## 3 Algorithm Flow
### Infrared Polarization Image Fusion
#### 3.2.3 High- and Low-Frequency Components' Fusion of the Polarization Image  
Fusion rules are crucial for image fusion, with the selection principle being to retain useful information while avoiding useless information and enhancing the fusion effect through complementarity and redundancy between images. After obtaining the high- and low-frequency components of the polarization intensity image and the polarization feature image, different fusion rules are used to fuse the high- and low-frequency components of the two images. (1) Low-Frequency Fusion of a Polarized Image  
The low-frequency part of the image represents the main information component of the image, represents the main energy distribution in the image, and reflects the main features of the image. In this paper, the low-frequency fusion rule of the region coefficient difference combined with the average gradient grayscale stretch (RCD-AG-GS) is adopted. While preserving the image information to the maximum extent possible, the clarity of the image is further enhanced. Firstly, the low-frequency components of the two images are grayscale stretched to enhance the contrast of the image. Then, the low-frequency fusion coefficients are preliminarily confirmed by using the difference of regional coefficients. The difference of regional coefficients is the difference between the grayscale value of the pixel and the average energy of the region in which it is located. The greater the difference is, the more useful information the pixel has. Its expression is [23]:  
\[\left\|e_{i}(x,y)\right\|_{M\times N}=\sum\limits_{m^{\prime\prime}=(1-M)/2}^{ (M-1)/2}\sum\limits_{m^{\prime\prime}=(1-N)/2}^{(N-1)/2}\left|\left|f_{i}(x+m ^{\prime\prime},y+n^{\prime\prime})\right|-\overline{e}(x,y)_{M\times N}\right| \tag{9}\]  
\[\overline{e}(x,y)_{M\times N}=\frac{\sum\limits_{m^{\prime\prime}=-(M-1)/2}^{ (M-1)/2}\sum\limits_{m^{\prime\prime}=-(N-1)/2}^{(N-1)/2}\left|f_{i}(x+m^{ \prime\prime},y+n^{\prime\prime})\right|}{M\times N} \tag{10}\]  
Here, \(i\) represents the PI and PF images, \(e\) represents the low-frequency component, \(f\) represents the grayscale value of a pixel, and \(\overline{e}(x,y)_{M\times N}\) represents the mean of the absolute grayscale value of the low-frequency part in an \(M\times N\) slider area. However, if the region coefficient difference is only used for fusion, useless information will also be fused into the image, resulting in a lower fusion effect and blurring. Therefore, the clarity needs to be improved. The average gradient can well reflect the tiny details and texture characteristics of the image, as well as the clarity of the image. Therefore, it is combined with the region coefficient difference to achieve effective fusion of the low-frequency components. The expression of the average gradient is as follows:  
\[\
abla\overline{g}_{i}=\frac{1}{(M-1)\times(N-1)}\sum\limits_{i=1}^{M-1}\sum \limits_{j=1}^{N-1}\sqrt{\frac{[f(i,j)-f(i+1,j)]^{2}+[f(i,j)-f(i,j+1)]^{2}}{2}} \tag{11}\]  
Here, \(f(i,j)\) is the grayscale value of the \(i\)-th row and \(j\)-th column of the image; \(M\) and \(N\) are the total number of rows and columns of the image, respectively. When the difference of the regional coefficients of the low-frequency components of two images is not equal, the grayscale value of the one with a larger difference is taken to determine the low-frequency fusion coefficient. Meanwhile, when the difference of the regional coefficients of the low-frequency components of two images is equal, the grayscale value of the one with a larger average gradient is taken as the low-frequency fusion coefficient. By combining Equations (9)-(11), the specific selection rule of the low-frequency fusion coefficient of the fused image can be obtained, as shown in Equation (12):  
\[f_{i}(x,y)=\left\{