## 5 Discussion and Operational Applications  
In this section, the spectral and polarization characteristics of space debris are studied using selected feature parameters. The characteristics of spectral polarization data are analyzed, with a focus on discussing the limitations of the HAC and FCM algorithms when used separately. The suitability of these algorithms for spatial target data characteristics is also examined. Finally, the limitations of the algorithm and possible future research directions are discussed. Figure 11 depicts the spectrum curves of six types of space debris materials. As shown in the figure, there is little variation in the wavelengths of the characteristic peaks obtained from different materials. Moreover, the commonly used characteristic parameter half-wave width cannot clearly distinguish the spectral characteristics of different materials. Therefore, spectral characteristic data cannot be used as a basis for directly distinguishing space debris materials. Figures 12 and 13 directly classify space debris materials based on their Stokes vector and degree of polarization, respectively. However, it can be observed that the classification results are not satisfactory. The spectral polarization characteristics of space debris possess multidimensional, hierarchical, and fuzzy features, posing a challenge for effective data clustering. In this regard, we propose the HAC-FCM algorithm, which is a clustering approach that integrates the HAC and FCM clustering techniques. The HAC algorithm is used to determine the initial cluster centers and cluster distances by recursively dividing the data into subsets until each subset contains only one data point. This hierarchical clustering strategy is well-suited for spatial fragmented spectral polarimetric data, which often exhibit complex hierarchical structures across multiple scales. Meanwhile, the FCM algorithm assigns data points to different categories with varying degrees of membership to address the fuzziness of the data. Figure 10: Image after edge processing. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Material**} & **Silver** & **Golden** & **SR107/S781** & **Aluminum Block** & **Iron Sheets** & **Background Material** & **Total** \\ \hline error points & 1 & 0 & 605 & 1314 & 3 & 0 & 1923 \\ false alarm rate & 0.0487\% & 0 & 13.67\% & 39.31\% & 0.12\% & 0 & 3.08\% \\ recognition rates & 77.73\% & 69.71\% & 86.60\% & 100\% & 100\% & 100\% & 96.92\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Analysis of various materials cluster results. By integrating the HAC and FCM clustering techniques, the HAC-FCM algorithm proposed in this study effectively addresses the multi-scale, hierarchical, and fuzzy features of spatial fragmented spectral polarimetric data. Specifically, the HAC algorithm is used to recursively divide the data into subsets, followed by FCM clustering on each subset to obtain the final clustering result. However, it is worth noting that the HAC-FCM algorithm may encounter computational efficiency issues when dealing with large data volumes and dimensions, highlighting the importance of carefully selecting the fuzzy index to achieve optimal clustering results. Moreover, expert interpretation may be required to fully understand the clustering outcomes in domain-specific applications. Figure 11: Spectral characteristic curve of space debris material. Figure 12: Spatial distribution of normalized Stokes vectors of space debris. Figure 13: Probability density distribution characteristics of the degree of polarization for different space debris materials. In order to address these limitations, further research could investigate the potential of integrating other clustering algorithms or including additional spectral and polarization features to enhance the accuracy and computational efficiency of clustering space debris data. It should be noted that the HAC-FCM algorithm has wide-ranging applications in diverse fields, such as remote sensing, image processing, and machine learning.