## 4 Image SR Reconstruction
### Optimization Solution  
We adopt the Lagrange multiplier method to transform Equation (20) into:  
\[ \begin{split}\text{arg}\,\,\underset{\widetilde{X}}{\text{min}} \,\lambda\,\parallel Ar\Big{(}D\widetilde{X}\Big{)}\parallel_{TV}\\ +\sum\limits_{i=1}^{I}\,\sum\limits_{j=1}^{j}\,\frac{\beta_{(i,j) }}{2}\parallel y_{(i,j)}-CHD\widetilde{X}_{(i,j)}\parallel_{2}^{2}+\parallel \widetilde{X}_{(i,j)}\parallel_{1}\end{split}  \tag{21}\]  
where \(\beta_{(i,j)}\) is the Lagrangian multiplier. We put forward the TwTVSI optimization method to solve Equation (21). Iterative solutions are expressed as follows:  
\[ \begin{split}\widetilde{X}^{(K+1)}=\widetilde{X}^{(K)}-\mu^{(K) }\bigg{\{}\lambda\frac{\partial\parallel Ar\big{(}D\widetilde{X}\big{)} \parallel_{TV}}{\partial\widetilde{X}^{(K)}}+\\ \frac{1}{2}\frac{\partial\sum\
olimits_{i=1}^{I}\sum\
olimits_{j= 1}^{J}\frac{\beta_{(i,j)}}{2}\parallel y_{(i,j)}-CHD\widetilde{X}_{(i,j)} \parallel_{2}^{2}+2\parallel\widetilde{X}_{(i,j)}\parallel_{1}}{\partial \widetilde{X}^{(K)}}\Bigg{\}}\\ =\widetilde{X}^{(K)}-\mu^{(K)}\left\{\lambda g_{1}\Big{(} \widetilde{X}^{(K)}\Big{)}+\frac{1}{2}g_{2}\Big{(}\widetilde{X}^{(K)}\Big{)} \right\}\end{split}  \tag{22}\]  
where \(\mu^{(K)}\) is the iteration step size; and \(g_{1}\Big{(}\widetilde{X}^{(K)}\Big{)}\) is the gradient derivative of \(Ar\Big{(}D\widetilde{X}\Big{)}_{TV}\) to the sparse coefficient matrix \(\widetilde{X}^{(K)}\), which can be solved by the gradient descent method directly. \[g_{1}\Big{(}\widetilde{X}^{(K)}\Big{)}=\frac{\partial\parallel Ar\Big{(}D \widetilde{X}\Big{)}\parallel_{TV}}{\partial\widetilde{X}^{(K)}}=\left\{ \frac{\partial\parallel Ar\Big{(}D\widetilde{X}\Big{)}\parallel_{TV}}{ \partial\widetilde{X}^{(K)}},\ldots,\frac{\partial\parallel Ar\Big{(}D \widetilde{X}\Big{)}\parallel_{TV}}{\partial\widetilde{X}^{(K)}_{(i,j)}}, \ldots,\frac{\partial\parallel Ar\Big{(}D\widetilde{X}\Big{)}\parallel_{TV} }{\partial\widetilde{X}^{(K)}_{(i,j)}}\right\} \tag{23}\]  
\[ \begin{split}\frac{\partial\parallel Ar\big{(}D\widetilde{X} \big{)}\parallel_{TV}}{\partial\widetilde{X}^{(K)}_{(i,j)}}&=\sum \limits_{m=1}^{S}\,\,\sum\limits_{n=1}^{S}\,\text{Pd}\Big{(}m,n,\widetilde{X}^ {(K)}_{(i,j)},\widetilde{X}^{(K)}_{(i,j)}\Big{)}\\ &+\sum\limits_{m=1}^{S}\,\,\sum\limits_{n=1}^{S}\,\,\text{Pd} \Big{(}m,n,\widetilde{X}^{(K)}_{(i,j+1)},\widetilde{X}^{(K)}_{(i,j)}\Big{)}\\ &+\sum\limits_{m=1}^{S}\,\,\sum\limits_{n=1}^{S}\,\,\text{Pd} \Big{(}m,n,\widetilde{X}^{(K)}_{(i+1,j)},\widetilde{X}^{(K)}_{(i,j)}\Big{)} \end{split}  \tag{24}\]  
\[ \begin{split}\text{Pd}\Big{(}m,n,\widetilde{X}_{a},\widetilde{X} _{b}\Big{)}&=\frac{\frac{\partial\text{Dr}\big{(}m,n,\widetilde{X} _{a}\big{)}}{\partial\widetilde{X}_{b}}\times\text{Dr}\Big{(}m,n,\widetilde{X} _{a}\big{)}+\frac{\partial\text{Dr}\big{(}m,n,\widetilde{X}_{a}\big{)}}{ \partial\widetilde{X}_{b}}\times\text{Dr}\Big{(}m,n,\widetilde{X}_{a}\big{)} }{\sqrt{\left\{\text{Dr}\Big{(}m,n,\widetilde{X}_{a}\big{)}\right\}^{2}+\, \text{Dr}\Big{(}m,n,\widetilde{X}_{a}\big{)}\,\right\}^{2}+10^{-6}}\end{split}  \tag{25}\]  
where \(\text{Dr}\Big{(}m,n,\widetilde{X}^{(K)}_{(i,j)}\Big{)}\) and \(\text{Dr}\Big{(}m,n,\widetilde{X}^{(K)}_{(i,j)}\Big{)}\) are, respectively, image row and column difference operation functions. Since we use the block SR model for the image, the cross operation of edge elements between adjacent image blocks is involved in the row-column direction difference operation. By derivation, we can use a periodically recurring sparse dictionary matrix to represent the difference operation between adjacent image patches. Let \(\theta\) be a cyclic matrix with a period of \(S\), i.e., \(\theta(m,n)=\theta(m+S,n)=\theta(m,n+S)\), and its matrix structure is as follows:  
\[\theta=\left( \begin{array}{cccc}1&2&\cdots&S\\ S+1&S+2&\cdots&2S\\ \vdots&\vdots&\ddots&\vdots\\ S^{2}-S+1&S^{2}-S+2&\cdots&S^{2}\end{array}