# Parameter Optimization Method for Metal Surface pBRDF Model Based on Improved Strawberry Algorithm
## 2 Polarization Bidirectional Reflection Distribution Function Model  
BRDF is defined as the ratio of the reflected radiance along the outgoing to the incident irradiance along the incoming direction, expressed [17] as follows:  
\[f_{r}(\theta_{i},\varphi_{i};\theta_{r},\varphi_{r};\lambda)=\frac{dL_{r}( \theta_{i},\varphi_{i};\theta_{r},\varphi_{r};\lambda)}{dE_{i}(\theta_{i}, \varphi_{i};\lambda)}(sr^{-1}) \tag{1}\]  
where \(dL_{r}\) is the reflected radiance, and \(dE_{i}\) is the incident irradiance. The parameters \(\theta_{i}\) and \(\varphi_{i}\) denote the incident polar and azimuth angles, respectively, and \(\theta_{r}\), \(\varphi_{r}\) denote the reflected polar and azimuth angles, respectively, as shown in Figure 1. The parameter \(\lambda\) is the transferring light wavelength. Considering the fact that the microstructure of metal surfaces is usually in the micrometer scale and random distribution pattern, the BRDF model can be established using microfacet theory as shown in Figure 1.  
In order to better analyze the optical properties of metal surfaces, the polarized bidirectional reflectance distribution function (pBRDF), usually reduced for BRDF with the help of the Muller matrix, is applied on many occasions. Assuming the pBRDF of a metal surface contains specular reflection, diffuse reflection and volume scattering denoted \(F_{j,k}^{s}\), \(F_{j,k}^{d}\) and \(F_{j,k}^{r}\), it can be represented as shown in the following:  
\[F=k_{s}F_{j,k}^{s}+k_{d}F_{j,k}^{d}+k_{v}F_{j,k}^{v} \tag{2}\]  
where \(k_{s},k_{d}\) and \(k_{v}\) denote the coefficients of the three components, respectively. Specular reflection, the main component of polarized reflecting light, represents the single reflection from the interaction between incident light and a metal surface, it can be denoted as [18]:  
\[F_{j,k}^{s}=\frac{\left(q-1\right)\left(\sqrt{2}\sigma\right)^{2q-2}\cdot\min(1.0,\frac{2 cos\alpha cos\theta_{r}}{cos\beta},\frac{2 cos\alpha cos\theta_{i}}{ cos\beta})}{4\pi cos\theta_{i}cos\theta_{r}cos^{4}\alpha[tan^{2}\alpha+\left( \sqrt{2}\sigma\right)^{2}]^{q}}\times M_{j,k}^{s} \tag{3}\]  
where \(\sigma\) is the surface roughness, \(\alpha\) is the angle between the normals of the microfacet and the macroscopic surface, \(\beta\) is the angle between the incident light and the normal of the microfacet, and \(q\) is the related constant about the properties of the metal material, reflecting the scattering inclusivity of the metal surface. Diffuse reflection and volume scattering are related to the texture and roughness of metal surfaces and are generally thought to be non-polarized for the randomness of the texture distribution. In this paper, the diffuse reflection from a metal surface was represented by the Minnaert model considering the light reversibility as follows [12]:  
\[F_{j,k}^{d}=\frac{1}{\pi}(cos\theta_{i}cos\theta_{r})^{c}M_{j,k}^{d}=\frac{1} {\pi}(cos\theta_{i}cos\theta_{r})^{c} \begin{bmatrix}1&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{bmatrix}  \tag{4}\]  
where \((cos\theta_{i}cos\theta_{r})^{c}\) reflects the variation in diffuse intensity with incident and reflection angles, and \(c\) is a constant in the range \((-1,0)\). Volume scattering is the multiple reflection result of incident light on a metal surface among adjacent microfacts with fine structure, whose magnitude is related to the reflection angle and generally considered to follow a Gaussian distribution [19]. In this paper, the volume scattering component was represented as follows:  
\[F_{j,k}^{v}=\frac{1}{\sqrt{2\pi}\sigma_{v}}exp\left(\frac{-\theta_{r}^{2}}{2 \sigma_{v}^{2}}\right)M_{j,k}^{v}=\frac{1}{\sqrt{2\pi}\sigma_{v}}exp\left( \frac{-\theta_{r}^{2}}{2\sigma_{v}^{2}}\right) \begin{bmatrix}1&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{bmatrix}  \tag{5}\]  
where \(\sigma_{v}\) is the undetermined coefficient.