# CMOS Fixed Pattern Noise Removal Based on Low Rank Sparse Variational Method
## 4 Proposed Model  
Based on the fidelity items analyzed above and the relevant priori constraint information, the optimization equation with constraints can be obtained as follows:  
\[ \begin{split}\mathrm{E}(\mathrm{U},S)=\min_{U,S}\frac{\alpha_{ 4}}{\alpha}\|Y-U-S\|_{2}^{2}+\tau\|S\|_{*}\\ \text{s.t.}\ \|\frac{\partial U}{\partial x}\|_{1}=0\\ \|\frac{\partial U}{\partial y}\|_{1}=0\\ \|\frac{\partial Y}{\partial y}-\frac{\partial U}{\partial y}\|_{ 1}=0\\ S\|_{2}^{2}=0\end{split}  \tag{3}\]  
Figure 8: Gradient probability distribution. (**a**) Horizontal gradient probability distribution; (**b**) vertical gradient probability distribution (x coordinates: horizontally or vertically gradient; y coordinates: probability distribution). where \(Y\) is the observed image, \(U\) is the clear image and S is the stripe noise. \(\|S\|_{*}\) is the kernel norm of S, which is used to constrain \(S\) with low rank. Then, according to the Lagrange multiplier method, the constrained Equation (3) becomes an unconstrained Equation (4). \[ \begin{array}{c}\mathrm{E}(\mathrm{U},S)=\frac{\alpha_{1}}{2}\|Y-U-S\|_{2}^{2} +\alpha_{2}\|\frac{\partial U}{\partial x}\|_{1}+\alpha_{3}\|\frac{\partial U} {\partial y}\|_{1}+\alpha_{4}\|\frac{\partial Y}{\partial y}-\frac{\partial U }{\partial y}\|_{1}\\ +\frac{\gamma}{2}\|S\|_{2}^{2}+\tau\|S\|_{*}\end{array}  \tag{4}\]  
where \(\alpha_{1},\alpha_{2},\ \alpha_{3},\ \alpha_{4},\gamma,\tau\) are Lagrange multipliers This is a multivariable convex optimization equation. At present, the commonly used methods for multivariable optimization are Bregman and ADMM. Through these optimization methods, the complex multivariable optimization process can be divided into more convenient sub optimization problems. The ADMM algorithm [38] is used in this paper. The specific optimization process is as follows: Three auxiliary variables H, J, K are introduced. Let \(\mathrm{H}=\frac{\partial\mathrm{U}}{\partial\chi}\), \(\mathrm{J}=\frac{\partial\mathrm{U}}{\partial y}\), \(\mathrm{K}=\frac{\partial Y}{\partial y}-\frac{\partial U}{\partial y}\), so Equation (4) equivalent to Equation (5)  
\[ \begin{array}{c}\mathrm{E}(\mathrm{U},S)=\frac{\alpha_{1}}{2}\|Y-U-S\|_{2}^{ 2}+\alpha_{2}\|\frac{\partial U}{\partial x}\|_{1}+\alpha_{3}\|\frac{\partial U }{\partial y}\|_{1}+\alpha_{4}\|\frac{\partial Y}{\partial y}-\frac{\partial U }{\partial y}\|_{1}\\ +\frac{\gamma}{2}\|S\|_{2}^{2}+\tau\|S\|_{*}\\ \mathrm{s.t.}\mathrm{H}=\frac{\partial\mathrm{U}}{\partial\chi}\\ \mathrm{J}=\frac{\partial\mathrm{U}}{\partial\gamma}\\ \mathrm{K}=\frac{\partial Y}{\partial y}-\frac{\partial U}{\partial y}\end{array}  \tag{5}\]  
According to the Augmented Lagrange multiplier method, the constraint equation of Equation (5) is changed into the unconstrained Equation (6)  
\[ \begin{array}{c}\mathrm{E}(\mathrm{U},\mathrm{S},\mathrm{H},\mathrm{J}, \mathrm{K})=\frac{\alpha_{1}}{2}\|Y-U-S\|_{2}^{2}+\alpha_{2}\|H\|_{1}+\alpha_{ 3}\|J\|_{1}+\alpha_{4}\|K\|_{1}+\frac{\gamma}{2}\|S\|_{2}^{2}+\tau\|S\|_{*}\\ +<R_{2},\ \mathrm{H}-\frac{\partial\mathrm{U}}{\partial\chi}>+<R_{3},\ \mathrm{J}-\frac{ \partial\mathrm{U}}{\partial\gamma}>+<R_{4},\mathrm{K}-\left(\frac{\partial Y}{ \partial y}-\frac{\partial U}{\partial y}\right)>\\ +\frac{\alpha_{2}}{2}\|\mathrm{H}-\frac{\partial\mathrm{U}}{\partial\chi}\|_{ 2}^{2}+\frac{\alpha_{3}}{2}\|J-\frac{\partial\mathrm{U}}{\partial y}\|_{2}^{2 }+\frac{\alpha_{4}}{2}\|\mathrm{K}-\left(\frac{\partial Y}{\partial y}-\frac{ \partial U}{\partial y}\right)\|_{2}^{2}\end{array}  \tag{6}\]  
where \(<\)A, B\(>\) is defined as the inner product of two variables. \(R_{2},\ R_{3},R_{4}\) are the regular coefficients of the regular terms \(\mathrm{H}-\frac{\partial\mathrm{U}}{\partial\chi},\mathrm{J}-\frac{\partial \mathrm{U}}{\partial\gamma}\), \(\mathrm{K}-\left(\frac{\partial Y}{\partial y}-\frac{\partial U}{\partial y}\right)\), respectively. Equation (6) can be changed into Equation (7) after the relevant terms are combined. \[