# Parameter Optimization Method for Metal Surface pBRDF Model Based on Improved Strawberry Algorithm
## 4 Simulation and Experiment  
### Simulation  
To validate the performance of the proposed method and its applicability, open-source data of copper, black paint and green paint materials from reference [1] were fitted.  
According to the analysis mentioned above, the pBRDF model of metal surface established in this paper has nine unknown parameters, namely \(n\), \(k\), \(\sigma\), \(k_{s}\), \(k_{d}\), \(k_{v}\), \(q\), \(c\) and \(\sigma_{v}\), shown in Equation (6). In this paper, the reflected polarization feature of the metal surface is characterized by a degree of polarization, and the optimization target is to select the optimal model parameters, minimizing the mean squared error \(\Delta E\) of the degree of polarization between the model predicted values and the experimentally measured ones. Therefore, the following optimization objective function can be constructed:  
\[min\Delta E(n,k,\sigma,k_{s},k_{d},k_{v},q,c,\sigma_{v})=\frac{\sum_{\theta_{i} }\sum_{\theta_{r}}\left[D(\theta_{i}\theta_{r},\Delta\varphi)-D_{m}(\theta_{i }\theta_{r},\Delta\varphi)\right]^{2}}{\sum_{\theta_{i}}\sum_{\theta_{r}} \left[D_{m}(\theta_{i}\theta_{r},\Delta\varphi)\right]^{2}} \tag{17}\]  
where \(D_{m}(\theta_{i}\theta_{r},\Delta\varphi)\) and \(D(\theta_{i}\theta_{r},\Delta\varphi)\) are the DOP values measured in the experiment and predicted by the proposed model, respectively.  
Table 1 shows the retrieved complex refractive index of copper, black paint and green paint surfaces with different models under the incident light at the wavelength of 650 nm.  
Figure 8: ISBA convergence curve.  
Figure 7: SBA convergence curve.  
It can be seen from Table 1 that the retrieved complex refractive index by the proposed method in this paper is closer to the true value taken from reference [1].  
By bringing the retrieved complex refractive index into Equation (9), the predicted degree of polarization, which is the function of the reflection angle, can be calculated. The curves fitted with the degree of polarization predicted by the proposed method and that taken from reference [1] were redrawn in Figure 9.  
From Figure 9, it can be seen that the referred DOP values are slightly higher than that predicted by the proposed method under small incident angles and reversed when the incident angle is beyond a certain value, while the fitting curve by the proposed method in this paper is closer to values in reference [1]. Compared to the model proposed in the reference, the multi-parameter pBRDF model optimized by ISBA takes into account the depolarization effect caused by the volume scattering by the roughness of the target surface and avoids obtaining a DOP value at virtual height. The inversion process of the pBRDF  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Sample**} & \multicolumn{2}{c}{**Truth Value**} & \multicolumn{2}{c}{**Reference Value**} & \multicolumn{2}{c}{**Estimated Value**} \\ \cline{2-7}  & \(n\) & \(k\) & \(n\) & \(k\) & \(n\) & \(k\) \\ \hline Copper & 0.40 & 2.95 & 0.54 & 3.19 & 0.51 & 2.90 \\ Black Paint & 1.40 & 0.22 & 1.46 & 1.32 & 1.26 & 0.27 \\ Green Paint & 1.39 & 0.34 & 1.47 & 0.47 & 1.37 & 0.34 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Retrieved complex refractive index.  
Figure 9: Curves fitted with predicted and referred DOP values under the incident angle of 60\({}^{\circ}\) (**a**)copper; (**b**) green paint; (**c**) black paint.  
model with the improved strawberry algorithm can keep from falling into local optimum, while the accuracy and reliability of the pBRDF model can be enhanced.