## 3 Single-Image Destriping Method
### Multi-Level Unidirectional Total Variation Stripe Noise Removal Model  
A remote sensing image with stripes can be expressed by a mathematical formula as follows:  
\[I_{s}(\mathrm{r},\mathrm{c})=u(r,c)+s(r,c). \tag{6}\]  
where \(r=1\), \(2\),..., \(\mathrm{R}\), \(c=1\), \(2\),..., \(\mathrm{C}\). \(\mathrm{R}\) and \(\mathrm{C}\) stand for the number of the rows and columns respectively. Here \(I_{s}\) represents the original image, and \(u\) represents the corrected image, and \(s\) represents the stripe noise in the image. Obviously, the stripe in the remote sensing image can be regarded as an additive structural noise, and the gradient change is mainly concentrated in the x-axis (let the direction perpendicular to the stripe be the x-axis), while the change in the y-axis (let the direction along the stripe be the y-axis) is much smaller than the x-axis. Figure 2 shows the gradient of the x-axis and y-axis. In this paper, we use this stripe feature to obtain the optimal solution of the model by minimizing the energy functional. The energy functional of the variational model is proposed as follows:  
\[E(u)= \begin{array}{c}\min\\ u\end{array}\begin{array}{l}\frac{1}{2}\left\|u-f\right\|_{2}^{2}+\lambda_{1} \left\|D_{y}(u-f)\right\|_{1}+\lambda_{2}\left\|D_{x}u\right\|_{1}\\ f=(I_{s}-B)./NG\end{array} . \tag{7}\]  
where the operators \(D_{x}\), \(D_{y}\) denote the first-order forward finite difference operators along the x-axis (horizontal direction) and y-axis (vertical direction). The first term of the energy functional variation model is the data fidelity term, which is to make the corrected image as close as possible to the image after adapted moment matching, and the second term is the gradient fidelity term to ensure the gradient information along the track direction. The third term is the regularization term, which is designed to maximize stripe removal. \(\lambda_{1}\), \(\lambda_{2}\) is the regularization parameter, which is used to adjust the proportion between the fidelity and the regularization term. Since the functional model contains a non-differentiable and inseparable L1 norm, this paper uses the split Bregman iterative algorithm to solve the problem. First, for the problem (7), two auxiliary variables \(\mathrm{d}_{ly}=D_{y}(u-f),d_{lx}=D_{x}u\) are introduced, and the unconstrained minimization problem is turned into the constrained minimization problem, that is, we can rewrite Equation (7) as Equation (8). \[\min_{u,d_{lx},d_{ly}}\ \ \frac{1}{2}\|u-f\|_{2}^{2}+\lambda_{1}\|d_{ly}\|_{1}+ \lambda_{2}\|d_{lx}\|_{1}+\frac{\alpha}{2}\|d_{ly}-D_{y}(u-f)-b_{ly}\|_{2}^{2}+ \frac{\beta}{2}\|d_{lx}-D_{x}u-b_{lx}\|_{2}^{2}. \tag{8}\]  
where \(\alpha\), \(\beta\) is a positive penalty parameter. The Bregman variables introduced by \(b_{lx}\) and \(b_{ly}\) are used to accelerate the iterative process. In this way, the minimization problem can be decomposed into three sub-problems of \(u\), \(d_{lx}\) and \(d_{ly}\). Sub-problems about \(u\):  
\[\min_{u}\ \ \frac{1}{2}\|u-f\|_{2}^{2}+\frac{\alpha}{2}\|d_{ly}-D_{y}(u-f)-b_{ly} \|_{2}^{2}+\frac{\beta}{2}\|d_{lx}-D_{x}u-b_{lx}\|_{2}^{2}. \tag{9}\]  
This equation is a least squares problem, which is equivalent to the following equation:  
\[(\alpha D_{y}^{T}D_{y}+\beta D_{x}^{T}D_{x}+1)u^{k+1}=f+\alpha D_{y}^{T}(d_{ly }^{k}+D_{y}f-b_{ly}^{k})+\beta D_{x}^{T}(d_{lx}^{k}-b_{lx}^{k}). \tag{10}\]  
The above problem can be solved by a fast Fourier transform. Sub-question about \(d_{lx}\):  
\[\min_{d_{lx}}\ \ \lambda_{2}\|d_{lx}\|_{1}+\frac{\beta}{2}\|d_{lx}-D_{x}u-b_ {lx}\|_{2}^{2}. \tag{11}\]  
Equation (11) can be solved by the soft-shrinkage operator, and the following equation can be obtained:  
\[d_{lx}^{k+1}=shrink(D_{x}u^{k}+b_{lx}^{k},\lambda_{2}/\beta). \tag{12}\]  
where  
\[shrink(x,r)=\frac{x}{|x|}\times\max(|x|-r,0). \tag{13}\]  
Figure 2: Partial derivative of band 16. (a) Gradient in the y direction; (b) Gradient in the x direction.