# CMOS Fixed Pattern Noise Removal Based on Low Rank Sparse Variational Method
## 4 Proposed Model
\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\|_{*}\\ +\frac{\alpha_{2}}{2}\|\mathrm{H}-\frac{\partial\mathrm{U}}{\partial\chi}+\frac{ R_{2}}{\alpha_{2}}\|_{2}^{2}+\frac{\alpha_{3}}{2}\|J-\frac{\partial\mathrm{U}}{ \partial\gamma}+\frac{R_{3}}{\alpha_{3}}\|_{2}^{2}+\frac{\alpha_{4}}{2}\| \mathrm{K}-\left(\frac{\partial Y}{\partial y}-\frac{\partial U}{\partial y} \right)+\frac{R_{4}}{\alpha_{4}}\|_{2}^{2}\end{array}  \tag{7}\]  
1. Sub-questions about U  
\[\begin{array}{c}\min_{U}\mathrm{E}(\mathrm{U},\mathrm{S},\mathrm{H},\mathrm{ J},\mathrm{K})=\frac{\alpha_{1}}{2}\|Y-U-S\|_{2}^{2}+\frac{\alpha_{2}}{2}\|H -\frac{\partial\mathrm{U}}{\partial\chi}+\frac{R_{2}}{\alpha_{2}}\|_{2}^{2}+ \frac{\alpha_{3}}{2}\|J\|-\frac{\partial\mathrm{U}}{\partial\gamma}+\frac{R_{3 }}{\alpha_{3}}\|_{2 To make full use of the advantages of fast FFT calculation, we transform the formulas above into frequency domain to solve the equation. The specific calculation process is as follows: Simultaneous Fourier Transform on Both Sides of Equation  
\[\mathcal{F}(\alpha_{1}U+\omega_{2}\frac{\partial^{2}U}{\partial x^{2}}+\omega_{3 }\frac{\partial^{2}U}{\partial y^{2}}+\omega_{4}\frac{\partial^{2}U}{\partial y ^{2}})=\]  
\[\mathcal{F}(\alpha_{1}Y-\alpha_{1}S+\omega_{2}\frac{\partial H}{\partial x}+ \frac{\partial R_{2}}{\partial x}+\omega_{3}\frac{\partial I}{\partial y}+ \frac{\partial R_{3}}{\partial y}-\omega_{4}\frac{\partial K}{\partial y}+ \omega_{4}\frac{\partial^{2}Y}{\partial y^{2}}-\frac{\partial R_{4}}{\partial y })\]  
\[\mathcal{F}(BU)=\mathcal{F}(A)\]  
where \(A\) and \(B\) are  
\[A=\alpha_{1}Y-\alpha_{1}S+\omega_{2}\frac{\partial H}{\partial x}+\frac{ \partial R_{2}}{\partial x}+\omega_{3}\frac{\partial I}{\partial y}+\frac{ \partial R_{3}}{\partial y}-\omega_{4}\frac{\partial K}{\partial y}+\omega_{ 4}\frac{\partial^{2}Y}{\partial y^{2}}-\frac{\partial R_{4}}{\partial y}\]  
\[B=a_{1}+\omega_{2}\frac{\partial^{2}}{\partial x^{2}}+\omega_{3}\frac{\partial ^{2}}{\partial y^{2}}+\omega_{4}\frac{\partial^{2}}{\partial y^{2}}\]  
\[\mathcal{F}(U)=\frac{\mathcal{F}(A)}{\mathcal{F}(B)}\]  
\[U=\mathcal{F}^{-1}\Bigg{(}\frac{\mathcal{F}(A)}{\mathcal{F}(B)}\Bigg{)} \tag{9}\]  
where \(\mathcal{F}(A)\) and \(\mathcal{F}(B)\) are  
\[\mathcal{F}(A)=\alpha_{1}\mathcal{F}(Y-S)+\omega_{2}\mathcal{F} \Big{(}\frac{\partial}{\partial x}\Big{)}\mathcal{F}(H)+\mathcal{F}\Big{(} \frac{\partial}{\partial x}\Big{)}\mathcal{F}(R_{2})+\omega_{3}\mathcal{F} \Big{(}\frac{\partial}{\partial y}\Big{)}\mathcal{F}(J)+\mathcal{F}\Big{(} \frac{\partial}{\partial y}\Big{)}\mathcal{F}(R_{3})\]  
\[-\omega_{4}\mathcal{F}\Big{(}\frac{\partial}{\partial y}\Big{)}\mathcal{F}(K) +\omega_{4}\mathcal{F}\Bigg{(}\frac{\partial^{2}}{\partial y^{2}}\Big{)} \mathcal{F}(Y)-\mathcal{F}\Big{(}\frac{\partial}{\partial y}\Big{)}\mathcal{F }(R_{4})\]  
\[\mathcal{F}(B)=\alpha_{1}+\omega_{2}\mathcal{F}\Big{(}\frac{\partial^{2}}{ \partial x^{2}}\Big{)}+\omega_{3}\mathcal{F}\Big{(}\frac{\partial^{2}}{ \partial y^{2}}\Big{)}+\omega_{4}\mathcal{F}\Big{(}\frac{\partial^{2}}{ \partial y^{2}}\Big{)}\]  
where \(\mathcal{F}\) is the forward Fourier transform, \(\mathcal{F}^{-1}\) is the inverse Fourier transform. 3. Sub-questions about \(S\) \[\mathcal{F}(\alpha_{1}Y-\alpha_{1}S+\omega_{2}\frac{\partial H}{\partial x}+ \frac{\partial R_{2}}{\partial x}+\omega_{3}\frac{\partial I}{\partial y}+ \frac{\partial R_{3}}{\partial y}-\omega_{4}\frac{\partial K}{\partial y}+ \omega_{4}\frac{\partial^{2}Y}{\partial y^{2}}-\frac{\partial R_{4}}{\partial y })\]0 (10)  
The extremum solution process of the function with kernel norm can be summed up in two steps. First, the extremum of the non-kernel norm is solved, and then the dimension of S is reduced by singular value decomposition.