# Research into Image Point Positioning Compensation of a High-Orbit Thermal Infrared Staring Camera
## 2 Virtual Control Points Calibration Scheme Based on Image Space Uniformity
### Calibration Principle  
To solve the problem of the motion parameter estimation error of high-value point targets caused by the lack of ground control points in the process of real-time geometric processing [18], this paper proposes a calibration method of virtual control points based on image space uniformity, with the image points of the target projected onto the Earth's surface through the photography center to obtain the corresponding projection point sequence of the Earth's surface, so as to obtain the virtual control points, as shown in Figure 1.  
At the observation time, the image points of the virtual control point sequence are completely coincident with the image points of the target point at that time, so a series of virtual control points are generated by using the intersection points obtained. Given the coordinates \((x,y)\) of the image plane of the target and the internal and external orientation elements at the observation time, the spatial coordinates \((X,Y,Z)\) of the corresponding virtual control points in the earth fixed coordinate system can be obtained from the geometric imaging model:  
\[\frac{x}{-f}=\frac{a_{1}X+b_{1}Y+c_{1}Z}{a_{2}X+b_{1}Y+c_{1}Z} \tag{4}\] \[\frac{y}{-f}=\frac{a_{2}X+b_{2}Y+c_{2}Z}{a_{3}X+b_{3}Y+c_{3}Z}\]  
\[\text{in which }\left\{ \begin{array}{l}a_{1}=\cos\varphi\cos\kappa-\sin\varphi \sin\omega\sin\kappa\\ a_{2}=-\cos\varphi\sin\kappa-\sin\varphi\sin\omega\cos\kappa\\ a_{3}=-\sin\varphi\cos\omega\\ b_{1}=\cos\omega\sin\kappa\\ b_{2}=\cos\omega\cos\kappa\\ b_{3}=-\sin\omega\\ c_{1}=\sin\varphi\cos\kappa+\cos\varphi\sin\omega\sin\kappa\\ c_{2}=-\sin\varphi\sin\kappa+\cos\varphi\sin\omega\cos\kappa\\ c_{3}=\cos\varphi\cos\omega\end{array} \right.\]  
Figure 1: Schematic diagram of ground virtual control points. For the generated virtual control image, \(N\) generated control points are selected as the calibration points, with the object space coordinates corresponding to the calibration points set as \((X_{i},Y_{i},Z_{i})\) and the corresponding image point coordinates set as \((x_{i},y_{i}),i=1,2,3,\cdots,N\), and then the calibration model is decomposed into the pointing angle error along the direction of \(x\) and the pointing angle error along the direction of y according to the two-dimensional pointing angle model:  
\[ \begin{cases}J(M_{W},W_{N})=\frac{a_{1}Q_{X}+b_{1}Q_{Y}+c_{1}Q_{Z}}{a_{3}Q_{X} +b_{3}Q_{Y}+c_{3}Q_{Z}}-\tan(\psi_{x}(x,y))\\ K(M_{W},W_{N})=\frac{a_{2}Q_{X}+b_{2}Q_{Y}+c_{2}Q_{Z}}{a_{3}Q_{X}+b_{3}Q_{Y}+c_{ 3}Q_{Z}}-\tan(\psi_{y}(x,y))\end{cases}  \tag{5}\]  
The initial values of the extrinsic calibration parameters are determined according to the initial installation matrix of the camera, and the initial values of the intrinsic calibration parameters are determined according to the parameters of the theoretical two-dimensional pointing angle model when the camera has no internal distortion. The current value of the intrinsic calibration parameter \(W_{N}\) is regarded as the true value, and the extrinsic calibration parameter \(M_{W}\) is regarded as the unknown parameter to be determined. For each virtual calibration control point, the above equation is linearized to establish the corresponding error equation:  
\[V_{i}=A_{i}X-L_{i} \tag{6}\]  
\[A_{i}= \begin{bmatrix}\frac{\partial I_{i}}{\partial M_{W}}\\ \frac{\partial K_{i}}{\partial M_{W}}\end{bmatrix} = \begin{bmatrix}\frac{ \partial I_{i}}{\partial x}&\frac{\partial I_{i}}{\partial x}&\frac{\partial I _{i}}{\partial x}\\ \frac{\partial K_{i}}{\partial M_{W}}\end{bmatrix} ,X=dX= \begin{bmatrix}d\alpha \\ d\beta\\ d\end{bmatrix} ,L_{i}= \begin{bmatrix}J(M_{Wi},W_{Ni})\\ K(M_{Wi},W_{Ni})\end{bmatrix} ;\]  
When solving the intrinsic calibration parameters, the extrinsic calibration parameter \(M_{W}\) obtained above is regarded as the true value, and the intrinsic calibration parameter \(W_{N}\) as the unknown parameter to be determined.