## 4 Image SR Reconstruction
### Objective Function  
On the basis of the principle of image degradation and compressed sensing, the objective function of SR can be written as follows [29].  
\[\begin{split}&\underset{\widetilde{x}}{\text{argmin}}\parallel \widetilde{x}\parallel_{1}\\ & s.t.\ y=CHD\widetilde{x}\end{split} \tag{16}\]  
Since the SR reconstruction of the whole image requires a large storage space and is very complicated to calculate, we choose the BCS method to reconstruct the image in blocks. We segment an image as follows:  
\[Y=\left(\begin{array}{cccc}Y_{b(1,1)}&Y_{b(1,2)}&\cdots&Y_{b(1,J)}\\ Y_{b(2,1)}&Y_{b(2,2)}&\cdots&Y_{b(2,J)}\\ \vdots&\vdots&\ddots&\vdots\\ Y_{b(I,1)}&Y_{b(I,2)}&\cdots&Y_{b(I,J)}\end{array}\right) \tag{17}\]  
where \(Y\) is an LR image. \(Y_{b(i,j)}\) is an LR image block; \(Y\) is divided into \(I\times J\) image blocks. \(X_{b(i,j)}\) is the HR image block at the corresponding position of \(Y_{b(i,j)}\), and the size of \(X_{b(i,j)}\) is \(S\times S\).  
The reconstruction of blocked compressed sensing approach will bring about the problem of \"block effect\", so we introduce a TV regular term into the objective function. In this way, the minimum variation constraint can be applied to the whole image, thereby eliminating the reconstructed image block effect. The image total variation is calculated as follows:  
\[\begin{split}\parallel X\parallel_{TV}=\\ \sum\limits_{i,j}\sqrt{\left(X(i,j)-X(i-1,j)\right)^{2}+\left(X(i,j)-X(i,j-1 )\right)^{2}}\end{split} \tag{18}\]  
where \(X(i,j)\) represents the value contained in row \(i\) and column \(j\) of the image matrix \(X\). The image is segmented and a TV regular term is introduced. So, the objective function becomes:  
\[\begin{split}&\underset{\widetilde{X}}{\text{argmin}}\lambda \parallel X\parallel_{TV}+\sum\limits_{i=1}^{I}\sum\limits_{j=1}^{J}\parallel \widetilde{X}_{(i,j)}\parallel_{1}\\ s.t.\ y_{(i,j)}=CHD\widetilde{X}_{(i,j)},\ \ i=1,2,\ldots I,\ \ j=1,2, \ldots J\end{split} \tag{19}\]  
where \(\widetilde{X}_{(i,j)}\) is the sparse representation of the column vector obtained by expanding \(X_{b(i,j)}\), and \(y_{(i,j)}\) is the row-expanded column vector corresponding to \(Y_{b(i,j)}\).  
In the solution process of Equation (19), it is a difficult problem to obtain \(\widetilde{X}_{(i,j)}\) from \(X\) when we perform sparse constraints on the updated \(X\) under the constraint of \(\parallel X\parallel_{TV}\). \(D\) as an overcomplete dictionary makes the sparse coefficient dimension of the image much larger than its own dimension. Therefore, we use the sparse coefficient matrix \(\widetilde{X}\) for theimage matrix \(X\) with the expression \(X=Ar\Big{(}D\widetilde{X}\Big{)}\). The objective function is improved and expressed as:  
\[\begin{split}\underset{\widetilde{X}}{\text{argmin}}\,\lambda\, \parallel Ar\Big{(}D\widetilde{X}\Big{)}\parallel_{TV}&+\sum \limits_{i=1}^{I}\,\,\sum\limits_{j=1}^{J}\,\parallel\widetilde{X}_{(i,j)} \parallel_{1}\\ s.t.&\,\,y_{(i,j)}=CHD\widetilde{X}_{(i,j)},\,\,\,\,\,i =1,2,\ldots,I,\,\,\,\,\,j=1,2,\ldots,J\end{split} \tag{20}\]  
where \(\widetilde{X}=\Big{(}\widetilde{X}_{(1,1)},\widetilde{X}_{(1,2)},\cdots, \widetilde{X}_{(1,J)},\widetilde{X}_{(2,1)},\cdots\widetilde{X}_{(I,J)}\Big{)}\) is the sparse coefficient matrix. The function \(Ar(z)\) indicates that the column vectors in \(z\) are sequentially stitched into corresponding images.