# 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
For each virtual calibration control point, the above equation is linearized to establish the corresponding error equation:  
\[V_{i}=B_{i}Y-L_{i} \tag{7}\]  
in which \(B_{i}= \begin{bmatrix}\frac{\partial I_{i}}{\partial W_{Ni}}\\ \frac{\partial K_{i}}{\partial W_{Ni}}\end{bmatrix} =[B_{1i},B_{2i}],Y=[Y1,Y2]=dY\);  
\[B_{1i}=\begin{bmatrix}\frac{\partial I_{i}}{\partial xu_{0}}&\frac{\partial I _{i}}{\partial xu_{1}}&\frac{\partial I_{i}}{\partial xu_{2}}&\frac{\partial I _{i}}{\partial xu_{3}}&\frac{\partial I_{i}}{\partial xu_{4}}&\frac{\partial I _{i}}{\partial xu_{6}}&\frac{\partial I_{i}}{\partial xu_{7}}&\frac{\partial I _{i}}{\partial xu_{8}}&\frac{\partial I_{i}}{\partial xu_{9 the internal parameter compensation model in real time using the external parameters after compensation, and then compensates the internal parameters using the updated internal parameter compensation model. On the other hand, the scheme of \"the internal first and then the external\" compensates the camera's internal parameters first, updates the external parameter compensation model in real time using the compensated internal parameters, and then compensates the external parameters using the updated external parameter compensation model. In order to better analyze the accuracy of parameter compensation, in the above-mentioned external-internal compensation scheme, after the first step of external parameter compensation, the statistical accuracy is only the accuracy of the external parameter compensation. Similarly, in the internal and external compensation scheme, the accuracy calculation is carried out after the internal parameter compensation, which is the accuracy of the internal parameter compensation. At this time, there are four kinds of compensation schemes (as shown in Figure 2) actually established: first outside and then inside, first inside and then outside, only inside compensation, and only outside compensation. The specific compensation process is shown in the following steps:  
1. The spatial motion model of the target and the strict imaging geometric model of the area array camera are established. The three-dimensional motion trajectory of the generated target space is observed and the imaging simulation results of the motion trajectory on the two-dimensional image are obtained. 2. In the two-dimensional imaging process, three-dimensional errors are added to the attitude parameters and orbital position parameters, and distortion errors are added to the camera parameters. The calculated position is used as the positioning result before error compensation. 3. The grid points with uniform and regular distribution are selected on the camera array. The pixel coordinates (0, 0) in the upper left corner are taken as the starting grid points, and the pixel coordinates (10,000, 10,000) in the lower right corner are taken as the ending grid points. The interval between the grid points in the X direction and the Y direction is 100 pixels. Figure 2: Design of internal and external parameter compensation process. (**a**) Internal compensation before external compensation; (**b**) external compensation before internal compensation. According to the pixel size design parameters, the pixel coordinates of these grid points are transformed into the coordinates in the camera coordinate system, and the two-dimensional pointing angle in the camera coordinate system is calculated as the initial coordinate value. 5. For the image point positioning error caused by the internal orientation elements, the actual two-dimensional pointing angle coordinates of each grid point in the camera coordinate system are calculated, and the difference between the actual two-dimensional pointing angle and the initial pointing angle is used to reflect the distortion inside the camera, and the internal parameter compensation model is constructed to calculate the internal calibration parameters. 6.