## 3 Acquisition of Spectral Polarization Information  
To eliminate the impact of extraneous light on the measurement results, the experiment was conducted out in a darkroom, and the experimental apparatus is depicted in Figure 2. A halogen lamp was used as the light source, and measurements were taken at different reflection angles around the specular angle. Ocean Optics fiber optic spectrometer USB2000+ serves as the receiver. The polarization state generator (PSG) consisted of a polarizer (P1) and a spectral filter (SF), while another polarizer (P2) functioned as the polarization state analyzer (PSA). The light emitted by the halogen lamp was transformed into a collimated spot using the beam expander system and then directed towards the sample surface via the PSG, with different polarization states. After being scattered by the surface, the light reached the spectrometer through the PSA. Following calibration with a standard whiteboard, the system was controlled to record the required polarization spectrum data. The test samples in this study comprise two thermally controlled coating materials (SR107 and S781), two commonly used metallic materials (aluminum and iron plates), and two insulation cladding layers (gold insulation cladding layer and silver insulation cladding layer). The experimental test band covers 370 nm-760 nm, the incident angle ranges from 10\({}^{\circ}\) to 60\({}^{\circ}\), and the interval is 10\({}^{\circ}\). A total of 14 sets of reflection angle data were collected, with the specular angle as the center and an interval of 3\({}^{\circ}\). The incident polarization angle and reflection polarization angle were both set to 0\({}^{\circ}\), 45\({}^{\circ}\), 90\({}^{\circ}\), and 135\({}^{\circ}\) for each set of data. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Symbol** & **Meaning** & **Symbol** & **Meaning** \\ \hline \(\delta\) & scaling factor & \(k\) & the number of iteration steps \\ \(\varepsilon\) & termination criterion & \(K\) & the set of cluster indices \\ \(c_{j}\) & center of the j cluster & \(m\) & operation’s index \\ \(C\) & cluster center matrix & \(N\) & initial data points \\ \(dist\) & distance metric function & \(t\) & the number of clusters \\ \(i\) & index variables & \(u_{ij}\) & the degree value of the \(j\) cluster that completely contains the \(i\) sample vector \\ \(j\) & index variables & \(U\) & the degree matrix of the initial sample \\ \(j\) & objective function & \(X_{i}\) & the data point for the index value \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Symbols and their meanings used in this section. Any beam's polarization state can be represented by the Stokes vector. The polarimetric bidirectional reflectance distribution function (pBRDF) is a Mueller matrix with spatial coordinates that is utilized as an intermediate matrix for the Stokes vector change as the beam traverses the sample. The Stokes vector of the reflected light of a single wavelength can be expressed in terms of the incident Stokes vector and the pBRDF matrix as follows [40]:  
\[ \begin{pmatrix}S_{0}^{\text{r}}\\ S_{1}^{\text{r}}\\ S_{2}^{\text{r}}\\ S_{3}^{\text{r}}\end{pmatrix} = \begin{pmatrix}f_{00}&f_{01}&f_{02}&f_{03}\\ f_{10}&f_{11}&f_{12}&f_{13}\\ f_{20}&f_{21}&f_{22}&f_{23}\\ f_{30}&f_{31}&f_{32}&f_{33}\end{pmatrix}\begin{pmatrix}S_{0}^{\text{i}}\\ S_{1}^{\text{i}}\\ S_{2}^{\text{i}}\end{pmatrix} = \begin{pmatrix}\sum\limits_{j=0}^{3}f_{0j}S_{j}^{ \text{i}}&\sum\limits_{j=0}^{3}f_{1j}S_{j}^{\text{i}}&\sum\limits_{j=0}^{3}f_{2 j}S_{j}^{\text{i}}&\sum\limits_{j=0}^{3}f_{3}S_{j}^{\text{i}}\end{pmatrix} ^{\text{T}} \tag{6}\]  
Here, \(f_{\text{ij}}\) (i, j, 0, 1, 2, 3), \(\left(S_{0}^{\text{i}},S_{1}^{\text{i}},S_{2}^{\text{i}},S_{3}^{\text{i}} \right)^{\text{T}}\), \(\left(S_{0}^{\text{r}},S_{1}^{\text{r}},S_{2}^{\text{r}},S_{3}^{\text{r}} \right)^{\text{T}}\) represents the pBRDF matrix element, incident Stokes vector, and reflected Stokes vector, respectively.