# Weighted Group Sparse Regularized Tensor Decomposition for Hyperspectral Image Denoising
## 3 Proposed Method
(3) Subproblem \(\mathcal{P}\):  
\[ \begin{split}&\mathcal{P}^{p+1}=\arg\min_{\mathcal{P}}\delta\| \mathcal{P}\|_{W,2,1}+\langle\Lambda_{3}^{p+\frac{1}{2}},D\mathcal{Q}^{p+1}- \mathcal{P}\rangle+\frac{\mu}{2}\|D\mathcal{Q}^{p+1}-\mathcal{P}\|_{F}^{2}\\ &=\arg\min_{\mathcal{P}}\delta\|\mathcal{P}\|_{W,2,1}+\frac{\mu} {2}\|\mathcal{P}-\left(D\mathcal{Q}^{p+1}+\frac{\Lambda_{3}^{p+\frac{1}{2}}}{ \mu}\right)\|_{F}^{2}.\end{split}  \tag{16}\]  
Using the soft threshold operator,  
\[R_{\Delta}(\mathbf{x})=\left\{ \begin{array}{ll}x-\Delta,&if\ \mathbf{x}>\Delta\\ x+\Delta,&if\ \mathbf{x}<\Delta\\ 0,&otherwise\end{array} \right.\]  
where \(x\in\mathbb{R}\), and \(\Delta>0\), we have:  
\[\mathcal{P}^{p+1}=R_{\frac{\mu}{p}}\left(D\mathcal{Q}^{p+1}+\frac{\Lambda_{3} ^{p+\frac{1}{2}}}{\mu}\right). \tag{17}\]  
(4) Subproblem \(\mathcal{S}\):  
\[ \begin{split}&\mathcal{S}^{p+1}=\arg\min_{\mathcal{S}}\lambda_{1 }\|\mathcal{S}\|_{1}+\langle\Lambda_{1}^{p+\frac{1}{2}},\mathcal{Y}-\mathcal{ X}^{p+1}-\mathcal{N}^{p}-\mathcal{S}\rangle+\frac{\mu}{2}\|\mathcal{Y}- \mathcal{X}^{p+1}-\mathcal{N}^{p}-\mathcal{S}\|_{F}^{2}\\ &=\arg\min_{\mathcal{S}}\lambda_{1}\|\mathcal{S}\|_{1}+\frac{\mu} {2}\|\mathcal{Y}-\mathcal{X}^{p+1}-\mathcal{N}^{p}-\mathcal{S}+\frac{\Lambda_ {3}^{p+\frac{1}{2}}}{\mu}\|_{F}^{2}\end{split}  \tag{18}\]  
By applying the aforementioned soft threshold operator, one can acquire the solution to the described problem:  
\[\mathcal{S}^{p+1}=R_{\frac{\lambda_{1}}{p}}\left(\mathcal{Y}-\mathcal{X}^{p+1 }-\mathcal{N}^{p}+\frac{\Lambda_{1}^{p+\frac{1}{2}}}{\mu}\right). \tag{19}\]  
(5) Subproblem \(\mathcal{N}\):  
\[ \begin{split}&\mathcal{N}^{p+1}=\arg\min_{\mathcal{N}}\lambda_{2 }\|\mathcal{N}\|_{F}^{2}+\langle\Lambda_{1}^{p+\frac{1}{2}},\mathcal{Y}- \mathcal{X}^{p+1}-\mathcal{N}-\mathcal{S}^{p+1}\rangle+\frac{\mu}{2}\| \mathcal{Y}-\mathcal{X}^{p+1}-\mathcal{S}^{p+1}-\mathcal{N}\|_{F}^{2}\\ &=\arg\min_{\mathcal{N}}\left(\lambda_{2}+\frac{\mu}{2}\right) \left\|\mathcal{N}-\frac{\mu\left(\mathcal{Y}-\mathcal{X}^{p+1}-\mathcal{S}^{ p+1}\right)+\Lambda_{3}^{p+\frac{1}{2}}}{\mu+2\lambda_{2}}\right\|_{F}^{2} \end{split}  \tag{20}\]  
By using a simple calculation, we can find the following solution:  
\[\mathcal{N}^{p+1}=\frac{\mu\left(\mathcal{Y}-\mathcal{X}^{p+1}-\mathcal{S}^{ p+1}\right)+\Lambda_{1}^{p+\frac{1}{2}}}{\mu+2\lambda_{2}} \tag{21}\]  
After solving these subproblems, we summarize those steps in Algorithm 1. ```
Input: The noisy hyperspectral image \(\mathcal{Y}\), the parameters \(\delta,\lambda_{1},\lambda_{2},\mu,r=[r_{1},r_{2},r_{3}]\), and constant \(\sigma,v_{1},v_{2},v_{3},v_{4},v_{5}\), and \(v_{6}\). Initialization:\(\mathcal{X}=\mathcal{N}=\mathcal{S}=\mathcal{Q}=\mathcal{P}=0\), \(p=1\). while not converged do  Update \(\mathcal{X}^{p+1}\) via (13). Update \(\Lambda_{1}^{p+\frac{1}{2}}\) via (10). Update \(\Lambda_{2}^{p+\frac{1}{2}}\) via (10). Update \(\mathcal{Q}^{p+1}\) via (15). Update \(\Lambda_{3}^{p+\frac{1}{2}}\) via (10). Update \(\mathcal{P}^{p+1}\) via (17). Update \(\mathcal{S}^{p+1}\) via (19). Update \(\mathcal{N}^{p+1}\) via (21). Update \(\Lambda_{1}^{p+1},\Lambda_{2}^{p+1},\Lambda_{3}^{p+1}\) via (10). Let \(\mu=max(1.5\mu,10^{6})\). Check the convergence condition \(\frac{\|\mathcal{X}^{p}-\mathcal{X}^{p+1}\|_{F}^{2}}{\|\mathcal{Y}\|_{F}^{2}} \leq 10^{-6}\). End while Output: The restored hyperspectral image \(\mathcal{X}\). ```  
**Algorithm 1** ADMM for HSI Denoising.