## 2 The Proposed Image Fusion Method
### Implementation of Fused Technique  
The scheme process of the proposed fusion method is discussed in detail, as shown in Algorithm 1. The fusion framework based on the NSCT domain includes image preprocessing and registration, decomposed acquisition of the low and high frequency sub-band image, coefficient selection of the sub-band image, reconstruction of the fused image, and quality evaluation. First, the adjacent sonar images are registered using the speed up robust features (SURF) [41] method to obtain the information \begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{2}{c}{**Objective Indexes**} & \multicolumn{2}{c}{**Mathematical Formulation**} \\ \hline \multirow{2}{*}{**AG**} & \(AG=\) \\  & \(\frac{1}{MCN}\sum\limits_{i=1}^{M}\sum\limits_{j=1}^{N}\sqrt{\left|\left[f(x,y) -f(x-1,y)\right]^{2}+\left|f(x,y)-f(x,y-1)\right|^{2}\right|}/2\) \\ \hline \multirow{2}{*}{**FD**} & \(\frac{FD}{MCN}\sum\limits_{x=1}^{M}\sum\limits_{y=1}^{N}\sqrt{\left|\left[f(x +1,y)-f(x,y)\right]^{2}+\left|f(x,y+1)-f(x,y)\right|^{2}\right|}/2\) \\ \hline \multirow{2}{*}{**E**} & E=\(-\sum\limits_{i=0}^{L-1}p_{i}\log_{2}p_{i}\) \\ \hline \multirow{2}{*}{**RCE**} & \(RCE=\sqrt{\left(CE_{A,F}^{2}+CE_{B,F}^{2}\right)/2}\), \(CE_{A/B,F}=\sum\limits_{i=0}^{L-1}p_{A/B}(i)\log_{2}\frac{p_{A/B}(i)}{p_{i}( i)}\) \\ \hline \multirow{2}{*}{**MI**} & \(MI=\sum\limits_{(a,b),f}p_{(A,B),F}(f,(a,b))\log_{2}\frac{p_{(A,B),F}(f,(a,b))} {p_{(F)}(f)(p_{A,B}(a,b))}\) \\ \hline \multirow{2}{*}{\(\mathbf{Q}^{AB/F}\)} & \(Q^{AB/F}=\frac{\sum_{m_{a}}\left(Q^{A}_{m_{a}}\mu^{A}_{a}+Q^{B}_{m_{a}}\mu^{B}_{a }\right)}{\sum_{m_{a}}\left(\mu^{A}_{a}+\mu^{B}_{a}\right)}\) \\ \hline \multirow{2}{*}{**SSIM**} & \(SSIM(A,B,F)=\left(SSIM(A,F)+SSIM(B,F)\right)/2\) \\  & \(SSIM(A/B,F)=\left(\frac{2\mu_{A/B}\mu+C_{1}}{\mu^{2}_{A/B}+C_{2}}\right)\left( \frac{2\mu_{A/B}\mu+C_{2}}{\mu^{2}_{A/B}+\mu^{2}_{1}+C_{1}}\right)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Objective index and mathematical description of the fused image. AG, average gradient; RCE, root mean square cross entropy; FD, figure definition; E, information entropy; MI, mutual information; Q\({}^{ABF}\), edge-based similarity measure; SSIM, structural similarity index. images of the overlapping area. NSCT is subsequently applied to decompose the pair of overlapping area images into low and high frequency sub-band images. Moreover, the sub-band coefficients of low and high frequency images are determined by the SMLF and IDPCNN model, respectively. However, after the above fusion criteria are applied, there are some random discrete or isolated pixels in the small neighborhood of sub-band coefficient matrices, which are obviously different from the adjacent pixel source images [42]. Therefore, consistency verification (CV) [43] is carried out through a window size of \(3\times 3\) to ensure the consistency relationship between adjacent coefficient sources. Finally, according to the selected sub-band images of low and high frequency, the fused image is reconstructed by inverse transformation of NSCT, and the effect of fused image is quantitatively analyzed and evaluated from subjective vision and multiple objective indexes. ```
0: Read in two sonar images of adjacent strip. Set some constant item values, \(Kn\), \(\alpha_{L}\), \(V_{L}\), \(\alpha_{\theta}\), \(V_{\theta}\), \(CR=1\). Initialize some parameters and matrices, \(Y_{ij}\) is zero matrix, \(\omega_{ij,pq}\) is ones matrix, the iteration number \(n=1\).