## 3 Our Proposed Method  
The fully variational regularization problem, which often leads to step effects, can be addressed by incorporating higher-order derivatives [30]. In the literature, several approaches are available for introducing higher-order derivatives. One commonly used method is the TGV proposed by [PERSON] [31], which combines first-order [32] and higher-order derivative information. TGV effectively preserves image edge details and suppresses step effects, making it successful in applications such as image denoising and MRI [33; 34]. Suppose that \(u\in W^{1,1}(\Omega)=\{u\in L^{1}(\Omega)\mid\
abla u\in L^{1}(\Omega,\mathbb{ R}^{n})\}\), and define the following:  
\[\Phi(u)=\sup_{\xi}\biggl{\{}\int_{\Omega}\langle\
abla u,\text{div}(\xi) \rangle_{\mathbb{R}^{n}}dx\mid\xi\in C_{c}^{2}\big{(}\Omega,\mathbb{R}^{n\times n }\big{)},\|\xi\|_{\infty}\leqslant 1\biggr{\}} \tag{17}\]  
where  
\[\xi=\left( \begin{array}{c}\xi_{1}\\ \xi_{2}\\ \vdots\\ \xi_{n}\end{array} \right)=\left( \begin{array}{cccc}\xi_{1,1}&\xi_{1,2}& \cdots&\xi_{1,n}\\ \xi_{2,1}&\xi_{2,2}&\cdots&\xi_{2,n}\\ \vdots&\vdots&&\vdots\\ \xi_{n,1}&\xi_{n,2}&\cdots&\xi_{n,n}\end{array} \right),\]  
\[\text{div}(\xi)=\left( \begin{array}{c}\text{div}(\xi_{1})\\ \text{div}(\xi_{2})\\ \vdots\\ \text{div}(\xi_{n})\end{array} \right)=\left( \begin{array}{c}\sum_{k=1}^{n} \frac{\partial\xi_{1,k}}{\partial x_{k}}\\ \vdots\\ \sum_{k=1}^{n}\frac{\partial\xi_{nk}}{\partial x_{k}}\end{array} \right)\]  
\[\xi_{i,j}:\Omega\rightarrow\mathbb{R},\|\xi\|_{\infty}=\sup_{x\in\Omega} \sqrt{\sum_{i,j=1}^{d}\xi_{i,j}^{2}(x)_{0}}.\]  
Equation (17) defines a full variational component for \(\
abla u\), which is considered as a second-order fully variational component. Additionally, based on Equation (17), the space of second-order bounded variational functions can be defined as follows:  
\[BV^{2}(\Omega)=\Big{\{}u\in W^{1,1}(\Omega)\mid\Phi(u)<+\infty\Big{\}} \tag{18}\]  
\[\Phi(u)=\sup_{\xi}\biggl{\{}\int_{\Omega}u\,\text{div}^{2}(\xi)\text{d}x\mid \xi\in C_{c}^{2}\big{(}\Omega,\mathbb{R}^{n\times n}\big{)},\|\xi\|_{\infty} \leqslant 1\biggr{\}} \tag{19}\]As a further generalization, for \(u(x)\in L^{1}(\Omega)\), its second-order generalized fully variational component can be defined as follows:  
\[\mathrm{TGV}^{2}_{\alpha}(u)=\sup_{\xi}\left\{\int_{\Omega}u\,\mathrm{div}^{2}( \xi)\mathrm{d}x\, \begin{vmatrix}\xi\in C^{2}_{c}\big{(}\Omega\mathrm{Sym}^{2}( \mathbb{R}^{n})\big{)}\\ \|\xi\|_{\infty}\leq\epsilon_{0,\|}\,\mathrm{div}(\xi)\|_{\infty}\leq\epsilon_ {\alpha}\end{vmatrix} \right\} \tag{20}\]  
\(\alpha=(\alpha_{0},\alpha_{1})^{\mathrm{T}}\in\mathbb{R}^{2}_{+}\) is the weight vector and \(C^{2}_{c}\Big{(}\Omega,\mathrm{Sym}^{2}(\mathbb{R}^{n})\Big{)}\) is a symmetric tensor field with a tight support. The generalized complete variational fraction of order \(k\) can be described more broadly as follows [31]. For a positive integer \(k\in\mathbb{Z}^{+}\) and \(\alpha=(\alpha_{0},\cdots,\alpha_{k-1})^{\mathrm{T}}\in\mathbb{R}^{k}_{+}\), we define a generalized fully variational fraction of order \(k\) as follows:  
\[\mathrm{TGV}^{k}_{\alpha}(u)=\sup_{\xi}\left\{\int_{\Omega}u\,\mathrm{div}^{k} (\xi)\mathrm{d}x\, \begin{vmatrix}\xi\in C^{k}_{c}\Big{(}\Omega,\mathrm{Sym}^{ k}(\mathbb{R}^{n})\Big{)},\\ \Big{\|}\mathrm{div}^{j}(\xi)\Big{\|}_{\infty}\leq\alpha_{1},j=0,1,\cdots,k-1 \end{vmatrix} \right\} \tag{21}\]  
Consequently, the following definition describes the space of generalized complete variational functions of order \(k\). \[\mathrm{BGV}^{k}_{\alpha}(\Omega)=\Big{\{}u\in L^{1}(\Omega)\Big{|}\mathrm{TGV }^{k}_{\alpha}(u)<+\infty\Big{|}\Big{\}} \tag{22}\]  
The definition of \(\mathrm{TGV}^{k}_{\alpha}(u)\) provides restrictions with various weights for each \(\mathrm{div}^{j}(\xi)\), as shown in Equation (21). The weights \(\alpha=(\alpha_{0},\cdots,\alpha_{k-1})^{\mathrm{T}}\) tend to have a significant influence on the outcomes yielded in practical computations; thus, it must be carefully chosen.