## 2 Theory and Methods
### TV-GPEF Algorithm  
The convergence of the GP-type extrapolation has been theoretically proven. However, the pure application of the GP-type extrapolation to compensate for the loss caused by the data deficiency is still not stable in practical for its weak noise robustness and frequent oscillations. Therefore, we combine the GPEF compensation method with the TV-based optimization to improve the stableness and the convergence of the algorithm. The variable splitting and Barzilai-Borwein based method is implemented to solve the optimization problem [52]. This method is known to have excellent performance in resolving the TV-based minimization problem. The optimization problem for the TV-based algorithm in Equation (9) can be rewritten as:  
\[ \begin{array}{l}\min_{\mathbf{u},\mathbf{A}}\left\{\alpha\underset{k}{ \sum}\left|\mathbf{u}_{k}\right|_{2}+\frac{\lambda}{2}\left|\mathbf{W}^{T} \cdot\mathbf{A}-\mathbf{g}\right|_{2}^{2}\right\}k=1,2...N_{x}N_{y},\\ \\ D_{k}\mathbf{A}=\mathbf{u}_{k}\end{array}  \tag{15}\]  
where \(\alpha\) and \(\lambda\) are weights of two parts of the objective function. By using the standard augmented Lagrangian method and incorporating the Barzilai-Borwein step size to obtain faster convergence [52; 53], the problem of Equation (15) can also be deduced as:  
\[ \begin{array}{l}(\mathbf{u}^{n+1},\mathbf{A}^{n+1})=\min_{\mathbf{u}, \mathbf{A}}\biggl{\{}\alpha\underset{i}{\sum}\left(\left|\mathbf{u}_{k} \right|_{2}+\frac{\lambda}{2}\left|\mathbf{u}_{k}-D_{k}\mathbf{A}^{n}-b_{k}^{ n}\right|_{2}^{2}\right)\\ \left.+\delta_{n}\left(\left|\mathbf{u}^{n+1}-\mathbf{u}^{n}\right|_{2}^{2}+ \frac{1}{2}\left|\mathbf{u}-\mathbf{u}^{n}+\delta_{n}^{-1}\mathbf{W}^{T} \big{(}\mathbf{W}^{T}\mathbf{A}^{n}-g\big{)}\right|_{2}^{2}\right)\right\} \end{array}  \tag{16}\]  
where \(b_{k}^{n}\) is the defined step parameter for the TV in the \(n\)th iteration and \(\delta_{n}\) is the defined Barzilai-Borwein step size in the \(n\)th iteration. The updating of \(\delta_{n}\) can be determined by the Barzilai-Borwein method. After using the variable splitting method, Equation (16) can be transformed into the following two sub-problems [41; 52]:  
\[ \begin{array}{l}\mathbf{u}_{k}^{n+1}=\min_{u_{k}}\biggl{\{}\left|\mathbf{u }_{k}\right|_{2}+\frac{\rho}{2}\left|\mathbf{u}_{k}-D_{k}\mathbf{A}^{n}-b_{k} ^{n}\right|_{2}^{2}+\frac{\delta_{n}}{\alpha}\left|\mathbf{u}_{k}-\mathbf{u}_ {k}^{n}\right|_{2}^{2}\biggr{\}},\\ \mathbf{A}^{n+1}=\min_{\mathbf{A}}\biggl{\{}\alpha\lambda\left|DA-\mathbf{u}^ {n+1}\right|_{2}^{2}+\delta_{n}\left|\mathbf{A}-\left(\mathbf{A}^{n}-\delta_ {n}^{-1}\mathbf{W}^{T}(\mathbf{W}\mathbf{A}^{n}-\mathbf{g})\right)\right|_{2}^ {2}\biggr{\}}\\ b_{k}^{n+1}=b_{k}^{n}-\left(\mathbf{u}_{k}^{n+1}-D_{k}\mathbf{A}^{n+1}\right),\\ \delta_{n+1}=\left(\left|\mathbf{W}(\mathbf{A}^{n+1}-\mathbf{A}^{n})\right|_{2 }^{2}/\left(\left|\mathbf{u}^{n+1}-\mathbf{u}^{n}\right|_{2}^{2}+\left| \mathbf{A}^{n+1}-\mathbf{A}^{n}\right|_{2}^{2}\right)\right.\end{array}  \tag{17}\]  
The two sub-problems can be solved as follows using the shrinkage operator method [41; 52]:  
\[\left\{ \begin{array}{l}\mathbf{u}_{k}^{n+1}=\max\biggl{\{}\|\frac{a_{1} \lambda+2\delta_{n}/\alpha}{a_{1}+a_{2}}\|-\frac{1}{a_{1}+a_{2}},0\biggr{\}} \frac{1/(a_{1}+a_{2})}{\|1/(a_{1}+a_{2})\|}\\ a_{1}=D_{i}\mathbf{A}^{n}+b_{k}^{n}\\ a_{2}=\mathbf{u}_{k}^{n}\end{array} \right. \tag{18}\]  
\[\mathbf{A}^{n+1}=F^{\mathrm{T}}\biggl{\{}\frac{F[\alpha\lambda\,D^{\mathrm{T}} \mathbf{u}^{n+1}+\delta_{n}\mathbf{A}^{n}-\mathbf{W}^{T}(\mathbf{W}\mathbf{A} ^{n}-\mathbf{g})]}{\alpha\rho\,F^{\mathrm{T}}D^{\mathrm{T}}D\mathrm{F}+\delta_ {n}\mathbf{I}}\biggr{\}} \tag{19}\]  
where \(F\) is the Fourier transform matrix. We apply the TV-based algorithm into the GPEF compensation method. In every iteration, the image is first updated by the TV optimization and then compensated by the GPEF method. The iteration steps of the TV-GPEF algorithm are summarized as follows:  
1. Initialization: Input **A**, \(\alpha\), \(\lambda\), \(\eta\). Set \(\delta_{0}=1\), \(b_{0}=0\).