## 2 Materials and Methods
### Water Spray Removal  
According to the CP characteristics, the CP light can be detected by setting up a measurement system as shown in Figure 4. Two key parts of the measurement system include a retarder and a linear polarizer, as shown in Figure 4. The imaging process of the system can be expressed by the Mueller matrix \(M_{R}(\phi,\theta_{R},\theta_{P})\) in Equation (3):  
\[ \begin{array}{l}M=M_{R}(\phi,\theta_{R})\cdot M_{P}(\theta_{P})\\ =\frac{1}{2}\left[\begin{array}{cccc}1&0&0&0\\ 0&\cos^{2}2\theta_{R}+&(1-\cos\phi)\times\\ &\cos\phi\sin^{2}2\theta_{R}&\sin 2\theta_{R}\cos 2\theta_{R}\\ &(1-\cos\phi)\times&\sin^{2}2\theta_{R}+\\ &\sin 2\theta_{R}\cos 2\theta_{R}&\cos\phi\cos^{2}2\theta_{R}\\ 0&\sin\phi\sin 2\theta_{R}&-\sin\phi\cos 2\theta_{R}&\cos\phi\end{array} \right]\cdot\left[\begin{array}{cccc}1&\cos 2\theta_{p}&\sin 2\theta_{p}&0\\ \cos 2\theta_{p}&\cos^{2}2\theta_{p}&\sin 2\theta_{p}\times\\ &\sin 2\theta_{p}&\sin 2\theta_{p}\\ &\sin 2\theta_{p}&\sin 2\theta_{p}\times\\ &\cos 2\theta_{p}&\sin^{2}2\theta_{p}&0\\ 0&0&0&0\end{array}\right]\,,\end{array}  \tag{3}\]  
where \(M_{R}(\phi,\theta_{R})\) and \(M_{P}(\theta_{P})\) represent the Mueller matrix of the retarder and the linear polarizer at an arbitrary angle, respectively; \(\phi\) is the total phase shift between the two orthogonal components of the incident field; \(\theta_{R}\) denotes the angle between the fast axis of the retarder and \(x\)-axis, and; the transmission axis of the polarizer is set at angle \(\theta_{P}\) to the \(x\)-axis. During measurement, \(\phi\) is fixed to \(90^{\circ}\) and \(\theta_{R}\) is set to \(45^{\circ}\) in order to derive Equation (4). Then, the transformation of CP to LP, its detection by rotating the linear polarizer to change \(\theta_{P}\), and two states (\(I_{B}(x,y,\theta_{P})\) and \(I_{W}(x,y,\theta^{\circ}p)\)) of the incident light--named the best state and the worst state--are determined [8]. When calculating the airlight \(I_{A}\cos(x,y)\), we choose the intensities of areas without any objects in the best state and worst state images and mark them by \(I_{B}(x,y,0^{\circ})\) and \(I_{W}(x,y,90^{\circ})\), respectively. Equation (5) gives the DoCP of the airlight, according to the definition of polarization degree. \[I(x,y,\theta_{P})=\frac{1}{2}\big{[}S_{0}(x,y)+S_{3}(x,y)\cdot\cos\big{(}2 \theta_{P}\big{)}\big{]}\, \tag{4}\]  
\[p_{c}=\left|\frac{I_{B}(x,y,0^{\circ})-I_{W}(x,y,90^{\circ})}{I_{B}(x,y,0^{ \circ})+I_{W}(x,y,90^{\circ})}\right|=\left|\frac{S_{3}(x,y)}{S_{0}(x,y)} \right|. \tag{5}\]  
As we know, a wavelet transform decomposes an image into low- and high-frequency components, where a low-frequency component denotes the approximation, and a high-frequency component characterizes the details of its last order image. Accordingly, the circular polarization information brought in by the interaction between raindrops and natural light is taken care of in the low-frequency parts [17]. In the captured images, the radiance \(I_{T}(x,y,\lambda)\) received by the detector can be expressed as Equation (6) in the low-spatial frequency parts. \[I_{T}(x,y,\lambda)=I_{D}(x,y,\lambda)+I_{A}(x,y,\lambda)\, \tag{6}\]  
where \(I_{D}(x,\,y,\,\lambda)\) represents the direct transmission of radiance from the imaged scene; \(I_{A}(x,y,\,\lambda)\) denotes the radiance of the airlight; \(I_{Q}(x,y,\,\lambda)\) stands for the object radiance in a  
Figure 4: The measurement diagram of CP states. clear-day image, as shown in Equation (7) [10]. Additionally, \(t(x,y,\lambda)=\mathrm{e}^{-\beta z(x,y,\lambda)}\) means the atmosphere-attenuated coefficient, where \(\beta\) is the scattering coefficient [4]; \(I_{A}{}^{\infty}(x,y,\lambda)\) is the airlight radiance corresponding to an object at infinity. \(\lambda\) is the wavelength.