## 4 Results and Discussion
### Spectral Polarization Image Clustering Rendering  
#### 4.2.1 Random Point Clustering of Spectral Polarization Images  
The laboratory experiments have demonstrated the effectiveness of the proposed algorithm in detecting and recognizing space debris targets, particularly for high-dimensional small data sets with large amounts of information. However, traditional single-spectrum, single-polarization images are unable to directly identify and classify targets. Therefore, field experiments are necessary to capture real spectral images of space targets in actual scenarios to validate the robustness of the algorithm.  
The algorithm proposed in this paper must rely on the material's own spectral polarization characteristics, making full use of characteristic parameters that can accurately characterize the spectral polarization characteristics of the space debris. Through the selection of different data combinations, it has been proven through experiments that the optimal data set combination is selected, which is consistent with the results of the characteristic data. Ultimately, the combination of Stokes and DOLP data is chosen as the final data set. Figure 6 depicts the hierarchical clustering tree and line graph of cluster distance in the experimental field spectral polarization characteristic data, from which the critical value of cluster distance \(\varepsilon=24.9133\) is derived. Using this threshold, clustering continues. However, due to the large amount of data, it is not intuitive to directly draw the cluster diagram. According to the clustering result itself, the number of clusters is calculated to be 6.  
The scatter diagram in Figure 7a shows the projection of the data vector's first dimension and 1560 th dimension based on the clustering algorithm, and it demonstrates that the six types of points can be separated normally. After setting the threshold, the objective function decreases with each iteration. When the FCM iteration threshold is set to 0.001, the change trend of the objective function J with the number of iterations is displayed in Figure 7b. From the figure, it is evident that the objective function eventually converges to the approximate global optimal value with the increase of the number of iterations, and the decline speed is fast. The convergence is accomplished in 21 iterations, demonstrating the effectiveness of the algorithm developed in this research.  
In this paper, the proposed data combination was compared with two classical clustering algorithms: the k-means clustering algorithm and the coalescent upward HAC clustering method. The parameters, such as prior cluster number and prior cluster center, were applied to the two classical algorithms, and the standardized Euclidean distance was used as the vector distance metric. The comparison of the running results of the three algorithms is shown in Table 2. The high accuracy of the fusion algorithm is due to the fact that it does not directly assign a certain numerical point to a certain division from the calculation of a certain level like the classical algorithms. Instead, it calculates the degree of membership of the point relative to the cluster after the final convergence. The algorithm flow structure reveals that the k-means algorithm has a time complexity of O(N), but the coalescent upward HAC algorithm has a time complexity of O(N3). With the sample size N being 726 in this paper, the number of iterations of the k-means algorithm should be  
Figure 6: Training set hierarchical clustering tree and cluster distance line diagram. (**a**) Stokes and DOLP combination data set hierarchical cluster tree (**b**) Data combination layer cluster distance folding line change diagram.  
an integer multiple of 726 (more than double), while the number of iterations of the HAC method is a multiple of the cube of 726. The proposed HAC-FCM algorithm only requires 21 iterations. Therefore, compared with the classical algorithms, the fusion algorithm can obtain clustering results more efficiently and accurately.