize the incremental Lagrangian function in (9), we need to fix the other variables and select the appropriate ones. In addition, we make use of the symmetric ADMM with the addition of the proximity point operator to improve the convergence speed of the algorithm. The specific algorithm framework is as follows:  
Figure 2: Framework of the proposed HSI denoising method. \[\left\{