# Weighted Group Sparse Regularized Tensor Decomposition for Hyperspectral Image Denoising
## 3 Proposed Method
\begin{array}{l}\mathcal{X}^{p+1}=\underset{\mathcal{X}}{\arg\min}\ \ \ \mathcal{L}_{\mu}\left(\mathcal{X},\mathcal{S}^{p},\mathcal{N}^{p},\mathcal{P}^{p },\mathcal{Q}^{p},\Lambda_{1}^{p},\Lambda_{2}^{p},\Lambda_{3}^{p}\right)\\ \Lambda_{1}^{p+\frac{1}{2}}=\Lambda_{1}^{p}+v_{1}\mu\left(\mathcal{Y}- \mathcal{X}^{p+1}-\mathcal{N}^{p}-\mathcal{S}^{p}\right)\\ \Lambda_{2}^{p+\frac{1}{2}}=\Lambda_{2}^{p}+v_{2}\mu\left(\mathcal{X}^{p+1}- \mathcal{Q}^{p}\right)\\ \mathcal{Q}^{p+1}=\underset{\mathcal{Q}}{\arg\min}\ \ \mathcal{L}_{\mu}\left( \mathcal{X}^{p+1},\mathcal{S}^{p},\mathcal{N}^{p},\mathcal{P}^{p},\mathcal{Q},\Lambda_{1}^{p+\frac{1}{2}},\Lambda_{2}^{p+\frac{1}{2}},\Lambda_{3}^{p}\right) +\frac{\sigma_{H}}{2}\|\mathcal{Q}-\mathcal{Q}^{p}\|_{\text{F}}^{2}\\ \Lambda_{3}^{p+\frac{1}{2}}=\Lambda_{3}^{p}+v_{3}\mu\left(D\mathcal{Q}^{p+1}- \mathcal{P}^{p}\right)\\ \mathcal{P}^{p+1}=\underset{\mathcal{P}}{\arg\min}\ \ \mathcal{L}_{\mu}\left( \mathcal{X}^{p+1},\mathcal{S}^{p},\mathcal{N}^{p},\mathcal{P},\mathcal{Q}^{p +1},\Lambda_{1}^{p+\frac{1}{2}},\Lambda_{2}^{p+\frac{1}{2}},\Lambda_{3}^{p+ \frac{1}{2}}\right)\\ \mathcal{S}^{p+1}=\underset{\mathcal{S}}{\arg\min}\ \ \mathcal{L}_{\mu}\left( \mathcal{X}^{p+1},\mathcal{S},\mathcal{N}^{p},\mathcal{P}^{p+1},\mathcal{Q}^{ p+1},\Lambda_{1}^{p+\frac{1}{2}},\Lambda_{2}^{p+\frac{1}{2}},\Lambda_{3}^{p+ \frac{1}{2}}\right)\\ \mathcal{N}^{p+1}=\underset{\mathcal{N}}{\arg\min}\ \ \mathcal{L}_{\mu}\left( \mathcal{X}^{p+1},\mathcal{S}^{p+1},\mathcal{N},\mathcal{P}^{p+1},\mathcal{Q} ^{p+1},\Lambda_{1}^{p+\frac{1}{2}},\Lambda_{2}^{p+\frac{1}{2}},\Lambda_{3}^{ p+\frac{1}{2}}\right)\\ \Lambda_{1}^{p+1}=\Lambda_{1}^{p+\frac{1}{2}}+v_{4}\mu\left(\mathcal{Y}- \mathcal{X}^{p+1}-\mathcal{N}^{p+1}-\mathcal{S}^{p+1}\right)\\ \Lambda_{2}^{p+1}=\Lambda_{2}^{p+\frac{1}{2}}+v_{5}\mu\left(\mathcal{X}^{p+1}- \mathcal{Q}^{p+1}\right)\\ \Lambda_{3}^{p+1}=\Lambda_{3}^{p+\frac{1}{2}}+v_{6}\mu\left(D\mathcal{Q}^{p+1} -\mathcal{P}^{p+1}\right)\end{array} \right. \tag{10}\]  
In the \(p+1\)th iteration, we update the variables involved in model (10) as follows:  
(11)  
(11) can be expressed as  
\[\min\mu\|\Phi([\mathcal{G}])-\tfrac{1}{2}\Big{(}\mathcal{Y}-\mathcal{N}^{p}- \mathcal{S}^{p}+\mathcal{Q}^{p}+\left(\Lambda_{1}^{p}-\Lambda_{2}^{p}\right)/ \mu\Big{)}\|_{F}^{2} \tag{12}\]  
Through tensor ring decomposition, we can easily obtain \(\mathcal{G}^{p+1}\), and \(\mathcal{X}\) can be updated as follows:  
\[\mathcal{X}^{p+1}=\Phi([\mathcal{G}^{p+1}]). \tag{13}\]  
(2) Subproblem \(\mathcal{Q}\):  
\[ \begin{array}{l}\mathcal{Q}^{p+1}=\underset{\mathcal{Q}}{\arg\min}\langle \Lambda_{2}^{p+\frac{1}{2}},\mathcal{X}^{p+1}-\mathcal{Q}\rangle+\langle \Lambda_{3}^{p},D\mathcal{Q}-\mathcal{P}^{p}\rangle\\ +\frac{\mu}{2}\big{(}\|\mathcal{X}^{p+1}-\mathcal{Q}\|_{\text{F}}^{2}+\|D \mathcal{Q}-\mathcal{P}^{p}\|_{\text{F}}^{2}\big{)}+\frac{\sigma_{H}}{2}\| \mathcal{Q}-\mathcal{Q}^{p}\|_{\text{F}}^{2}\end{array}  \tag{14}\]  
This problem can be optimized with the following linear system:  
\[ \begin{array}{l}(\mu I+\mu D^{*}D+\sigma\mu)\mathcal{Q}=\mu\mathcal{X}^{p+1}+ \mu D^{*}(\mathcal{P}^{p})+\Lambda_{2}^{p+\frac{1}{2}}-D^{*}\Big{(}\Lambda_{3} ^{p}\Big{)}+\sigma\mu\mathcal{Q}^{p}\end{array} \]  
where \(D^{*}\) is the adjoint operator of \(D\). The matrix corresponding to operator \(D^{*}D\) has a block cycle structure, which can be diagonalized by the 3-D Fast Fourier Transform (FFT) matrix. Thus, we have  
\[\left\{ \begin{array}{l}H_{\mathcal{Q}}=\mu\mathcal{X}^{p+1}+\mu D^{*}( \mathcal{P}^{p})+\Lambda_{2}^{p+\frac{1}{2}}-D^{*}\Big{(}\Lambda_{3}^{p}\Big{)} +\sigma\mu\mathcal{Q}^{p},\\ T_{\mathcal{Q}}=W_{x}^{2}|fftn(D_{x})|^{2}+W_{y}^{2}|fftn(D_{y})|^{2},\\ \mathcal{Q}^{p+1}=ifftn\Big{(}\frac{fftn(H_{\mathcal{Q}})}{\mu I+\mu I_{ \mathcal{Q}}+\sigma_{H}}\Big{)}\end{array} \right. \tag{15}\]  
where \(fftn\) and \(ifftn\) correspond to the rapid 3-D Fourier transformation and its inverse conversion, respectively. The squared component is denoted by \(|\cdot|^{2}\), with division being executed in an element-wise manner.