## 3 Proposed Method  
In this study, we propose an SRR method for infrared images based on the quaternion FTV with the \(L_{p}\) quasinorm:  
\[\mathbf{F}=\arg\min_{\mathbf{F}}\frac{1}{2}\|\mathbf{SHF}-\mathbf{G}\|_{2}^{2}+\mu\text{FTV4L}_{p }\quad(\mu>0), \tag{7}\]  
where \(\mathbf{H}\) is a circulant matrix that represents the convolution for the anti-aliasing filter. \(\mathbf{S}\) is a binary sampling matrix, where the rows are subsets of the identity matrix. Further, \(\mathbf{G}\) is an observation image, while \(\mathbf{F}\) represents the corresponding original image. FTV4L\({}_{p}(\mathbf{F})\) represents the quaternion FTV with the \(L_{p}\) quasinorm, which is defined as follows:  
\[\text{FTV4L}_{p}(\mathbf{F})=\left\|\mathbf{K}_{1}*\mathbf{F}\right\|_{p}^{p}+\left\|\mathbf{K }_{2}*\mathbf{F}\right\|_{p}^{p}+\left\|\mathbf{K}_{3}*\mathbf{F}\right\|_{p}^{p}+\left\| \mathbf{K}_{4}*\mathbf{F}\right\|_{p}^{p}, \tag{8}\]  
where \(0<p<1\) represents the \(L_{p}\) quasinorm. \(\mathbf{K}_{i}\)\((i=1,2,3,4)\) represents the convolution kernels along the horizontal, vertical, back diagonal, and diagonal directions, respectively; these are defined as follows:  
\[\mathbf{K}_{1}=\mathbf{c},\;\mathbf{K}_{2}=\mathbf{c}^{\text{T}},\;\mathbf{K}_{3}=diag(\mathbf{c}),\; \mathbf{K}_{4}=flipidr(diag(\mathbf{c})), \tag{9}\]  
where \(\mathbf{c}\in\mathbf{R}^{1,N}\) and \(\mathbf{c}_{k}^{v}=(-1)^{k}\mathbf{c}_{k}^{v}\); in addition, \(flipidr(\mathbf{A})\) is a function that returns A with its columns flipped in the left-right direction. The coefficient \(\mathbf{c}_{k}^{v}\) can be obtained as follows:  
\[\mathbf{c}_{0}^{v}=1,\;\mathbf{c}_{k}^{v}=(1-\frac{v+1}{k})\mathbf{c}_{k-1}^{v},\quad k=1, 2,... \tag{10}\]  
Thus, the proposed SRR method can be represented by  
\[\mathbf{F}=\arg\min_{\mathbf{F}}\frac{1}{2}\|\mathbf{SHF}-\mathbf{G}\|_{2}^{2}+\mu(\left\|\bm {K}_{1}*\mathbf{F}\right\|_{p}^{p}+\left\|\mathbf{K}_{2}*\mathbf{F}\right\|_{p}^{p}+\left\| \mathbf{K}_{3}*\mathbf{F}\right\|_{p}^{p}+\left\|\mathbf{K}_{4}*\mathbf{F}\right\|_{p}^{p}). \tag{11}\]  
Furthermore, it can be rewritten as  
\[\mathbf{F}=\arg\min_{\mathbf{F}}\frac{1}{2}\|\mathbf{SHF}-\mathbf{G}\|_{2}^{2}+\mu\sum_{i=1}^ {4}\left\|\mathbf{K}_{i}*\mathbf{F}\right\|_{p}^{p}. \tag{12}\]  
To solve the FTV4L\({}_{p}(\mathbf{F})\) model in the framework of the plug-and-play ADMM, an assistant variable \(\mathbf{Z}\) is required to convert the unconstrained problem given by Equation (12) into the constrained problem:  
\[(\mathbf{F},\mathbf{Z})=\arg\min_{\mathbf{F},\mathbf{Z}}\frac{1}{2}\|\mathbf{SHF}-\mathbf{G}\|_{2}^{2} +\mu\sum_{i=1}^{4}\left\|\mathbf{K}_{i}*\mathbf{Z}\right\|_{p}^{p},\quad\mathbf{s.t.}\quad \mathbf{Z}=\mathbf{F}. \tag{13}\]  
Consequently, the corresponding augmented Lagrangian function is as follows:\[ \begin{split}\mathbf{L}_{\rho}(\mathbf{F},\mathbf{Z},\mathbf{Y})&=\frac{1 }{2}\|\mathbf{SHF}-\mathbf{G}\|_{2}^{2}+\mu\sum_{i=1}^{4}\|\mathbf{K}_{i}*\mathbf{Z}\|_{p}^{p} \\ &+\langle\mathbf{Y},(\mathbf{F}-\mathbf{Z})\rangle+\frac{\rho}{2}\|\mathbf{F}- \mathbf{Z}\|_{2}^{2},\end{split}  \tag{14}\]  
where \(\langle\mathbf{A},\mathbf{B}\rangle=\sum_{i=1}^{n}\sum_{i=1}^{n}a_{ij}b_{ij}\) is the inner product of matrices \(\mathbf{A}\) and \(\mathbf{B}\). \(\mathbf{Y}\) is a Lagrange multiplier, and \(\rho>0\) is a penalty parameter. The minimizer of Equation (13) is the saddle point of \(L_{\rho}\), which can be found by solving the following sequence of sub-problems:  
\[ \begin{split}\mathbf{F}^{(k+1)}&=\arg\min_{\mathbf{F}} \frac{1}{2}\|\mathbf{SHF}-\mathbf{G}\|_{2}^{2}+\langle\mathbf{Y}^{(k)},(\mathbf{F}-\mathbf{Z}^{(k) })\rangle+\frac{\rho}{2}\|\mathbf{F}-\mathbf{Z}^{(k)}\|_{2}^{2}\\ &=\arg\min_{\mathbf{F}}\frac{1}{2}\|\mathbf{SHF}-\mathbf{G}\|_{2}^{2}+\frac{ \rho}{2}\|\mathbf{F}-(\mathbf{Z}^{(k)}-\frac{\mathbf{Y}^{(k)}}{\rho})\|_{2}^{2},\end{split}