# Real-Time Vision through Haze Based on Polarization Imaging
## 3 Real-Time Optical Sensing and Detection System
### Polarization Image Acquisition Method Based on Stokes Vectors  
In 1852, [PERSON] proposed a detection method that utilized light intensity to describe the polarization characteristics. This method, known as the Stokes vector [10] method, allows for a more intuitive and convenient observation and acquisition of the polarization information of light. The Stokes matrix \(S=[S_{0},\,S_{1},\,S_{2},\,S_{3}]^{T}\) is commonly used to describe the polarization state of a light wave, where \(S_{0}\) denotes the total intensity of radiation, \(S_{1}\) the intensity difference between vertically and horizontally polarized light waves, \(S_{2}\) the intensity difference between two 45\({}^{\circ}\)-polarized light waves, and \(S_{3}\) the intensity difference between right-handed and left-handed circularly polarized light waves. The expression \(S\) is as follows:  
\[S=\left[\begin{array}{c}S_{0}\\ S_{1}\\ S_{2}\\ S_{3}\end{array}\right]=\left[\begin{array}{c}I_{0^{\circ}}+I_{90^{\circ}} \\ I_{0^{\circ}}-I_{90^{\circ}}\\ I_{45^{\circ}}-I_{135^{\circ}}\\ 2I_{\lambda/4A5^{\circ}}-\left(I_{0^{\circ}}+I_{90^{\circ}}\right)\end{array}\right] \tag{6}\]  
where \(I_{0^{\circ}},\,I_{45^{\circ}},\,I_{90^{\circ}},\) and \(I_{135^{\circ}}\) denote the intensities of light corresponding to polarization directions of 0\({}^{\circ},\,45^{\circ},\,90^{\circ},\) and \(135^{\circ}\); these intensities of light are obtained by installing polarizers that have different light transmission directions (including 0\({}^{\circ},\,45^{\circ},\,90^{\circ},\) and \(135^{\circ}\)) in front of the detector. \(I_{\lambda/4,\,45^{\circ}}\) means the right-handed circularly polarized light. This is achieved by installing a polarizer that transmits 45\({}^{\circ}\) light in front of the detector, and then installing a wave plate whose fast axis direction is 0\({}^{\circ}\) in front of the polarizer.  
Under natural conditions, the circular component can be neglected because there is no circular polarization properties in nature. Hence, the stokes vector can be reduced to a vector containing only linear polarization components \(S^{\prime}=[S_{0},\,S_{1},\,S_{2}]^{T}\). Therefore, only the intensities of light corresponding to different polarization directions, namely \(I_{0^{\circ}}\), \(I_{45^{\circ}}\), \(I_{90^{\circ}}\), and \(I_{135^{\circ}}\), need to be measured. Subsequently, linear components \(S_{0}\), \(S_{1}\), and \(S_{2}\) of the target-light Stokes matrix can be obtained using Equation (7). The polarization state of the light wave can be expressed as follows:  
\[S^{\prime}=\left[\begin{array}{c}S_{0}\\ S_{1}\\ S_{2}\end{array}\right]=\left[\begin{array}{c}I_{0^{\circ}}+I_{90^{\circ}} \\ I_{0^{\circ}}-I_{90^{\circ}}\\ I_{45^{\circ}}-I_{135^{\circ}}\end{array}\right] \tag{7}\]  
After the polarization information of the light wave is obtained through measurement, it is visualized using polarization degree images \(P\) or polarization angle images \(\theta\). In general, the degree of polarization (DoP) can be calculated by means of the Stokes vectors, and the degree of polarization of the actual measurement, that is, the degree of linear polarization (DoLP), can be calculated using Equation (8):  
\[DoLP=\frac{\sqrt{S_{1}^{2}+S_{2}^{2}}}{S_{0}} \tag{8}\]  
Further, \(I_{\text{max}}\) and \(I_{min}\), that is, the maximum and minimum light intensities, respectively, required for the polarization-based imaging model, can be calculated using Equation (9):  
\[\begin{array}{l}I_{\text{max}}=\frac{(1+DoLP)S_{0}}{2}\\ I_{\text{min}}=\frac{(1-DoLP)S_{0}}{2}\end{array} \tag{9}\]