## 3 Proposed Method
 \tag{15}\]  
\[\mathbf{Z}^{(k+1)}=\arg\min_{\mathbf{Z}}\mu\sum_{i=1}^{4}\|\mathbf{K}_{i}*\mathbf{Z}\|_{p}^{p} +\frac{\rho}{2}\|\mathbf{Z}-(\mathbf{F}^{(k+1)}+\frac{\mathbf{Y}^{(k)}}{\rho})\|_{2}^{2}, \tag{16}\]  
\[\mathbf{Y}^{(k+1)}=\mathbf{Y}^{(k)}+\rho(\mathbf{F}^{(k+1)}-\mathbf{Z}^{(k+1)}). \tag{17}\]  
To solve the sub-problem of \(\mathbf{F}\), let \(\mathbf{W}=\mathbf{SH}\). Then, Equation (15) can be represented as follows  
\[\mathbf{F}^{(k+1)}=\arg\min_{\mathbf{F}}\frac{1}{2}\|\mathbf{W}\mathbf{F}-\mathbf{G}\|_{2}^{2}+ \frac{\rho}{2}\|\mathbf{F}-(\mathbf{Z}^{(k)}-\frac{\mathbf{Y}^{(k)}}{\rho})\|_{2}^{2}. \tag{18}\]  
By setting the first-order derivative of \(\mathbf{F}\) in Equation (18) as zero, we have  
\[\mathbf{0}=\mathbf{W}^{\rm T}(\mathbf{W}\mathbf{F}-\mathbf{G})+\rho[\mathbf{F}-(\mathbf{Z}^{(k)}-\frac\mathbf{{Y}^{(k)}}{\rho})]. \tag{19}\]  
Accordingly, the solution is  
\[\mathbf{F}^{(k+1)}=(\mathbf{W}^{\rm T}\mathbf{W}+\rho\mathbf{I})^{-1}(\mathbf{W}^{\rm T}\mathbf{G}+ \rho(\mathbf{Z}^{(k)}-\frac{\mathbf{Y}^{(k)}}{\rho})). \tag{20}\]  
To solve the sub-problem \(\mathbf{Z}\), let \(\widehat{\mathbf{Z}}=\mathbf{F}^{(k+1)}+\frac{\mathbf{Y}^{(k)}}{\rho}\). Then, Equation (16) can be transformed as follows:  
\[ \begin{split}\mathbf{Z}^{(k+1)}&=\arg\min_{\mathbf{Z}} \frac{\rho}{2}\|\mathbf{Z}-\widehat{\mathbf{Z}}\|_{2}^{2}+\mu\sum_{i=1}^{4}\|\mathbf{K}_{i }*\mathbf{Z}\|_{p}^{p}\\ &=\arg\min_{\mathbf{Z}}\frac{1}{2}\|\mathbf{Z}-\widehat{\mathbf{Z}}\|_{2}^{2} +\frac{\mu}{\rho}\sum_{i=1}^{4}\|\mathbf{K}_{i}*\mathbf{Z}\|_{p}^{p}.\end{split}  \tag{21}\]  
Let \(\lambda=\frac{\mu}{\rho}\); then, Equation (21) can be rewritten as  
\[\mathbf{Z}^{(k+1)}=\arg\min_{\mathbf{Z}}\frac{1}{2}\|\mathbf{Z}-\widehat{\mathbf{Z}}\|_{2}^{2}+ \lambda\sum_{i=1}^{4}\|\mathbf{K}_{i}*\mathbf{Z}\|_{p}^{p}. \tag{22}\]  
By treating \(\mathbf{Z}\) and \(\widehat{\mathbf{Z}}\) as the original and noisy images, respectively, Equation (22) is the standard denoising model based on FTV4L\({}_{p}(\mathbf{Z})\). Consequently, it can be replaced by using an image denoising algorithm to yield  
\[\mathbf{Z}^{(k+1)}=\mathbf{D}_{\lambda}(\widehat{\mathbf{Z}}^{(\mathbf{k+1})}), \tag{23}\]where \(\mathbf{D}_{\lambda}\) indicates the denoising process with the regularity parameter \(\lambda\). The proposed SRR algorithm is summarized in Algorithm 1. ```
Initialize:\(\rho,\mu,tol\) While\(\|\text{PSNR}(\mathbf{F}^{(k+1)},\mathbf{G})-\text{PSNR}(\mathbf{F}^{(k)},\mathbf{G})\|_{2}/\| \text{PSNR}(\mathbf{F}^{(k)},\mathbf{G})\|_{2}>tol\)do
1. \(\mathbf{F}^{(k+1)}=\arg\min_{\mathbf{F}}\frac{1}{2}\|\mathbf{SHF}-\mathbf{G}\|_{2}^{2}+\frac{ \rho}{2}\|\mathbf{F}-(\mathbf{Z}^{(k)}-\frac{Y^{(k)}}{\rho})\|_{2}^{2}\)
2. \(\mathbf{Z}^{(k+1)}=\mathbf{D}_{\lambda}(\widehat{\mathbf{Z}}^{(k+1)})\), where \(\widehat{\mathbf{Z}}=\mathbf{F}^{(k+1)}+\frac{Y^{(k)}}{\rho}\), \(\lambda=\frac{\mu}{\rho}\)
3. \(\mathbf{Y}^{(k+1)}=\mathbf{Y}^{(k)}+\rho(\mathbf{F}^{(k+1)}-\mathbf{Z}^{(k+1)})\)
4. \(k=k+1\)
```  
**Algorithm 1** Super-resolution using the plug-and-play ADMM (alternating direction method of multipliers)  
In addition, based on the fast ADMM algorithm proposed by [PERSON] et al. (2015), we adopt the accelerated step \(\alpha^{(k+1)}\), variables \(\widehat{\mathbf{Z}}^{(k+1)}\), and dual variables \(\widehat{\mathbf{Y}}^{(k+1)}\) as  
\[\alpha^{(k+1)}=\frac{1+\sqrt{1+4(\alpha^{(k)})^{2}}}{2}, \tag{24}\]  
\[\widehat{\mathbf{Z}}^{(k+1)}=\mathbf{Z}^{(k+1)}+\frac{\alpha^{(k)}-1}{\alpha^{(k+1)}}( \mathbf{Z}^{(k+1)}-\mathbf{Z}^{(k)}), \tag{25}\]  
\[\widehat{\mathbf{Y}}^{(k+1)}=\mathbf{Y}^{(k+1)}+\frac{\alpha^{(k)}-1}{\alpha^{(k+1)}}( \mathbf{Y}^{(k+1)}-\mathbf{Y}^{(k)}). \tag{26}\]  
Compared with the accelerated step proposed by [PERSON] et al. (2016, 2017) that utilizes an accumulated history of the past iterates recursively, the abovementioned accelerated step only adopts the traditional projection-like step that is evaluated at an auxiliary point obtained from the previous two iterates and an explicit dynamically updated step size.