## 3 Proposed Method for Zenith Angle Estimation from Mixed Reflected Light  
### Principle of Polarized 3D Reconstruction  
Generally, natural light is assumed to be completely unpolarized. However, after reflection on the surface of targets, light is partially polarized. The polarization state of the reflected light is related to the material ingredients, surface roughness, and angle of incidence [21].  
According to the law of reflection, the angle of incidence equals the angle of reflection, denoted as \(\theta\) in Figure 1. The propagation direction of the reflected light is along the \(Z\)-axis, whereas the XY plane is the plane of the detector. The projection of the object surface normal \(\overrightarrow{n}\) onto the XY plane is \(\overrightarrow{n}\), and the angle between \(\overrightarrow{n}\) and the \(X\)-axis is the azimuth angle \(\varphi\). \(\theta_{pol}\) represents the linear polarization direction of the polarizer [20].  
The zenith angle \(\theta\) and the azimuth angle \(\varphi\) determine the surface normal, which can be expressed as follows [22]:  
\[\overrightarrow{n}=\begin{bmatrix}\overrightarrow{n}\\ \overrightarrow{n}\\ \overrightarrow{n}\\ \overrightarrow{n}\end{bmatrix}=\begin{bmatrix}\cos\theta\cos\varphi\\ \cos\theta\sin\varphi\\ \cos\theta\end{bmatrix}=\begin{bmatrix}\tan\theta\cos\varphi\\ \tan\theta\sin\varphi\\ 1\end{bmatrix} \tag{1}\]  
where \(\theta\in[0^{\circ},\,90^{\circ}]\), \(\varphi\in[0^{\circ},\,360^{\circ}]\), \(\overrightarrow{n_{x}}\), \(\overrightarrow{n_{y}}\), \(\overrightarrow{n_{z}}\), and \(\overrightarrow{n}\) represent the components in the three directions, respectively. The surface normal vector represents the gradient information related to changes in the shape of an object's surface. \(p\) and \(q\) denote the gradient fields of the surface normal in the \(x\) and \(y\) directions, respectively [23].  
\[\begin{cases}p=\tan\theta\cos\varphi=\frac{\partial z}{\partial x}\\ q=\tan\theta\sin\varphi=\frac{\partial z}{\partial y}\end{cases} \tag{2}\]  
The degree of polarization of the diffuse reflected light \(P_{t}\) and specular reflected light \(P_{r}\) with respect to the zenith angle \(\theta\) and the refractive index \(n\) of the object surface are expressed as follows [24]:  
\[\begin{cases}P_{t}=\frac{\left(n-\frac{1}{\theta}\right)^{2}\sin^{2}\theta}{2+2 n^{2}-\left(n+\frac{1}{\theta}\right)^{2}\sin^{2}\theta+4\cos\theta\sqrt{n^{2}- \sin^{2}\theta}}\\ P_{r}=\frac{2\sin^{2}\theta\cos\theta\sqrt{n^{2}-\sin^{2}\theta}}{\left(n^{2} -\sin^{2}\theta\right)\cos^{2}\theta+\sin^{4}\theta}\end{cases} \tag{3}\]  
The azimuth angle \(\varphi\) can be calculated as follows [23]:  
\[\varphi=\frac{1}{2}\begin{cases}\arctan(\frac{S_{2}}{S_{1}})+0^{\circ},\ S_{1}>0;S_{2}\geq 0 \\ \arctan(\frac{S_{2}}{S_{1}})+180^{\circ},\ S_{1}>0;S_{2}<0\\ \arctan(\frac{S_{2}}{S_{1}})+90^{\circ},\ S_{1}<0\\ \frac{\pi}{3},\ S_{1}=0;S_{2}>0\\ \frac{3\pi}{3},\ S_{1}=0;S_{2}<0\end{cases} \tag{4}\]  
In practical applications, the polarization state of a light wave is often represented by the Stokes vector, which consists of the Stokes parameters. \(I_{0}\) denotes the intensity of the  
Figure 1: Schematic diagram of the surface normal.  
horizontally polarized component of the light beam, \(I_{90}\) is the intensity of the vertically polarized component, \(I_{45}\) is the intensity of the component polarized at \(45^{\circ}\), and \(I_{135}\) is the intensity of the component polarized at \(135^{\circ}\). The Stokes parameters are defined as follows:  
\[\left\{\begin{aligned} S_{1}=I_{0}-I_{90}\\ S_{2}=I_{45}-I_{135}\end{aligned}\right. \tag{5}\]  
The surface normal at each point can be determined by substituting the obtained zenith angle \(\theta\) and azimuth angle \(\varphi\) into Equation (1). Thereafter, the target surface depth function can be evaluated using the Frankot-Chellappa integration algorithm [25].