# 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  
### Compensation Scheme  
In this paper, we construct an error compensation model for the image point positioning of early warning area array cameras according to the two-dimensional pointing angle model, including an external parameter error compensation model and an internal parameter error compensation model.  
The external parameter compensation model is as follows:  
\[\begin{bmatrix}\tan(\psi_{x}(x,y))\\ \tan(\psi_{y}(x,y))\\ -1\end{bmatrix}=\lambda R^{CAM}_{INS}(\alpha,\beta,\theta)_{(t)}\begin{bmatrix} Q_{X}\\ Q_{Y}\\ Q_{Z}\end{bmatrix} \tag{1}\]  
in which \(\begin{bmatrix}Q_{X}\\ Q_{Y}\\ Q_{Z}\end{bmatrix}=R^{INS}_{J2000}R^{I2000}_{WGS84}\begin{bmatrix}X-X_{S}\\ Y-Y_{S}\\ Z-Z_{S}\end{bmatrix}\), \(R^{CAM}_{INS}(\alpha,\beta,\theta)_{(t)}=\begin{bmatrix}a1&b1&c1\\ a2&b2&c2\\ a3&b3&c3\end{bmatrix}\), wherein \((x,y)\) is the probe coordinate corresponding to the image point on the image, \(\psi_{x}(x,y)\), \(\psi_{y}(x,y)\) are the sight directions of the \(x\) direction and the \(y\) direction, respectively, in the camera coordinate system, \((X_{S},Y_{S},Z_{S})\) is the position coordinate of the satellite in the earth fixed coordinate system, \((X,Y,Z)\) is the position coordinate of the object point corresponding to the image point in the earth fixed coordinate system, \(R^{I2000}_{WGS84}\) is the conversion matrix from the earth fixed coordinate system to the J2000 coordinate system, \(R^{CAM}_{INS}(\alpha,\beta,\theta)_{(t)}\) refers to the time-varying conversion matrix between the star sensor and the camera under different temperature conditions, and \((\alpha,\beta,\theta)\) refers to the relative installation angle between the star sensor and the camera, which is also the compensation object, and \(\lambda\) is the scale coefficient.  
The calibration results of the installation angle at different temperatures are calculated using the simulation data samples. We establish the variation curve of the installation angle and time, and then apply it to the compensation of the external parameters. Finally, the image point positioning error before and after compensation is directly calculated to verify the accuracy of the external parameter error compensation.  
The internal parameters compensation model is as follows:  
\[\begin{array}{c}(\frac{x}{f},\frac{y}{f},-1)^{T}=(\tan(\psi_{x}(x,y)),\tan( \psi_{x}(x,y)),-1)^{T}\end{array} \tag{2}\]  
\[\begin{array}{c}\tan(\psi_{x}(x,y))=\frac{x}{f}=xa_{0}+xa_{1}l+xa_{2}s+xa_{3 }ls+xa_{4}l^{2}+xa_{5}s^{2}\\ +xa_{6}l^{2}s+xa_{7}ls^{2}+xa_{8}l^{3}+xa_{9}s^{3}\\ \tan(\psi_{y}(x,y))=\frac{y}{y}=ya_{0}+ya_{1}l+ya_{2}s+ya_{3}ls+ya_{4}l^{2}+ya_{ 5}s^{2}\\ +ya_{6}l^{2}s+ya_{7}ls^{2}+ya_{8}l^{3}+ya_{9}s^{3}\end{array} \tag{3}\]  
in which \(xa_{0}\sim xa_{9}\), \(ya_{0}\sim ya_{9}\) are internal compensation parameters.  
For the internal parameter compensation, the two-dimensional pointing angle corresponding to the probe on the area array camera can be calculated by using the calculated internal calibration parameters, and the deviation between the two-dimensional pointing angle of the sample probe and the two-dimensional pointing angle after adding the distortion parameters can be counted, and the pointing angle after calibration can be displayed intuitively. Or the two-dimensional pointing angle of the probe element can be calculated according to the calculated internal calibration parameters, and compared with the image point coordinates calculated according to the original internal parameters to compare the image point positioning errors before and after calibration.