## 4 Image Motion Modeling and Analysis
### Image Motion Analysis  
#### 4.1.1 Image Motion Model  
The mapping relation of the airborne area camera based on geometrical optics is demonstrated in Figure 5. The origin \(O_{W}\) of the W frame is the projection of \(O_{L}\) on local, level ground. \(O_{L}\) is the center of the exit pupil of the camera lens. The supposition is made that the aircraft stands still and the scene on the ground moves at the velocity of \(v^{W}\), while the aircraft flies at the velocity of \(-v^{W}\). \(A_{S}\) is a random point on the sensor plane, and \(A\) is the corresponding point on the ground. \(A\) moves at \(v^{W}\) to \(B\) on the ground during a short time interval \(\Delta t\). \(B_{S}\) is the corresponding point of \(B\) on the sensor. Therefore, \(A_{S}^{\rightarrow}B_{S}\) is the image motion vector of \(A_{S}\) during \(\Delta t\).  
Figure 5: Schematic diagram of airborne area camera photography.  
\(O_{L}^{\rightarrow}O_{S}\) is first rotated about \(y_{S}\) by the angle \(\alpha\) and then rotated around \(x_{S}\) by the angle \(\beta\). Then, \(O_{L}^{\rightarrow}O_{S}\) is transformed to the vector which has the same direction as the primary light vector \(O_{L}^{\rightarrow}A_{S}\). The angle between \(O_{L}^{\rightarrow}A_{S}\) and \(v^{W}\), represented by \(\theta_{A}\), can be calculated as follows:  
\[\theta_{A}=\cos^{-1}\left(\left(r_{A_{S}}^{S}\cdot v^{S}\right)/\left(\left|r_ {A_{S}}^{S}\right|\cdot\left|v^{S}\right|\right)\right) \tag{9}\]  
where  
\[r_{A_{S}}^{S}=R_{x}^{roll}(\beta)^{T}\cdot R_{y}^{pitch}(\alpha)^{T}\cdot\left[ \begin{array}{cccc}0&0&1\end{array}\right]^{T}\]  
\[v^{S}=R_{x}^{roll}(\Phi_{T})\cdot R_{y}^{pitch}(\Theta_{T})\cdot R_{W}^{N} \cdot v^{W}\]  
Therefore, the image motion angular rate \(\omega_{A}\) of A can be calculated as follows:  
\[\omega_{A}=\left|v^{W}\right|\cdot\sin(\theta_{A})/\left|r_{A}^{W}-r_{O_{L}}^ {W}\right| \tag{10}\]  
where \(r_{O_{L}}^{W}=\left[\begin{array}{cccc}o_{1}&o_{2}&o_{3}\end{array}\right]^ {T}\) represents the coordinate of \(O_{L}\) in the W frame, \(r_{A}^{W}=\left[\begin{array}{cccc}o_{1}-o_{3}\frac{m_{1}}{m_{3}}&o_{2}-o_{3 }\frac{m_{2}}{m_{3}}&0\end{array}\right]^{T}\) represents the coordinate of A in the W frame, and \(r_{A_{S}}^{W}=R_{S}^{W}\cdot r_{A_{S}}^{S}=\left[\begin{array}{cccc}m_{1}&m_ {2}&m_{3}\end{array}\right]^{T}\).  
According to geometrical optics law, the direction of the intersecting line between the \(O_{L}AB\) plane and the sensor plane represented by \(r_{A_{J}}^{S}\) is the direction of the image motion's linear velocity \(v_{A_{S}}^{S}\) of \(A_{S}\). According to the vector cross product law, \(r_{A_{J}}^{S}\) can be described as in the formula below:  
\[r_{A_{J}}^{S}=v^{S}\times r_{A_{S}}^{S}\times\left[\begin{array}{cccc}0&0&1 \end{array}\right]^{T} \tag{11}\]  
Then,  
\[\theta_{A_{S}}=\cos^{-1}\left(\frac{\left(r_{A_{S}}^{S}\cdot r_{A_{J}}^{S} \right)}{\left(\left|r_{A_{S}}^{S}\right|\cdot\left|r_{A_{J}}^{S}\right|\right)}\right) \tag{12}\]  
The magnitude of the image motion's angular rate of \(A_{S}\) equals that of \(A\). Therefore, \(v_{A_{S}}^{S}\) can be expressed as  
\[v_{A_{S}}^{S}=\frac{r_{A_{J}}^{S}}{\left|r_{A_{J}}^{S}\right|}\cdot\left(\frac {\omega_{A}\cdot\left|O_{L}^{\rightarrow}A_{S}\right|}{\sin(\theta_{A_{S}})} \right)=\frac{r_{A_{J}}^{S}\cdot f\cdot\omega_{A}}{\left|r_{A_{J}}^{S}\right| \cdot\cos(\alpha)\cdot\cos(\beta)\cdot\sin(\theta_{A_{S}})} \tag{13}\]