# Star Identification Algorithm Based on Dynamic Distance Ratio Matching
## 3 Star Identification Algorithm
### Algorithm Robustness Analysis  
In this section, we illustrate the robustness of the algorithm through a typical identification process. In Figure 3, the filled circle represents the real observed star image, \(l_{c}\) is the center of FOV of star tracker. The star closest to \(l_{c}\) is selected as the reference star \(R\) to be identified. In fact, the star pattern feature \(R\) is represented as a dashed circular area. As seen in the Figure 3, the current observed star image contains neighboring stars 1, 2, 3, 6, 7, and the false star \(F\). Comparing the ideal neighboring stars 1, 2, 3, 4, 5, 6, and 7 used to construct the star pattern feature \(R\), we find that stars 4 and 5 are missing from the observed star image, while the false star \(F\), which does not exist in the ideal star pattern, is present. The current scene involves the issues of lost stars and multiple stars in stellar identification, which can be used to validate the robustness of the algorithm presented in this paper in handling situations involving missing stars and false stars.  
The ideal star patterns of \(R\) are \((\theta_{1},\theta_{2},\theta_{3},\theta_{4},\theta_{5},\theta_{6},\theta_{7})\) and \((r_{1},r_{2},r_{3},r_{4},r_{5},r_{6},r_{7})\). The actual extracted star patterns are \((\theta_{1},\theta_{2},\theta_{3}+\theta_{4}+\theta_{5},\theta_{6},\theta_{F7},\theta_{7})\) and \((r_{1},r_{2},r_{3},r_{6},r_{F},r_{7})\). We randomly select the angle \(\theta_{1}\) formed by stars 1 and 2 to create \((\theta_{1},\frac{r_{1}}{r_{2}})\) as the initial matching feature. This angle \(\theta_{1}\) will serve as the starting angle for accumulating angles in the matching degree calculation. The cumulative angle feature of the ideal star model is obtained as \((\theta_{12},\theta_{13},\theta_{14},\theta_{15},\theta_{16},\theta_{17},360)\), and the real pattern is \((\theta_{12},\theta_{13},\theta_{16},\theta_{17},\theta_{17},360)\).  
As shown in Figure 4, the current number of neighbor star feature matches is 5. Due to the absence of stars 4 and 5, the accumulated angles \(\theta_{14}\) and \(\theta_{15}\) have not completed matching. Additionally, the presence of the false star \(F\) has affected the matching of the accumulated angle \(\theta_{1F}\), which also remains unmatched. However, these two situations do not affect the subsequent matching of neighboring star features. Both stars 6 and 7 complete their matching. Therefore, the proposed algorithm demonstrates strong tolerance to interference from missing stars and false stars.  
In addition, to prevent excessive redundancy during the matching process, it is necessary to further verify the dynamic distance ratio after each pair of accumulated angles completes matching. According to previous sections, the distance ratio is independent of focal length, allowing the use of the distance ratio to replace actual distances, thus enhancing the robustness of the identification algorithm against variations in the focal length of the star sensor. When selecting neighboring stars to form the ratio, a dynamic selection method should also be considered.  
Figure 5 shows the matching situation using the distance ratios of neighboring angles, with a feature match count of 3. Figure 6 illustrates the method using dynamic distance ratios, where each time the ratio between the current neighboring star's distance and the distance of the previously matched neighboring star is calculated, resulting in a feature match count of 5. It can be seen that the use of dynamic distance ratios effectively reduces interference from false stars and missing stars. Furthermore, transforming the ratio into a dynamic process enhances the robustness of the algorithm.  
Figure 4: Schematic diagram of cumulative angle matching.  
Figure 5: Distance proportional verification.  
Figure 3: Graphical representation of a typical star identification process.  
Figure 6: Dynamic distance proportional verification.