## 3 Weighted Wavelet Fusion Approach
### Weighted Wavelet Fusion Based on Structure-Aware Sobel Operator
\begin{array}{l}\lambda_{H1}=\frac{G_{HH}}{G_{HH1}+G_{HH2}}\\ \\ \lambda_{V1}=\frac{G_{VH1}}{G_{VH1}+G_{VH2}}\\ \\ \lambda_{D1}=\frac{G_{DH1}}{G_{DH1}+G_{DH2}}\end{array} \right. \tag{7}\]  
where \(G_{VH},G_{HH},G_{DH}\) denote the Sobel operator for the vertical high-frequency component, the Sobel operator for the horizontal high-frequency component, and the Sobel operator for the diagonal component, respectively. \(\lambda_{H},\lambda_{V},\lambda_{D}\) denote the weight coefficients of the horizontal, vertical, and diagonal high-frequency components, respectively. The weighting coefficients for the high-frequency sub-bands (detail coefficients) are calculated to enable frequency-adaptive fusion. These coefficients simultaneously govern the integration of both high-frequency texture details and low-frequency structural information during reconstruction. For the low-frequency sub-band (approximation coefficients), weights are assigned proportional to the regional energy content \(E\), ensuring the preferential retention of illumination-stable structural profiles critical for underwater infrastructure assessment. The energy metric \(E\) for an image region is defined as follows:  
\[E=\sum_{x=1}^{M}\sum_{y=1}^{N}I(x,y)^{2} \tag{8}\]  
where \(I(x,y)\) is the pixel value of the image in the \(y\) column of the \(x\) row, and \(M\) and \(N\) are the height and width of the image, respectively. The fusion weights of the low-frequency components are denoted as follows:  
\[\lambda A_{1}=E_{1}/(E_{1}+E_{2}) \tag{9}\]  
where \(\lambda A_{1}\) denotes the fusion weights of the low-frequency components of the image, and \(E_{1}\) and \(E_{2}\) are the energy values of the image computed by Equation (8). The fused low-frequency component \(cA_{fused}\) is denoted as follows:  
\[cA_{fused}=\lambda A_{1}\cdot cA_{1}+(1-\lambda\text{A}_{1})\cdot cA_{2} \tag{10}\]  
where \(cA_{fused}\)\(cA_{1}\), and \(cA_{2}\) are the low-frequency components obtained from the wavelet decomposition of the two input images, respectively. The horizontal, vertical, and diagonal high-frequency components of the fusion, \(cH_{fused},cV_{fused},cD_{fused}\), are denoted as follows:  
\[\left\{ \begin{array}{l}cH_{fused}=\lambda_{H1}\cdot cH_{1}+(1-\lambda_{H1} )\cdot cH_{2}\\ cV_{fused}=\lambda_{V1}\cdot cV_{1}+(1-\lambda_{V1})\cdot cV_{2}\\ cD_{fused}=\lambda_{D1}\cdot cD_{1}+(1-\lambda_{D1})\cdot cD_{2}\end{array} \right. \tag{11}\]  
where \(cH,cV,cD\) denote the horizontal, vertical, and diagonal high-frequency components, respectively; and \(\lambda_{H},\lambda_{V},\lambda_{D}\) denote the horizontal, vertical, and diagonal high-frequency component weighting coefficients, respectively. Finally, the high- and low-frequency component images obtained above are wavelet-inverted to reconstruct the final fused image \(I_{fused}\):  
\[I_{fused}=IDWT(cA_{fused},cH_{fused},cV_{fused},cD_{fused}) \tag{12}\]  
where \(IDWT\) is the wavelet inversion function, which is used to fuse the high- and low-frequency components of the image. The low-frequency sub-bands are fused with the maximum-energy rule. For every location, the coefficient whose energy is larger is retained, because a higher low-band energy more completely conveys the scene's overall luminance distribution and dominant contours. For the high-frequency sub-bands, fusion weights are derived from the Sobel gradient magnitude. The Sobel operator estimates the local edge strength; a larger response implies richer detail and texture, so its normalized value is used as the weight assigned to that coefficient set. Consequently, sub-bands with stronger Sobel responses contribute more to the final reconstruction.