## 3 Algorithm Flow
### Infrared Polarization Image Fusion
#### 3.2.3 High- and Low-Frequency Components' Fusion of the Polarization Image
\begin{array}{c}f_{Pl}(x,y),\ \|e_{Pl}(x,y)\|_{M\times N}>\|e_{ pF}(x,y)\|_{M\times N}\\ f_{PF}(x,y),\ \|e_{Pl}(x,y)\|_{M\times N}<\|e_{PF}(x,y)\|_{M\times N}\\ f_{Pl}(x,y),\ \|e_{Pl}(x,y)\|_{M\times N}=\|e_{PF}(x,y)\|_{M\times N^{\prime}}\| \
abla\overline{g}_{Pl}\|_{M\times N}\geq\|\
abla\overline{g}_{PF}\|_{M\times N }\\ f_{PF}(x,y),\ \|e_{Pl}(x,y)\|_{M\times N}=\|e_{PF}(x,y)\|_{M\times N}\|\
abla \overline{g}_{Pl}\|_{M\times N}<\|\
abla\overline{g}_{PF}\|_{M\times N}\\ \end{array} \right. \tag{12}\]  
2. High-Frequency Fusion of a Polarized Image  
The high-frequency component of an image generally reflects the part of the image with a sharp grayscale transformation, contains a large number of image details and characteristic information, and can distinguish the object and scene in the image well. The high-frequency part after wavelet decomposition is divided into vertical, horizontal, and diagonal components. In this paper, the fusion rules of taking the maximum energy of the weighted region are used to fuse the three components. This method can effectively highlight the image texture and detailed information. The energy algorithm of an image calculates the grayscale value of an image or a certain region, while the energy of a region is found from calculating the grayscale value of a single pixel and its neighborhood within the range. The square neighborhood (\(x-k,y-k,x+k,y+k\)) of a pixel (\(x\),\(y\)) is defined as the region \(\Omega\), then the energy of the region can be expressed as:  
\[E(x,y)=\sum\limits_{m}\sum\limits_{n}\Omega(h,l)f^{2}(h,l) \tag{13}\]  
Here, \((h,l)\)\(\in\)\(\Omega\), \(\Omega(h,l)\) is the weighted value, taking the pixel point (\(x\),\(y\)) as the center, and the weight gradually decreases away from the center. \(f(h,l)\) is the grayscale value at (\(h,l\)). The regional energy matching degree is defined as:  
\[M_{PIF}(x,y)=\frac{2\sum\limits_{m}\sum\limits_{n}\Omega(h,l)f_{Pl}(h,l)f_{ PF}(h,l)}{E_{PI}(x,y)+E_{PF}(x,y)} \tag{14}\]  
Here, the value range of \(M_{PIF}(x,y)\) is [0,1], and the higher the value is, the higher the regional energy matching degree is and the smaller the energy difference is. The matching degree threshold is set as \(\delta\). When \(M_{PIF}(x,y)\)\(<\)\(\delta\), the grayscale value of the high-frequency component of the polarization intensity image and polarization characteristic image with larger regional energy is taken as the high-frequency fusion coefficient. When \(M_{PIF}(x,y)\)\(\geq\)\(\delta\), the weighted fusion is carried out according to the size of the regional energy and the size of the regional energy matching degree, and the obtained value is taken as the high-frequency fusion coefficient. In summary, by combining Equations (13) and (14), the specific selection rule of the high-frequency fusion coefficient of the fused image can be obtained as shown in Equation (15):  
\[F_{i}(x,y)= \begin{cases}f_{PI}(x,y),&M_{PIF}(x,y)<\delta,E_{PI}(x,y)\geq E_{PF}(x,y)\\ f_{PF}(x,y),&M_{PIF}(x,y)<\delta,E_{PI}(x,y)<E_{PF}(x,y)\\ (1-M_{PIF}(x,y))f_{PI}(x,y)+M_{PIF}(x,y)f_{PF}(x,y),&M_{PIF}(x,y)\geq\delta,E_{ PI}(x,y)<E_{PF}(x,y)\\ M_{PIF}(x,y)f_{PI}(x,y)+(1-M_{PIF}(x,y))f_{PF}(x,y),&M_{PIF}(x,y)\geq\delta,E_{ PI}(x,y)\geq E_{PF}(x,y)\end{cases}  \tag{15}\]