# Hyperspectral Image Super-Resolution Based on Spatial Group Sparsity Regularization Unmixing
## 3 Proposed Method
### Problem Solution
However, considering that hexagonal grids have more non-diagonal neighbors than square grids, which leads to less distance distortion loss when processing boundary pixels, a hexagonal grid is introduced to promote the accuracy of clustering. In addition, as the number of spectral bands in the HSI is much more than the number of channels in the conventional color image, we directly use the spectral channels in the clustering process while the color channels of the color image are used for color space transformation. This has the benefit of minimizing the loss of spectral information. The cluster center is defined by the following formula:  
\[C_{i}=[x_{1i},\cdots,x_{Li},m^{i},n^{i}]^{T}\in\mathbb{R}^{L+2}, \tag{11}\]  
where \([x_{1i},\cdots,x_{Li}]^{T}\) is the average spectral reflectance of the \(i^{\text{th}}\) cluster and \([m^{i},n^{i}]^{T}\) denotes the location of the cluster center. Step 3: Suppose \(d=||\Phi(x,y)-\Phi(x_{k},y_{k})||\) is the distance between a pixel \((x,y)\) and the cluster center. Assign the pixel to the closest cluster center. Step 4: Update cluster centers based on the spatial centroid of the clustered pixels under the current iteration. Step 5: Repeat until the distance is less than a pre-determined threshold. Step 6: Assign the disjoint segments to the largest nearby cluster. The Alternating Direction Method of Multipliers (ADMM) [9] algorithm is used to solve the optimization problem shown in Equation (7):  
\[ \begin{gathered}\underset{A}{\text{min}}\ ||EA-Y||_{2,1}+\lambda||A||_{2,1}+ \mathfrak{I}_{R_{+}}(A)\\ \mathfrak{I}_{R_{+}}(A)=\sum_{i=1}^{P}\mathfrak{I}_{R_{+}}(A_{i}) \end{gathered} , \tag{12}\]  
where \(\mathfrak{I}_{R_{+}}(A)\) is the indicator function for the non-negative orthant \(R_{+}\) and \(A_{i}\) is the \(i^{\text{th}}\) atom of the abundance vector. It should be noted that when \(\mathfrak{I}_{R_{+}}(A_{i})=0\), \(A_{i}\) is in the non-negative orthant or \(+\infty\)Adding three auxiliary matrices V1, V2, and V3, the problem in Equation (12) can be reformulated as follows:  
\[\min_{A}||V_{1}||_{2,1}+\lambda||V_{2}||_{2,1}+\mathrm{i}_{R_{+}}(A)\quad s.t.V_ {1}=EV_{3}-Y\quad V_{3}=V_{2}\quad V_{3}=A. \tag{13}\]  
Equation (13) can be compacted as:  
\[\min_{I,V_{3}}g(I)\quad s.t.GV_{3}+HI=J\]  
\[g(I)=|V_{1}||_{2,1}+\lambda||V_{2}||_{2,1}+\mathrm{i}_{R_{+}}(A)\]  
\[G= \begin{bmatrix}E\\ I\\ I\end{bmatrix} \quad H= \begin{bmatrix}-I&0&0\\ 0&-I&0\\ 0&0&-I\end{bmatrix} \quad J= \begin{bmatrix}Y\\ 0\\ 0\end{bmatrix} \cdot \tag{14}\]  
\[I\equiv(V_{1},V_{2},A)\]  
Thus, the augmented Lagrangian function is:  
\[\mathfrak{L}(I,V_{3},\Lambda)=g(I)+\frac{\mu}{2}||GV_{3}+HI-J-\Lambda||_{F}^{2}, \tag{15}\]  
where \(\Lambda/\mu\) are the Lagrange multipliers and \(\mu>0\) is the Lagrange multiplier regularization parameter. The \(\mathfrak{L}\) can be optimized by updating \(V_{1}\), \(V_{2}\), A, and \(V_{3}\). (1) To update \(V_{1}\),  
\[V_{1}^{k+1}=arg\min_{V_{1}}||V_{1}||_{2,1}+\frac{\mu}{2}||EV_{3}^{k}-Y-V_{1}- \Lambda_{1}^{k}||_{F}^{2}. \tag{16}\]  
Introducing the vect-soft threshold [10]:  
\[ \begin{array}{l}V_{1}^{k+1}(r,:)=vect-soft(\xi(r,:),\frac{1}{\mu}),\\ \xi=EV_{3}^{k}-Y-\Lambda_{1}^{k}\end{array}  \tag{17}\]  
where \(r\) represents the row of \(V_{1}\) and vect-soft\((*,*)\) is the vect-soft threshold function. (2) To update \(V_{2}\), a formula similar to Equation (17) needs to be solved. The difference is that \(\mathfrak{L}\) is replaced by \(\gamma=V_{3}-\Lambda_{2}^{k}\). (3) To update \(V_{3}\),  
\[ \begin{array}{l}V_{3}^{k+1}=arg\min_{V_{3}}||EV_{3}-Y-V_{1}^{k+1}-\Lambda_{ 3}^{k}||_{F}^{2}+||V_{3}-V_{2}^{k+1}\\ \qquad\qquad\qquad-\Lambda_{2}^{k}||_{F}^{2}+||V_{3}-A^{k+1}-\Lambda_{3}^{k} ||_{F}^{2}\end{array} .