# Single-Lens Imaging Spectral Restoration Method Based on Gradient Prior Information Optimization
## 4 Simulation  
The system parameters were set as follows: lens diameter 30 mm, focal length 60 mm @ 587.6 nm, and CCD pixel size 12.4 um. In this study, the PaviaU dataset [35] was used for simulation, which consists of 103 spectral bands covering a range from 0.430 um to 0.860 um. Spectral data from bands 23 to 31 were selected for simulation, and an image size of 250 \(\times\) 250 was cropped. During the simulation, the point spread function (PSF) was automatically generated using a Gaussian operator, and the blur parameter was set to \(3\times 3\). The 23 rd band image from the PaviaU dataset was selected as the original image and was degraded to calculate the gradient distribution, as shown in Figure 2(a). The degraded image's gradient was then optimized using the optimization method proposed in this study, as shown in Figure 2(b). It can be clearly observed that the optimized gradient distribution is much closer to the original image's gradient distribution, enabling better image restoration. Figure 3(a)-i shows the simulation results for different spectral bands. From left to right, it respectively represents the original image, the degraded image, the MTV-restored image, and the image restored using the algorithm proposed in this study.  
Figure 3: _Cont._  
Figure 4: _Cont_.  
Figure 3: Gradient distribution maps: (**a**) spatial gradient distribution maps of the original image and the degraded image; (**b**) spatial gradient distribution maps of the original image, degraded image, and the optimized image after restoration.  
Figure 4: _Cont._  
A quality evaluation of the three restoration methods is shown in Table 1.  
Based on a comparison of the quality evaluation metrics mentioned above, it can be observed that the restoration quality of the algorithm proposed in this study is superior to that of the MTV algorithm. Taking the original image of band 23 as an example, points A and B were selected as feature points, as shown in Figure 5.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Band**} & **Restoration Method** & \multirow{2}{*}{**RMSE**} & **PSNR** & **SSIM** \\ \hline \multirow{2}{*}{23} & MTV & 0.1837 & 23.7500 & 0.7796 \\  & Ours & 0.0332 & 38.6151 & 0.9394 \\ \hline \multirow{2}{*}{24} & MTV & 0.1852 & 23.6229 & 0.7786 \\  & Ours & 0.0334 & 38.5035 & 0.9407 \\ \hline \multirow{2}{*}{25} & MTV & 0.1868 & 23.4442 & 0.7766 \\  & Ours & 0.0334 & 38.4020 & 0.9394 \\ \hline \multirow{2}{*}{26} & MTV & 0.1885 & 23.2302 & 0.7766 \\  & Ours & 0.0335 & 38.2492 & 0.9400 \\ \hline \multirow{2}{*}{27} & MTV & 0.1894 & 23.1542 & 0.7771 \\  & Ours & 0.0336 & 38.1643 & 0.9397 \\ \hline \multirow{2}{*}{28} & MTV & 0.1904 & 23.1282 & 0.7768 \\  & Ours & 0.0338 & 38.1394 & 0.9411 \\ \hline \multirow{2}{*}{29} & MTV & 0.1916 & 23.0859 & 0.7782 \\  & Ours & 0.0340 & 38.1012 & 0.9412 \\ \hline \multirow{2}{*}{30} & MTV & 0.1929 & 22.9958 & 0.7769 \\  & Ours & 0.0341 & 38.0360 & 0.9421 \\ \hline \multirow{2}{*}{31} & MTV & 0.1943 & 22.8818 & 0.7803 \\  & Ours & 0.0343 & 37.9550 & 0.9425 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparison of the quality of restoration methods.  
Figure 4: Comparison of image restoration results.  
Figure 5: Spectral feature points of simulation image.  
A comparison between the spectral quality of restoration using the MTV algorithm and the algorithm proposed in this study is shown in Figure 6. The horizontal axis represents the spectral bands, and the vertical axis represents the normalized intensity values. Figure 5(a) shows the spectral curve for point A, and b shows the spectral curve for point B.  
A quality evaluation of the restored spectral data is shown in Table 2.  
Based on a comparison of the results, it can be concluded that for both spectral similarity and spectral root-mean-square error, the restoration quality of the algorithm proposed in this study is superior to that of the MTV algorithm for points A and B.