# Visual Image Dehazing Using Polarimetric Atmospheric Light Estimation
## 4 Polarimetric Dark Channel Dehazing System
### Polarization Calibration  
Due to the incomplete polarization-preserving characteristics of the imaging system, the polarization state of the incident light will change after being transmitted through the imaging system. Therefore, the polarization state calculated from the polarization images is not the same as the polarization state of the incident light. To address this issue, polarization calibration is necessary to accurately calibrate the polarization characteristics of the imaging system. This involves measuring and characterizing the system's polarization response using known polarized light sources and using the calibration data to correct the detected polarization images. By performing polarization calibration, we can obtain more accurate polarization information and ensure the reliability of the polarization-based imaging system. The effect of the transmission medium on the polarization state of light can be represented by the Mueller matrix \(M\). For the incident light with polarization state \(S\), after passing through the telescope imaging, the polarization state becomes \(S^{\prime}\), and it can be expressed as  
\[S^{\prime}=M\times S \tag{13}\]  
where \(M\) is the Mueller matrix of the imaging system, representing the deviation of polarization information introduced by the imaging system itself. From Equation (13), we can see that the polarization state we collected is the polarization state changed by the system from the original polarization state, resulting in a deviation from the true polarization information of the incident light. To obtain the incident light's \(S\), we need to calculate the Mueller matrix \(M\) through calibration and then use Equation (14) to solve for the incident light's \(S\):  
\[S=M^{-1}\times S^{\prime} \tag{14}\]  
We used the four-point calibration algorithm to calibrate the system. As indicated by Equation (13), to calculate the Mueller matrix \(M\) of the system, we need a known polarization state of the incident light. Therefore, we placed a linear polarizer between the light source and the system to modulate the incident light \(S\), obtaining the known Stokes vector \(S_{m}\). The energy then passes through our imaging system and reaches the polarized detector, resulting in the measured Stokes vector \(S^{\prime}\). The calibration setup is illustrated in Figure 8. The calibration unit in the figure represents the linear polarizer used to modulate the incident light \(S\) and obtain the known polarization state light source \(S_{m}\). By changing the rotation angle of the linear polarizer, we can acquire various known polarization states. Next, utilizing Equation (15), we can solve for the system's Mueller matrix \(M\):  
\[ \begin{split} S^{\prime}=M\times S_{m}\\ M=S^{\prime}\times S_{m}^{-1}\end{split}  \tag{15}\]  
We modulate the incident light four times by adjusting the angle of the linear polarizer to obtain the known polarization states \(S_{m1}\), \(S_{m2}\), \(S_{m3}\), and \(S_{m4}\) as the calibration light sources. Next, we use the polarization camera to capture four images, resulting in the measured polarization states \(S^{\prime}_{1}\), \(S^{\prime}_{2}\), \(S^{\prime}_{3}\), and \(S^{\prime}_{4}\) after passing through the telescope system. We can solve for the system's Mueller matrix \(M\), as shown in Equation (16):  
\[M=\left[ \begin{array}{cccccc}S^{\prime}_{1}&S^{\prime}_{2}&S^{\prime}_{3}& S^{\prime}_{4}\end{array} \right]\left[ \begin{array}{cccccc}S_{m1}&S_{m2}&S_{m3}&S_{m4} \end{array} \right]^{-1} \tag{16}\]  
Figure 8: Four-point calibration process. The Mueller matrix of the polarizer is  
\[M_{p}=\frac{1}{2}\left[ \begin{array}{cccc}M_{p11}&M_{p12}&M_{p13}&0\\ M_{p21}&M_{p22}&M_{p23}&0\\ M_{p31}&M_{p32}&M_{p33}&0\\ 0&0&0&M_{p44}\end{array}