## 3 Infrared and Visible Image Fusion
### DT-CWT Multiscale Decomposition  
The dual-tree-complex wavelet transform (DT-CWT) is an improved method of discrete wavelet transform (DWT) that solves the frequency aliasing problem of the DWT and has translation invariance. The DT-CWT is constructed by two juxtaposed DWTs, realizing a low-pass filter and a high-pass filter. The two wavelet functions \(\psi_{h}(t)\) and \(\psi_{g}(t)\) areused as the real and imaginary parts of the complex wavelet, respectively, as expressed in Equation (6).  
\[\psi(t)=\psi_{h}(t)+\mathrm{i}\psi_{g}(t) \tag{6}\]  
When transforming the original signal \(s(t)\), the wavelet coefficients \(d\) and scale coefficients \(c\) in the real and imaginary parts are calculated by Equation (7) and Equation (8), respectively.  
\[\begin{array}{c}d_{\mathrm{Re}}^{j}=2^{j/2}\int_{-\infty}^{+\infty}s(t)\psi_{ h}(2^{j}t-k)\mathrm{d}t\\ d_{\mathrm{Im}}^{j}=2^{j/2}\int_{-\infty}^{+\infty}s(t)\psi_{g}(2^{j}t-k) \mathrm{d}t\end{array} \tag{7}\]  
\[\begin{array}{c}c_{\mathrm{Re}}^{l}(k)=2^{l/2}\int_{-\infty}^{+\infty}s(t) \varphi_{h}(2^{l}t-k)\mathrm{d}t\\ c_{\mathrm{Im}}^{l}(k)=2^{l/2}\int_{-\infty}^{+\infty}s(t)\varphi_{g}(2^{l}t-k )\mathrm{d}t\end{array} \tag{8}\]  
where \(j\) is the scale factor, \(j\) is the number of decomposition layers, and \(k\) is the filter length. \(\varphi_{h}\) and \(\varphi_{g}\) are the scale functions of the real and imaginary trees, respectively. The scale function in the context of wavelet analysis represents the low-frequency components of the original signal, while the wavelet function captures the high-frequency components. By performing multiscale decomposition on the signal, the scale coefficients from each scale form the low-frequency subbands, while the wavelet coefficients form the high-frequency subbands. These coefficients are computed and combined to yield the wavelet coefficients \(d\) and scale coefficients \(c\) required for signal reconstruction. The reconstructed signal can be mathematically expressed using Equation (9).  
\[\hat{s}(t)=\sum_{j=1}^{J}d^{j}+\hat{c}^{j} \tag{9}\]  
The visible and infrared images are independently decomposed using DT-CWT to obtain their respective low-frequency and high-frequency components at different scales. Typically, the low-frequency component of an image contains the primary target information, while the high-frequency component captures the edge and texture details of the target. However, in the case of a crack target image, the low-frequency component primarily represents the background information, while the high-frequency component contains both the target (crack) and background texture information, owing to the small scale of the crack. The presence of high-contrast texture details can potentially lead to misclassification by the semantic segmentation network.  
Hence, a crucial objective in image fusion is to enhance the crack features and suppress the background texture. In this research paper, multiscale fusion rules are specifically designed based on the characteristics of crack images, aiming to preserve the high contrast from the infrared images and the high resolution from the visible images. The fusion process is visually illustrated in Figure 3.