## 3 Results
### Remote Sensing Image Simulation  
In this section, the mechanism of image distortion of the curvature map and image degradation, caused by satellite random vibration, have been explained. Firstly, the high-resolution ground image is converted by the ground curvature mapping Formula (15), and the distorted image on the camera focal plane is obtained; secondly, the distorted image is filtered in the frequency domain by the filter template (18) and, then, by the Fourier inverse transform (19). Finally, the degraded image is obtained.  
In the coaxial plane optical system, the image degradation result, in the direction along the track and the vertical track, is the same. Then, the system filter template (18) can be expressed as:  
\[\mathrm{MTF}(v)=\exp\Bigl{(}-2\pi^{2}d_{\mathrm{is}}{}^{2}v_{x}{}^{2}\Bigr{)} \cdot\exp\Bigl{(}-2\pi^{2}d_{\mathrm{is}}{}^{2}v_{y}{}^{2}\Bigr{)} \tag{23}\]  
Figure 9: MTF curves of dynamic frame-by-frame imaging.  
where the error of velocity, caused by random vibration, is set to 0.00076\({}^{\circ}\)/s. Then, the frequency domain-filtered template images, at attitude angles of \(0^{\circ}\), \(30^{\circ}/-30^{\circ}\), and \(50^{\circ}/-50^{\circ}\), are simulated, as shown in Figure 10. It can be seen that the image filtering has the same effect when the attitude angle is symmetrical about the position of the satellite nadir.  
In order to intuitively describe the actual image degradation phenomenon, the high-resolution ground target image, with a size of 2200 pixel \(\times\) 400 pixel, is used as the original image. The sub-frame images at attitude angles of \(-50^{\circ}\), \(-30^{\circ}\), \(0^{\circ}\), \(30^{\circ}\), and \(50^{\circ}\) are assumed to be the stitching units for wide-format imaging. After being filtered by the template shown in Figure 10, the images interfered by random vibration at different attitude angles can be obtained, as shown in Figure 11. The first line is the original image, the second line is the distorted output image on the camera focal plane, and the third line is the degraded image caused by the satellite vibration.  
From the first line of Figure 11, it can be seen that the imaging range of the satellite nadir is the smallest. As the angle between the optical axis of the payload and the position of the satellite nadir increases, the imaging range of the area-array camera on the ground becomes wider. From the second line of Figure 11, it can be seen that the degree of image distortion far from the satellite nadir is greater, while the size and scale of the target image get bigger, and the disturbance factor has more influence on the image clarity. From the third line of Figure 11, it can be seen that satellite vibration seriously affects the quality of remote sensing images. Image sharpness evaluation indicators, SSIM (structural similarity) and MSE (mean square error), are usually used to evaluate the degree of image degradation. The higher the value of MSE is, the greater the degree of distortion is. The value range of SSIM is [0, 1]. The larger the value of SSIM is, the smaller the image distortion is. The data of SSIM and MSE, at different attitude angles, are listed in Table 2. When the attitude angle is \(0^{\circ}\), the SSIM value is 0.3136, and the MSE value is 76.7977. With the attitude angle increasing to \(30^{\circ}\) and \(50^{\circ}\), SSIM decreases, and the MSE increases, which indicates that, when the error of velocity is the same, the image sharpness decreases with the increasing of the deviation angle between the payload optical axis and the satellite nadir position.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Attitude Angle** & **SSIM** & **MSE** \\ \hline \(-50^{\circ}\) & 0.2125 & 89.3903 \\ \(-30^{\circ}\) & 0.2958 & 79.9711 \\ \(0^{\circ}\) & 0.3136 & 76.7977 \\ \(30^{\circ}\) & 0.2509 & 84.1601 \\ \(50^{\circ}\) & 0.2041 & 94.5861 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The results of image sharpness evaluation.