## 2 Object Models and Algorithms  
The CCD images taken by the Chinese Academy of Sciences Ground-based Optical Telescope with a 3.2\({}^{\circ}\)\(\times\) 3.2\({}^{\circ}\) field of view, an 800 mm aperture and a 2\({}^{\prime\prime}\) angular resolution- can be expressed as  
\[I=bg+s+ob+n, \tag{1}\]  
where \(I\) represents astronomical gray values of a CCD image, \(bg\) are deep space backgrounds, \(s\) are stars, \(ob\) are usually satellites and space debris, and \(n\) mainly includes Gaussian noise from the circuit and Poisson noise from dark current and background light. In different observation modes, objects or stars appear in different shapes on star images. When the telescope tracks the motion of stars, the stars are relatively stationary and appear as points, whereas the target appears as a streak. In this mode, the target streak needs to be extracted. On the contrary, when the telescope moves with the target, the target appears as a point and the stars appear as streaks. In this mode, the star streaks should be removed first. Therefore, the recognition of target streaks and star streaks is the focus of our work. We uniformly call such targets or stars with point shapes \"point targets\" and call those with long streak shapes \"target streaks\", as shown in Figure 1. In the two observation modes, either the target or the stars are stationary relative to the star image. Therefore, the direction of the streaks is consistent with the direction of relative motion, and the streaks are straight. The length of the streak is related to the target velocity relative to the stars and exposure time. We can appropriately increase exposure time to make the dark target brighter and easier to identify. However, with the increase of exposure time, the bright stars become saturated and the bright spots spread to a very large area around, which affects target recognition. Therefore, exposure time should be adjusted dynamically according to the use requirements and weather conditions, and for observing high-orbit space objects, usually exposure time is at second level. We take various noises, influence factors and deep space backgrounds together as undulating image backgrounds. In addition, we first rotate star images according to streak parameters and adjust the direction of the streaks to horizontal (the \(X\)-axis direction of the image is horizontal, and the \(Y\)-axis direction of the image is vertical). On this basis, we select 1D Gaussian function to lengthshways fit them. \[\widetilde{y}_{k}=BG+A\exp(\frac{-x_{k}^{2}}{2\sigma^{2}}) \tag{2}\]  
where \(\widetilde{y}_{k}\) are object gray values of the constructed model, \(BG\) are local image backgrounds, \(A\) is the object amplitude above the local background intensity, \(\sigma^{2}\) is the variance of the Gaussian function, and \(x_{k}\) are image ordinates. In order to ensure that the model can fit the image data better, mean square error should be minimized. \[f\left(x_{k};\sigma^{2}\right)=\frac{1}{n}\underset{k}{\overset{n}{\sum}}( \widetilde{y}_{k}-y_{k})^{2}\rightarrow\min \tag{3}\]  
where \(f\) is the mean square error function, and \(y_{k}\) are observed object gray values. We find \(\sigma_{0}^{2}\) when \(f\) is the minimum value, and the Gaussian function with \(\sigma_{0}^{2}\) is the optimal fitting model for the group of data. The derivative of \(f\) is calculated as  
\[\frac{A}{\sigma^{4}}\underset{k}{\overset{n}{\sum}}\left(Ax_{k}^{2}\exp \left(\frac{-x_{k}^{2}}{\sigma^{2}}\right)-\left(y_{k}-BG\right)x_{k}^{2}\exp \left(\frac{-x_{k}^{2}}{2\sigma^{2}}\right)\right)\to 0 \tag{4}\]  
The pixel numbers occupied by dim objects in star images are different and unstable [29]. For star images taken by our wide field telescope, dim objects or stars usually occupy several pixels, and the bright stars usually occupy more pixels.