## 3 Model of Image Motion  
### Coordinate Systems  
In the process of transforming the coordinate systems from object remote sensors to optical remote sensors in a space camera, the right-hand Cartesian coordinate systems was adopted. Using homogeneous coordinate transformation matrices is a convenient method for expressing coordinate transformations. The position relationship between each coordinate system is shown in Figure 2. The coordinate systems involved in mapping from the object to the image plane are defined in the following section. 1. Geocentric inertial coordinate system \(O_{I}-x_{I}y_{I}z_{I}\): The origin \(O_{I}\) is geocentric, \(x_{I}\) points to the intersection of the orbital plane and the equatorial plane, \(z_{I}\) points to the  
Figure 1: Structure diagram of a space camera with a two-axis pointing platform. Figure 2: Imaging coordinate system of a space camera. north pole, \(y_{1}\) is perpendicular to \(x_{I}\) and \(z_{I}\), and the coordinate system maintains the inertial space. 2. Spacecraft orbit coordinate system \(O_{b}-x_{b}y_{b}z_{b}\): The origin \(O_{b}\) is in orbit, with the \(x_{b}\) pointing forward along the orbit and the \(z_{b}\) pointing toward space. It also passes through the origin of the geocentric inertial coordinate system. The \(x_{b}\) and \(z_{b}\) axes are located within the track plane, while the \(y_{b}\) axis is perpendicular to the track plane. The spacecraft orbit coordinates are located in the geocentric inertial coordinate system and move in orbit at an angular velocity \(\Omega\). 3. Spacecraft coordinate system \(O_{s}-x_{s}y_{s}z_{s}\): The origin \(O_{s}\) is the same as the origin \(O_{b}\) of the spacecraft orbit coordinate system. When the spacecraft is stationary without motion, the spacecraft coordinate system \(O_{s}-x_{s}y_{s}z_{s}\) coincides with the spacecraft orbit coordinate system \(O_{b}-x_{b}y_{b}z_{b}\). When the spacecraft has an attitude motion, the spacecraft coordinate system can be obtained by rotating the spacecraft orbit coordinate system. The roll angle is \(\varphi\), the pitch angle is \(\theta\), and the yaw angle is \(\psi\). 4. Camera installation coordinate system \(O_{c}-x_{c}y_{c}z_{c}\): The origin of the camera installation coordinate system is the main point of the camera objective. When the error between the camera and the spacecraft is within the installation allowable range, the camera installation coordinate system completely coincides with the spacecraft coordinate system. 5. Camera object plane coordinate system \(O_{d}-x_{d}y_{d}z_{d}\): The origin is located at the object main point of the camera. The plane \(O_{d}x_{d}y_{d}\) is the object principal plane of the camera, and the axis \(z_{d}\) is the optical axis direction, forming the right-hand coordinate system. When the positioning star is located on the optical axis, the camera object coordinate system coincides with the camera installation coordinate system. If the positioning star is not on the optical axis, the camera needs to rotate the two-axis pointing platform to make the positioning star on the optical axis. The camera object plane coordinate system can be obtained by rotating the camera installation coordinate system. The azimuth angle of the turntable is \(A_{t}\), and the elevation angle is \(E_{t}\). 6. Camera image plane coordinate system \(O_{p}-x_{p}y_{p}z_{p}\): The origin is the main point of the image plane. The plane \(O_{p}x_{p}y_{p}\) is the main plane of the image plane of the camera, and the axis \(x_{p}\) points toward the direction of the imaging element integration sequence and is parallel to the axis \(x_{d}\). The axis \(z_{p}\) is the normal direction of the main plane of the image plane. For coaxial optical systems, the normal direction of the image plane is the same as the optical axis direction, so the axis \(z_{b}\) and axis \(z_{p}\) directions are the same, forming a right-hand coordinate system.