## 4 Hyperspectral Image Destriping Method
### Multi-Level Unidirectional Total Variation Stripe Noise Removal Model  
For hyperspectral images, the model (6) can be extended to the following model:  
\[I_{s}(r,c,b)=u(r,c,b)+s(r,c,b). \tag{22}\]  
where \(b\) = 1, 2,..., B, and B is the number of image bands. Considering the smooth constraints of spectral dimensions, we can extend the UTV model in (7) to define the spectral-spatial UTV model as follows:  
\[E(u)= \begin{array}{c}\min\\ u\end{array}\begin{array}{c}\frac{1}{2}\|u-f\|_{2}^{2}+\lambda_{1}\|D_{y}(u-f )\|_{1}+\lambda_{2}\|D_{x}u\|_{1}+\lambda_{3}\|D_{2}u\|_{1}\\ f=(I_{s}-B)./NG)\end{array} . \tag{23}\]where the \(D_{z}\) represents the first-order forward finite-difference operators along z-axis (spectral direction), and other parameters are the same as the model (7). The difference between (7) and (23) is the extra spectral smoothness along the z-axis. For the formula (23), three auxiliary variables \(\mathrm{d}_{ly}=D_{y}(u-f)\), \(d_{lx}=D_{x}u\), \(d_{iz}=D_{z}u\) are introduced, and then the problem (23) can be transformed into (24). \[ \begin{array}{c}\min\\ u,d_{lx},d_{ly},d_{lx}\\ \frac{\mathrm{d}}{2}\left\|u-f\right\|_{2}^{2}+\lambda_{1}\left\|d_{ly}\right\| _{1}+\lambda_{2}\left\|d_{lx}\right\|_{1}+\lambda_{3}\left\|d_{iz}\right\|_{1 }+\\ \frac{\mathrm{d}}{2}\left\|d_{ly}-D_{y}f-b_{ly}\right\|_{2}^{2}+\frac{\beta}{2 }\left\|d_{lx}-D_{x}u-b_{lx}\right\|_{2}^{2}+\frac{\gamma}{2}\left\|d_{iz}-D _{z}u-b_{tz}\right\|_{2}^{2}\end{array} . \tag{24}\]  
where the parameter definition is the same as Equation (8). The minimization problem can be decomposed into four sub-problems of \(u\), \(d_{lx}\), \(d_{ly}\) and \(d_{iz}\). Sub-problems about \(u\):  
\[ \begin{array}{c}\min\\ u\end{array} \frac{1}{2}\left\|u-f\right\|_{2}^{2}+\frac{\alpha}{2}\left\|d_{ly }-D_{y}(u-f)-b_{ly}\right\|_{2}^{2}+\frac{\beta}{2}\left\|d_{lx}-D_{x}u-b_{lx }\right\|_{2}^{2}+\frac{\gamma}{2}\left\|d_{lx}-D_{z}u-b_{tz}\right\|_{2}^{2}.\end{array} \tag{25}\]  
This equation is a least squares problem, which is equivalent to the following equation:  
\[(\alpha D_{y}^{T}D_{y}+\beta D_{z}^{T}D_{x}+\gamma D_{z}D_{z}^{T}+1)u^{k+1}=f+ \alpha D_{y}^{T}(d_{ly}^{k}+D_{y}f-b_{ly}^{k})+\beta D_{x}^{T}(d_{lx}^{k}-b_ {lx}^{k})+\lambda D_{z}^{T}(d_{lx}^{k}-b_{lx}^{k}). \tag{26}\]  
The above problem can be solved by n-D FFT. Sub-question about \(d_{lx}\) and \(d_{ly}\):  
The problem is the same as in Section 3.2. Sub-question about \(d_{lx}\):  
\[ \begin{array}{c}\min\\ d_{ly}\end{array} \lambda_{3}\left\|d_{lx}\right\|_{1}+\frac{\gamma}{2}\left\| d_{lx}-D_{z}u-b_{tz}\right\|_{2}^{2}. \tag{27}\]  
Equation (27) can be solved by the soft-shrinkage operator. Finally, update the Bregman variable \(b_{lx}\), \(b_{ly}\), \(b_{tz}\) as follows. \[ \begin{array}{c}b_{ly}^{k+1}=b_{ly}^{k}+D_{y}(u^{k+1}-f)-d_{ly}^{k+1}\\ b_{lx}^{k+1}=b_{lx}^{k}+D_{x}u^{k+1}-d_{lx}^{k+1}\\ b_{tz}^{k+1}=b_{tz}^{k}+D_{z}u^{k+1}-d_{tz}^{k+1}\end{array} . \tag{28}\]  
Finally, we use the multi-level iterative method proposed(LRSID) [21] and Statistical linear destriping model (SLD) [23],. Finally we compare our method with three hyperspectral destriping methods: The anisotropic spectral-spatial TV model (ASSTV) [28], the image decomposition based band-by-band low-rank regularization and spatial-spectral TV model (LRMID) [21] and lastly the low-rank tensor decomposition model (LRTD) [29]. In the following experiments, the parameters in compared methods are manually tuned according to the rules recommended by their papers to get the possibly good performance. For the parameters of our method, we would like to show the detailed discussion in Section 5.4.1. By using the above method, stripe noise removal experiments were performed on an OHS image, and their effects were evaluated by subjective and objective evaluation criteria.