## 3 Our Proposed Method
Although the definition in Equation (21) is used for general values of \(k\in\mathbb{Z}^{+}\), for two-dimensional images, \(\mathrm{TGV}^{1}_{\alpha}(u)\) or \(\mathrm{TGV}^{2}_{\alpha}(u)\) is more commonly utilized. In fact, we have that \(\mathrm{Sym}^{k}(\mathbb{R}^{n})=\mathbb{R}^{n}\) and \(\alpha\in\mathbb{R}^{+}\) when \(k=1\):  
\[\mathrm{TGV}^{1}_{\alpha}(u)=\alpha\mathrm{TV}(u) \tag{23}\]  
The following second-order generalized full-variance regularization model is suggested using the second-order generalized full variance concept described above:  
\[\underset{u}{\text{min}}\bigg{\{}E(u)=\mathrm{TGV}^{2}_{\alpha}(u)+\frac{ \lambda}{2}\|f-Au\|_{2}^{2}\bigg{\}} \tag{24}\]  
where \(f\in L^{2}(\Omega)\) is the observed image with noise; \(A\) is the degeneracy operator; and \(\Omega\subset\mathbb{R}^{n}\) is the image support domain. The SOGTV, as defined in Equation (14), is challenging to compute. Unlike the first-order generalized TV \(\mathrm{TGV}^{1}_{\alpha}(u)\), which focuses solely on variance [32], the second-order generalized TV also accounts for covariance and constant terms [35]. This necessitates the calculation of additional derivatives and second-order partial derivatives, making its formulation more intricate and computationally demanding. Specifically, \(\mathrm{TGV}^{2}_{\alpha}(u)\) involves determining the Hessian matrix of the image function and, subsequently, computing its eigenvalues and eigenvectors. This computational process requires a significant amount of computation and may not be feasible for large-scale problems. Moreover, the \(\mathrm{TGV}^{2}_{\alpha}(u)\) regularization term might result in excessively smooth solutions or introduce artifacts in the reconstructed image. However, the second-order generalized full variance can offer more comprehensive information and effectively capture the intricate details and variations in the images and data. As a result, to achieve enhanced computational efficiency, a simplified and easily computable representation of the second-order generalized full variance is commonly employed in practical calculations, as explained in [36]. For the sake of simplicity, we denote \(U=C^{2}_{c}(\Omega,\mathbb{R})\), \(\mathbf{W}=\mathbf{C}^{2}_{c}(\Omega,\mathbb{R}^{2})\),\[E=C_{c}^{2}\Big{(}\Omega,\text{Sym}^{2}(\mathbb{R}^{n})\Big{)},\text{ and div}\,\xi=w.\text{ Then,}\] \[\text{TGV}_{\alpha}^{2}(u)=\max_{u\in V_{c},\xi\in E}\{\langle u, \text{div}\,w\rangle\ |\ \text{div}\,\xi=w,\|\xi\|_{\infty}\leqslant\alpha_{0},\|w\|_{\infty}\leqslant \alpha_{1}\} \tag{25}\] \[\text{where}\ w=\left( \begin{array}{c}w_{1}\\ w_{2}\end{array} \right),\xi=\left( \begin{array}{cc}\xi_{11}&\xi_{12}\\ \xi_{21}&\xi_{22}\end{array} \right),\text{div}\,\xi=\left( \begin{array}{c} \partial_{x}\xi_{11}+\partial_{x}\xi_{12}\\ \partial_{x}\xi_{21}+\partial_{x}\xi_{22}\end{array} \right).\]  
For any closed set \(B\), if its demonstrative function is defined as  
\[\chi_{B}(x)= \begin{cases}0,&x\in B\\ \infty,&x\
otin B\end{cases}  \tag{26}\]  
\(\chi_{|0|}(\cdot)=-\min_{v}\langle y,\cdot\rangle\), we have  
\[\text{TGV}_{\alpha}^{2}(u)=\max_{ \begin{subarray}{c}\|\xi\|_{ \infty}\leqslant\alpha_{0},\xi\in E\\ \|\boldsymbol{v}\|_{\infty}\leqslant\alpha_{1},\boldsymbol{w}\in W \end{subarray} }\left(\langle u,\text{div}\,\boldsymbol{w}\rangle+\min_{p\in \mathbb{W}}\langle\boldsymbol{p},w-\text{div}\,\xi\rangle\right)\] \[=\min_{p\in\mathbb{W}}\max_{ \begin{subarray}{c}\|\xi\|_{\infty} \leqslant\alpha_{0},\xi\in E\\ \|\boldsymbol{v}\|_{\infty}\leqslant\alpha_{1},\boldsymbol{w}\in W \end{subarray} }\left(u,\text{div}\,w\right)+\langle\boldsymbol{p},w-\text{ div}\,\xi\rangle\] \[=\min_{p\in\mathbb{W}}\max_{ \begin{subarray}{c}\|\xi\|_{\infty} \leqslant\alpha_{0},\xi\in E\\ \|\boldsymbol{v}\|_{\infty}\leqslant\alpha_{1},\boldsymbol{w}\in W \end{subarray}