## 4 Image Motion Modeling and Analysis
### Image Motion Analysis
#### 4.1.2 Image Motion Distribution  
The distribution of the image motion's linear velocity (IMLV) on the sensor plane could be analyzed by Equation (13), which is illustrated in Figure 6. The parameters were set as follows: \(f\)= 100 mm, \(H\) = 2000 m, \(v^{W}=\left[\begin{array}{cccc}150&0&0\end{array}\right]^{T}\) m/s, the pixel size is 10 \(\upmu\)m, the camera array scale is 6000 \(\times\) 6000, the cross-flight angle of view is 33.4\({}^{\circ}\), the along-flight angle of view is 33.4\({}^{\circ}\), \(\Phi_{T}=-30^{\circ}\), and \(\Theta_{T}=-20^{\circ}\). The green square \(E_{S}F_{S}G_{S}H_{S}\) represents the sensitive area of the camera. The blue line with an arrow represents the vector of the image motion's linear velocity. The starting point of each vector is the corresponding image point of the vector. The image projection area on the ground is shown in Figure 7, which is indicated by the green quadrangle EFGH. The green line segment in the middle of quadrangle EFGH represents the ground projection of \(Y_{S}\), which is perpendicular to \(v^{W}\). The red line with an arrow represents the primary light vector of the ground point which is the ending point of the vector. The corresponding image points for E, F, G, and H are \(E_{S}\), \(F_{S}\), \(G_{S}\), and \(H_{S}\), respectively. It can be determined the IMLV varied significantly on the sensor plane. The magnitude of the IMLV increased with increasing values of \(Y_{S}\), and the phase angle of the IMLV increased with increasing values of \(X_{S}\). \(E_{S}\) had the minimum magnitude of the IMLV, which was 3.81 mm/s, and had the maximum phase angle of the IMLV, which was 16.98\({}^{\circ}\). had the maximum magnitude of the IMLV, which was 8.18 mm/s, and had the minimum phase angle of the IMLV, which was 4.57\({}^{\circ}\).  
The variation of the IMLV on the sensor plane will degrade the image motion's compensation performance, so it is necessary to study the variation laws of the maximum deviation of the IMLV. The variations of the maximum deviation of the IMLV with the target roll and pitch angles are shown in Figure 8. The parameters were the same as those in Figure 6, except for the target roll angle and target pitch angle. \(\Phi_{T}\) increased from \(-30^{\circ}\) to \(-5^{\circ}\), while \(\Theta_{T}=-5^{\circ}\). \(\Theta_{T}\) increased from \(-30^{\circ}\) to \(-5^{\circ}\), while \(\Phi_{T}=-5^{\circ}\). The magnitude of the maximum deviation of the IMLV would increase with \(|\Phi_{T}|\) or \(|\Theta_{T}|\). The absolute value of the phase angle of the maximum deviation of the IMLV would increase with \(|\Phi_{T}|\) but decrease when \(|\Theta_{T}|\) increased. The attitude of the camera provided by the POS had measuring error; the azimuth, pitch, and roll measuring errors were 0.08\({}^{\circ}\), 0.05\({}^{\circ}\), and 0.05\({}^{\circ}\), and 0.05\({}^{\circ}\).  
Figure 6: Image motion’s linear velocity distribution on the sensor plane.  
Figure 7: Ground imaging area and primary light distribution.  
respectively, in the system. The sensitivity of the maximum deviation of the IMLV for the attitude error is illustrated in Figure 9. The magnitude error of the maximum deviation of the IMLV would decrease with \(|\Phi_{T}|\) or \(|\Theta_{T}|\), and the maximum magnitude error was 0.011 mm/s. If the exposure duration was 10 ms, the maximum image motion arising from the attitude error was 0.11 \(\upmu\)m. That is one hundredth the size of one pixel, which has little impact on the imaging quality.