## 2 Necessity Analysis of Image-Motion Compensation  
In order to fulfill the requirements of wide-width measurement imaging, the aerial camera employs the scanning imaging mode. This mode ensures that the exposure time and scanning speed of the camera correspond to the necessary requirements for forward overlap rate and lateral overlap rate, which are essential for scale mapping. In order to ascertain the necessity of image-motion compensation, some flight parameters of an aerial camera were selected as examples, as shown in Table 1, for analysis and calculation of the image motion of the aerial camera in the scanning imaging mode.  
During the aircraft's forward flight, the aerial camera scans and captures images. When the camera is oriented vertically for photography, combined with the parameters in Table 1, the forward image motion and scanning image motion of the aerial camera at the camera exposure time can be calculated theoretically [9; 10; 11].  
\[S_{x}=f\dot{\alpha}\tau, \tag{1}\]  
\[S_{y}=f\frac{V_{s}}{H}\tau, \tag{2}\]  
where \(S_{x}\) is the scanning image motion of the aerial camera, \(S_{y}\) is the forward image motion of the aerial camera, \(f\) is the focal length, \(\dot{\alpha}\) is the scanning angular velocity, \(\tau\) is the exposure time, \(V_{s}\) is the flight speed, and \(H\) is the flight altitude. \(\dot{\alpha}\) can be calculated according to the lateral overlap rate, the scanning FOV, and the scanning imaging time.  
According to Equations (1) and (2) and the parameters in Table 1, the image motions in both directions at different speed-to-height ratios could be calculated as shown in Table 2.  
As illustrated in Table 2, the forward and scanning image motions are positively correlated with the speed-to-height ratio. The maximum forward image motion is 0.73 pixels, while the maximum scanning image motion is 62.34 pixels. These values are greater when the exposure time is longer or when the speed-to-height ratio is greater. In general, when the image motion of the system exceeds 0.5 pixels, the image quality is severely degraded and needs to be compensated for.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & \multicolumn{2}{c}{**Values**} \\ \hline Aircraft platform parameters & Flight altitude & 7.50 km \\  & Flight speed & 180 km/h-280 km/h \\ \hline  & Spectrum & 435 nm\(\sim\)900 nm \\  & Focal length (\(f\)) & 450.00 mm \\  & Pixel size (\(ps\)) & 6.40 \(\upmu\)m \\  & FOV & \(4.17^{\circ}\times 3.13^{\circ}\) \\ Camera parameter & F number & 4.20 \\  & MTF & \(\geq\)0.20 (80 lp/mm @ all FOV) \\  & Exposure time & 1.00 ms \\  & Scanning range & \(-60^{\circ}\)\(\sim\)\(+60^{\circ}\) \\  & Forward overlap rate & 78\% \\  & Lateral overlap rate & 30\% \\ \hline \hline \end{tabular}
\end{table}
Table 1: The flight parameters of an aerial camera.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Image Motions**Speed-to-Height Ratios** & **0.0067** & **0.0074** & **0.0081** & **0.0089** & **0.0096** & **0.0104** \\ \hline Forward image motions & \(\upmu\)m & 3.02 & 3.34 & 3.64 & 4 & 4.32 & 4.68 \\  & pixel & 0.47 & 0.52 & 0.57 & 0.63 & 0.68 & 0.73 \\ \hline Scanning image motions & \(\upmu\)m & 256.8 & 285.2 & 313.4 & 341.6 & 370.8 & 399 \\  & pixel & 40.13 & 44.56 & 48.97 & 53.38 & 57.94 & 62.34 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Image motions at different speed-to-height ratios.