# Real-Time Vision through Haze Based on Polarization Imaging
## 2 Physical Model for Polarization Image  
To construct our physical model, we consider the situation in which two degraded orthogonally polarized azimuth images are first acquired by rotating the polarizer installed in front of the detector in bad weather that has various gases and aerosols present. A clear scene is subsequently acquired by effectively splitting the azimuth images based on the differences of polarization properties between the atmospheric light and the target light [8]. The total intensity incident on the detector can be expressed as:  
\[I_{Total}=T+A=L_{Object}e^{-\beta z}+A_{\infty}\Big{(}1-e^{-\beta z}\Big{)} \tag{1}\]  
The total light intensity \(I_{Total}\) received by the detector consists of the target light \(T\) and the atmospheric light \(A\). Target light \(T\) is the target radiation \(L_{Object}\) that reaches the detector after being scattered by various gases and aerosols, and it exponentially decays during transmission [9], as shown in the attenuation model in Figure 1a. Atmospheric light \(A\) refers to the sunlight received by the detector after the scattering of various gases and aerosols, and it increases with the detection distance, as shown in the atmospheric light model in Figure 1b. The relationship between \(I_{Total}\)\(T\), \(A\), and \(L_{Object}\) is given by Equation (1), where \(A_{\infty}\) denotes the atmospheric light intensity from infinity, \(\beta\) denotes the atmospheric attenuation coefficient, and \(z\) denotes the distance between the target and the detector.  
During polarization imaging, two orthogonally polarized azimuth images are acquired at the detector by rotating the polarizer. In particular, \(I_{\text{min}}\) denotes the minimum atmospheric light intensity, whereas its orthogonal counterpart, \(I_{\text{max}}\), denotes the maximum atmospheric light intensity [6]. The expressions for \(I_{\text{min}}\), \(I_{\text{max}}\), and \(I_{Total}\) are as follows:  
\[I_{\text{max}}=I_{T}/2+I_{\text{B}}^{\text{max}} \tag{2}\]  
\[I_{\text{min}}=I_{T}/2+I_{\text{B}}^{\text{min}} \tag{3}\]  
Figure 1: (**a**) Attenuation model, and (**b**) atmospheric light model.  
Upon selecting an area with uniform atmospheric light without targets and setting \(A_{\max}(x,y)\) and \(A_{\min}(x,y)\), the polarization of the atmospheric light can be expressed as Equation (4), where \(A_{\max}(x,y)\) and \(A_{\min}(x,y)\) denotes the brightest and darkest atmospheric light, respectively, and \((x,y)\) denotes image pixel point coordinate:  
\[p=\frac{A_{\max}(x,y)-A_{\min}(x,y)}{A_{\max}(x,y)+A_{\min}(x,y)} \tag{4}\]  
Substitution of Equations (2)-(4) into Equation (1) yields the mathematical model for a high-definition scenario, as follows:  
\[L_{Object}=\frac{\left(I_{Total}-\left(I_{\max}-I_{\min}\right)/p\right)}{1-A/A _{\infty}} \tag{5}\]  
From Equation (5), we note that the polarization image model uses the vector vibration direction difference between the target and the atmospheric light to filter the effects of the atmospheric light. High-contrast and high-definition images of targets under smoggy conditions are acquired by comparing the differences of the orthogonally linear-polarized azimuth images acquired by rotating the polarizers.