# Underwater Degraded Image Restoration by Joint Evaluation and Polarization Partition Fusion
## 2 Underwater Degraded Image Restoration Methods
### Underwater Polarization Image Restoration Model
Using the multi-channel real-time polarization detection system to obtain \(0^{\circ}\), \(45^{\circ}\), \(90^{\circ}\), \(135^{\circ}\), four different polarization angle images combined with the Stokes vector can be solved for \(I\), \(Q\), \(U\). Substituting these into Formula (7), the different polarization angle can be found when the light intensity image is \(I(\alpha)\). The two polarization images with the maximum and minimum light intensity can be obtained by following \(I(\alpha)\), and their polarization angles are orthogonal, which are written as \(I_{max}(x,y)\) and \(I_{min}(x,y)\), according to the intensity, respectively, as shown in Figure 1b. Combined with Equation (1), the two polarized images with the maximum and minimum light intensities can also be represented by backscattered light and light reflected from the target:  
\[I_{max}(x,y)=S_{max}(x,y)+B_{max}(x,y) \tag{8}\]  
\[I_{min}(x,y)=S_{min}(x,y)+B_{min}(x,y) \tag{9}\]  
\(S_{max}(x,y)\) and \(B_{max}(x,y)\) in Equation (8) are the target reflected and backscattered light at the maximum light intensity, respectively. Similarly, \(S_{min}(x,y)\) and \(B_{min}(x,y)\) in Equation (9) are the target reflected light and backward-scattered light when the light intensity is at the minimum, respectively. Based on the definition of polarizability, the polarizability of backscattered light \(P_{scat}\) can be derived as follows:  
\[I=I(0^{\circ},\ 0)+I(90^{\circ},\ 0)\]0  
Similarly, the target light polarization \(P_{tar}\) can be expressed as Equation (11) as follows:  
\[I=I(0^{\circ},\ 0)+I(90^{\circ},\ 0)\]1Associating Equations (8) and (9) yields the total light intensity of the image expressed as \(I_{total}\), and the coordinate quantity \((x,y)\) is omitted for the sake of concise representation:  
\[I=I(0^{\circ},\ 0)+I(90^{\circ},\ 0)\]2  
The association of Equations (8)-(11) yields:  
\[I=I(0^{\circ},\ 0)+I(90^{\circ},\ 0)\]3  
Therefore, according to Equations (12) and (13), we can solve the target reflected light \(S\) and backscattered light \(B\):  
\[I=I(0^{\circ},\ 0)+I(90^{\circ},\ 0)\]4  
\[I=I(0^{\circ},\ 0)+I(90^{\circ},\ 0)\]5  
After the above derivation, according to Equations (14) and (15), based on the original information of the underwater polarization image, the recovered image can be derived by solving \(P_{tar}\), \(P_{scat}\), \(I_{min}\), \(I_{max}\). We aim to recover the overall region efficiently when the target polarization characteristics are complex. This paper further builds on reference [16]. First, we use histogram stretching methods for preprocessing two polarized orthogonal images, this increases the image contrast and enhances the image detail information. Then, the target is partitioned according to the values of each pixel point of the polarization image, and the low and high polarization target regions are extracted based on the polarization values. To address the practical problem, the low polarization region is recovered using the polarization difference method, and the high polarization region is recovered using the joint estimation of multiple optimization metrics. Finally, the low polarization and the high polarization regions are fused, as shown in \[I=I(0^{\circ},\ 0)+I(90^{\circ},\ 0)\]6