## 3 Infrared and Visible Image Fusion
### Fusion Rules for Low-Frequency Subbands  
The low-frequency component of the image carries crucial information about the structure of the lining and the location, as well as the direction of crack expansion. When allocating weights to the background component, priority should be given to the infrared image in order to enhance the brightness of the lining area and improve the contrast of the cracks. Since the brightness of the crack location differs significantly from that of the background, the saliency of each pixel can be computed to predict the crack location. The saliency of a pixel is determined by taking the difference between the brightness of the pixel itself and the average brightness of its surrounding neighborhood, as calculated using Equation (10).  
\[S_{IR(VIS)}(x,y)=I_{IR(VIS)}(x,y)-\mu_{IR(VIS),\mu}(x,y) \tag{10}\]  
In the equation, \(I_{IR}\) and \(I_{VIS}\) represent the brightness values of the infrared and visible images, respectively, and \(\mu\) represents the average brightness within a neighborhood \(\delta\) of radius \(n\). When considering the background location, the difference between \(S_{IR}\) and \(S_{VIS}\) is relatively small. However, at the crack locations, \(S_{IR}\) is significantly larger than \(S_{VIS}\). Based on the analysis mentioned above, the low-frequency coefficients (\(L_{F}\)) obtained after fusion at each scale can be represented as follows:  
\[L_{F}(x,y)=\left\{\begin{aligned} & w_{VIS}L_{VIS}(x,y)+(1-w_{VIS})L_{ IR}(x,y)\ |S_{IR}(x,y)-S_{VIS}(x,y)|\geq S_{th}\ \text{and}\ S(x,y)\leq 0\\ & w_{IR}L_{IR}(x,y)+(1-w_{IR})L_{VIS}(x,y)\ |S_{IR}(x,y)-S_{VIS}(x,y)|<S_{th}\\ & 0.5[L_{IR}(x,y)+L_{VIS}(x,y)]\ \text{else}\end{aligned}\right. \tag{11}\]  
In this paper, the threshold value \(S_{th}\) is selected as \(5\sigma^{2}(\epsilon^{0.5}-1)\). Here, \(L_{IR}\) and \(L_{VIS}\) represent the low-frequency coefficients obtained from the DT-CWT decomposition of the infrared and visible images, respectively. \(w_{IR}\) and \(w_{VIS}\) denote the weight coefficients assigned to the two images, which are adjusted based on the difference in saliency. The adjustment is performed according to the equation expressed as Equation (12).  
\[\begin{aligned} & w_{VIS}{=}\log\left(1+\frac{|S_{IR}-S_{VIS}|}{2 \sigma_{VIS}^{2}}\right)\\ & w_{IR}=\exp\left(\frac{|S_{IR}-S_{VIS}|}{2\sigma_{IR}^{2}} \right)\end{aligned} \tag{12}\]  
where \(\sigma^{2}\) is the brightness variance in neighborhood \(\delta\). \(w_{VIS}\) is positively correlated with the saliency difference, meaning that visible images are assigned higher weights when the difference in saliency for a pixel exceeds the threshold \(S_{th}\) and the brightness is lower than the average value within the neighborhood. Conversely, \(w_{IR}\) is negatively correlated with the saliency difference, indicating that infrared images are assigned higher weights when the saliency difference is below the \(S_{th}\) threshold. In cases where neither condition is met, both infrared and visible images are assigned equal weights of 0.5 for the fusion process.  
Figure 3: The proposed method for infrared image and visible image fusion.