## 4 Gloss-Bridge: The Glossiness Mapping Method  
In Experiment 1, we obtain psychophysical scales for glossiness perception in both real and virtual environments. However, the independent variable in the real environment is the glossiness units (GU) measured by an instrument, whereas, in the virtual environment, the independent variable is the dimensionless smoothness parameter of PBR materials. To compare the differences in glossiness perception between VR materials and real-world materials, it is necessary to convert the smoothness parameter of PBR materials into glossiness units (GU). ISO 7668:1986 [32] provides a strict definition for specular glossiness. Under specified conditions of the light source and receiver angles, the ratio of the reflected light flux of the sample in the specular reflection direction to that of the standard glass sample in the same direction is defined as the specular glossiness. A black glass standard sample with a refractive index of 1.567 is used, and its glossiness is set as 100 (glossiness units). Therefore, the formula for calculating the glossiness units of a certain test sample is shown in Formula (3). In the formula, \(\Phi\) represents the measured luminous flux (lm), and \(\theta\) represents the selected measurement angle (which can be chosen from \(20^{\circ}\), \(45^{\circ}\), \(60^{\circ}\), and \(85^{\circ}\)). When the unit solid angle is determined, the formula can be equivalent to the ratio of calculated emissivity. \[GU=\frac{\Phi_{test}^{\theta}}{\Phi_{standard}^{\theta}}\times 100 \tag{3}\]  
For the Real Group stimuli in Experiment 1, a measurement is taken using a 60-degree incidence angle chosen through the application of the Fresnel equations. This enables us to calculate the corresponding reflectance of the standard glass sample as 0.100056032172. We then utilize the rendering equation to calculate the reflectance radiance of the standard sample as well as each smoothness material in the virtual reality platform. To simplify the calculation process, we fix the incident radiance at 1. The rendering equation is displayed in Equation (4), where l represents the unit vector of the incident light direction and is set to \(\left(1,\frac{\pi}{3},0\right)\). Furthermore, the integration domain is set to \(\theta\in\left[\frac{\pi}{3}-e,\frac{\pi}{3}+e\right]\)\(\phi\in\left[\pi-e,\pi+e\right]\)  
Figure 8: The results of differential distribution. and to ensure that the contribution of specular reflection is taken into account, a small value of \(\epsilon=0.05\) is chosen. \[L_{\rm o}=\int_{\Omega}f_{r}\times\mathbf{n}\cdot Id\omega=\int_{\frac{\pi}{4}- \epsilon}^{\frac{\pi}{3}+\epsilon}\int_{\pi-\epsilon}^{\pi+\epsilon}f_{r} \times\mathbf{n}\cdot\mathbf{l}\times sin\theta\mathrm{d}\theta\mathrm{d}\phi \tag{4}\]  
In Experiment 1, the VR group uses the BRDF of the Unity Standard Material shader for rendering which is divided into a diffuse term \(f_{df}\) and a specular term \(f_{spec}\). An analysis of the source codes reveals the calculation formulas, presented in Equations (5)-(13). \[f_{df}=\left(1+(f_{d90}-1)(1-\mathbf{n}\cdot\mathbf{l})^{5}\right)\times\left(1+(f_{d9 0}-1)(1-\mathbf{n}\cdot\mathbf{v})^{5}\right) \tag{5}\]  
\[f_{d90}=0.5+2\times\mathbf{l}\cdot\mathbf{n}\times\mathbf{l}\cdot\mathbf{h}\times(1-smoothness) \tag{6}\]  
\[f_{spec}=F\cdot D\cdot V \tag{7}\]  
\[F=f_{0}+(1-f_{0})\times(1-\mathbf{h}\cdot\mathbf{l})^{5} \tag{8}\]  
\[D=\frac{\alpha^{2}}{\pi\times\left(\left((\mathbf{n}\cdot\mathbf{h}\times\alpha^{2}- \mathbf{n}\cdot\mathbf{h})\times\mathbf{n}\cdot\mathbf{h}+1\right)^{2}\right)} \tag{9}\]  
\[V=\frac{0.5}{\lambda_{\rm\varphi}+\lambda_{l}} \tag{10}\]  
\[\lambda_{\rm\varphi}=\mathbf{n}\cdot\mathbf{l}\times(\mathbf{n}\cdot\mathbf{v}\times(1-\alpha )+\alpha) \tag{11}\]  
Figure 8:0  
Figure 8:1  
The calculation of the complete BRDF is a highly complex process, and finding the antiderivative can be a challenging task. To address this issue, we utilize MATLAB as an aid to perform numerical integration.