## 3 Our Proposed Method
}\left(\boldsymbol{p}-\
abla u,w\right)+\langle\xi(\boldsymbol{ p}),\xi\rangle \tag{27}\] \[=\min_{p\in\mathbb{W}}\alpha_{1}\|\
abla u-p\|_{1}+\alpha_{0}\| \varepsilon(\boldsymbol{p})\|_{1}\] \[=\min_{p\in\mathbb{W}}\alpha_{1}\int_{\Omega}\sqrt{\sum_{j=1}^{ 2}\left(\
abla_{j}u(x)-p_{j}(x)\right)^{2}}\,\text{d}x+\alpha_{0}\int_{\Omega }\sqrt{\sum_{j,k=1}^{2}\left(\bar{\varepsilon}(\boldsymbol{p})(x)\right)_{l,k }^{2}}\,\text{d}x\]  
This leads to a relatively concise equivalent expression for \(\text{TGV}_{\alpha}^{2}(u)\), namely  
\[TGV_{\alpha}^{2}(u)=\min_{p\in C_{c}^{2}(\Omega,\mathbb{R}^{n})}\alpha_{1}\| \
abla u-\boldsymbol{p}\|_{1}+\alpha_{0}\|\varepsilon(\boldsymbol{p})\|_{1} \tag{28}\]  
where  
\[\varepsilon(\boldsymbol{p})=\left[ \begin{array}{cc}\frac{\partial\boldsymbol {p}_{1}}{\partial x}&\frac{1}{2}\Big{(}\frac{\partial\boldsymbol{p}_{1}}{ \partial y}+\frac{\partial\boldsymbol{p}_{2}}{\partial x}\Big{)}\\ \frac{1}{2}\Big{(}\frac{\partial\boldsymbol{p}_{1}}{\partial y}+\frac{ \partial\boldsymbol{p}_{2}}{\partial x}\Big{)}&\frac{\partial\boldsymbol{p}_{2 }}{\partial y}\end{array} \right],\boldsymbol{p}=(\boldsymbol{p}_{1},\boldsymbol {p}_{2})\in C_{c}^{2}\Big{(}\Omega,\mathbb{R}^{2}\Big{)} \tag{29}\]  
Equation (28) reveals that \(\text{TGV}_{\alpha}^{2}(u)\) can be understood as a combination of two regularization terms. The first term in Equation (28) demonstrates that \(\boldsymbol{p}\) approximates \(\
abla u\) and acts as a constraint on the first-order derivative of \(u\). The second term imposes a constraint on the first-order derivative of \(\boldsymbol{p}\) in all directions, which is equivalent to imposing a constraint on the second-order derivative of \(u\). In the smoothed region, the second term suppresses the step effect, while the first term enables the preservation of edges at the image's boundaries. The parameters \(\alpha_{0}\) and \(\alpha_{1}\) represent the weights assigned to the two canonical terms in the composite function. However, the selection of these weights in practical problems remains an open question. The second-order generalized fully variational regularization model can be expressed using the above second-order generalized fully variational equivalence expression, which helps with solving the model. We have proposed an improved alternating direction of multipliers method (ADMM) algorithm for solving the SOGTV problem by combining the accelerated Nesterov [37] and the Fast-ADMM [38]. \[\min_{u,p}\alpha_{1}\|\
abla u-\mathbf{p}\|_{1}+\alpha_{0}\|\varepsilon(p)\|_{1}+ \frac{\lambda}{2}\|f-Au\|_{2}^{2} \tag{30}\]  
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. \[\mathbf{\
abla}u=\big{[}\mathbf{Q}_{x}u,\mathbf{Q}_{y}u\big{]}^{\mathrm{T}},\mathbf{p }=[p_{1},p_{2}]^{\mathrm{T}}, \tag{31}\]  
\[\overline{\varepsilon}(p)=\left[ \begin{array}{cc}Q_{x}p_{1}&\frac{1}{2} \big{(}Q_{x}p_{2}+Q_{y}p_{1}\big{)}\\ \frac{1}{2}\big{(}Q_{x}p_{2}+Q_{y}p_{1}\big{)}&Q_{y}p_{2}\end{array} \right] \tag{32}\]  
where \(Q_{x}\) and \(Q_{y}\) denote the first-order derivative operators along the \(x\) and \(y\) directions, respectively. Variables \(\mathbf{y}=[y_{1},y_{2}]^{\mathrm{T}}=\
abla u-\mathbf{p}\) and \(\mathbf{z}=\left[ \begin{array}{cc}z_{1}&z_{3}\\ z_{3}&z_{2}\end{array} \right]=\overline{\varepsilon}(p)\) are introduced, and then, the above optimization problem can be equivalently expressed as  
\[\left\{ \begin{array}{ll}\min_{u,p}&\alpha_{1}\|\mathbf{y}\|_{1}+\alpha_{0}\|z\|_ {1}+\frac{\lambda}{2}\|f-Au\|_{2}^{2}\\ \text{s. t.}&\mathbf{y}=\
abla u-\mathbf{p},z=\overline{\varepsilon}(p)\end{array} \right. \tag{33}\]  
Then, the incremental Lagrange function is  
\[