## 2 Materials and Methods
### Iterative Optimization Process  
Using the Alternating Direction Method of Multipliers (ADMM) approach [29], we solve Equation (21) by fixing all other variables while solving for one variable at a time. This method introduces auxiliary variables and Lagrange multipliers, decomposing the original problem into a series of smaller subproblems, which are then alternately updated to progressively approach the optimal solution of the original problem. By introducing auxiliary variables \(G_{1},\ G_{2}\), we reformulate Equation (21) as follows:  
\[ \begin{array}{l}\min_{U,V,T}\alpha_{1}\|G_{1}\|_{1}+\alpha_{2}\|G_{2}\|_{1}+ \lambda\|w\bigodot T\|_{1}\\ s.t.D=UV^{T}+T,V^{T}V=I,\
abla_{i}(\mathrm{U})=G_{i}\end{array}  \tag{22}\]  
We employ the Augmented Lagrange Multiplier Method (ALMM) [30] to solve the convex optimization problem involving the combination of nuclear norm and \(l_{1}\)-norm minimization, reformulating the above equation as:  
\[L(U,V,T,G_{1},G_{2},\beta_{1},\beta_{2},\beta_{3})=\sum\limits_{i=1}^{2}\left( \alpha_{i}\|G_{i}\|_{1}+\frac{\mu}{2}\Big{\|}\
abla_{i}(U)-G_{i}+\frac{\beta_ {i}}{\mu}\Big{\|}_{F}^{2}\right)+\lambda\|w\bigodot T\|_{1}+\frac{\mu}{2}\Big{\|} D-UV^{T}-T+\frac{\beta_{3}}{\mu}\Big{\|}_{F}^{2} \tag{23}\]  
In the equation, \(\beta_{1},\ \beta_{2},\ \text{and}\ \beta_{3}\) represent the Lagrange multipliers, and \(\mu\) denotes the penalty parameter. Subsequently, the ADMM method is employed to iteratively solve Equation (23). The complete solution process is outlined as follows. 1. Updating \(G_{1},G_{2}\): By fixing all variables except \(G_{1},G_{2}\) in Equation (23), we obtain the following subproblem: \[G_{i}=\underset{G_{i}}{\text{argmin}}\sum\limits_{i=1}^{2}\left(\alpha_{i}\|G _{i}\|_{1}+\frac{\mu}{2}\bigg{\|}\
abla_{i}(U)-G_{i}+\frac{\beta_{i}}{\mu} \bigg{\|}_{F}^{2}\right)\] (24) The soft thresholding function [31] is employed to solve: \[G_{i}=Soft\left(\
abla_{i}(U)+\frac{\beta_{i}}{\mu},\frac{\alpha_{i}}{\mu} \right)\,\ (i=1,\ 2)\] (25)
2. Updating \(V\): By fixing all variables except \(V\) in Equation (23), we obtain the following subproblem: \[V=\underset{V}{\text{argmin}}\frac{\mu}{2}\bigg{\|}D-UV^{T}-T+\frac{\beta_{3}} {\mu}\bigg{\|}_{F}^{2}\] (26) The calculation is performed using the theorem mentioned in [32]: \[[\ A,\ \sim,C]=svd\left(\left(D-T+\frac{\beta_{3}}{\mu}\right)^{T}U\right)\] (27)
3. Updating \(U\): By fixing all variables except \(U\) in Equation (23), we obtain the following subproblem: \[\underset{U}{\text{argmin}}\frac{\mu}{2}\sum\limits_{i=1}^{2}\left(\bigg{\|} \
abla_{i}(U)-G_{i}+\frac{\beta_{i}}{\mu}\bigg{\|}_{F}^{2}\right)+\frac{\mu}{ 2}\bigg{\|}D-UV^{T}-T+\frac{\beta_{3}}{\mu}\bigg{\|}_{F}^{2}\] (28) Taking the derivative of the above equation: \[\left(I+\sum\limits_{i=1}^{2}\
abla_{i}^{T}\
abla_{i}\right)(U)=\left(D-T+ \frac{\beta_{3}}{\mu}\right)V+\sum\limits_{i=1}^{2}\
abla_{i}^{T}\left(G_{i}- \frac{\beta_{i}}{\mu}\right)\] (29)In the equation, \(\
abla_{i}^{T}\) represents the \"transpose\" operator of \(\
abla_{i}\). Treating the difference operation \(\
abla_{i}(U)\) as the convolution of the difference filter \(O_{i}\)\(\otimes\)\(fold\)\((U)\), where \(O_{i}\) is the corresponding difference filter, the closed-form solution for \(U\) can be easily derived by applying the Fourier transform to both sides of the equation and utilizing the convolution theorem [33]. \[U=unfold_{3}\left(\mathcal{F}^{-1}\left(\frac{\mathcal{F}(fold((\mu(D-T)+\beta _{3})V))+\sum\limits_{i=1}^{2}\mathcal{F}(O_{i})^{*}\bigodot\mathcal{F}(fold( \mu G_{i})-\beta_{i})}{\mu(ten(1))+\mu\Big{(}\left|\mathcal{F}(O_{1})\right|^{ 2}+\left|\mathcal{F}(O_{2})\right|^{2}\Big{)}}\right)\right) \tag{30}\]  
Here, \(F(\cdot)\) and \(\left|{}_{\cdot}\right|^{2}\) denote the Fourier transform and element-wise square operation, respectively, while \(ten(1)\) represents a tensor with all elements equal to 1. 4.