## 2 The Proposed Image Fusion Method
### DPCNN and Its Improved Model
#### Improved DPCNN Model  
(1) Theory of improved DPCNN model  
By analyzing the mechanism of the DPCNN model, it can be found that the setting of some default constants is very skillful for activating all neurons [33], which often affects the fusion performance and running time, such as maximum iteration \(K\!n\), \(\alpha_{\theta}\), \(V_{\theta}\), and so on. Therefore, an improved DPCNN (IDPCNN) model is proposed to select the coefficients of high frequency sub-band images. The improved pulse activation process mainly occurs in the information judgment and pulse activation  
Figure 2: DPCNN model. Multiplication, summation, and difference between two sides are represented by symbols \(\otimes\), \(\oplus\), and \(\ominus\), respectively.  
domain, as shown in Figure 3. The amplitude value \(Re_{ij}\) (k = 1, 2) is calculated by the soft limit function [34], which is in conjunction with the dynamic threshold to determine whether the pulse is activated and record each ignition time. After \(Kn\) iterations, the channel coefficients corresponding to the larger cumulative output time \(\sum T_{ij}^{k}\) are served as the fused high frequency sub-band coefficients.  
\[\begin{split}& Re_{ij}^{k}(n)=1/\left(1+e^{(\theta_{ij}(n-1)-U_{ij} ^{k}(n))}\right)\,\ \ \ k=\{1,2\}\\ & Y_{ij}(n)=\begin{cases}1&\max\{Re_{ij}^{k}(n)\}\geq 0.5\\ 0&\text{others}\\ \end{cases}\\ & T_{ij}^{k}(n)=\begin{split}& Re_{ij}^{k}(n)Y_{ij}(n),\ \ \ k=\{1,2\}\\ \end{split}\end{split} \tag{8}\]  
where \(T_{ij}^{k}\) denotes as the pulse ignition time.  
(2)Key parameter settings of IDPCNN model  
In the traditional DPCNN model, the parameters mainly depend on experience and massive attempts [32]. A set of parameters that achieves good performance may be unsuitable for other data. Therefore, key points in the IDPCNN model are as follows for the image fusion of side-scan sonar, 1) pulse ignition time \(T_{ij}^{k}\); 2) link strength \(\beta_{ij}^{k}\); and 3) external excitation \(S_{ij}^{k}\).  
In this IDPCNN model, the soft limiting function is applied to reflect the amplitude difference of the total ignition time (see Equation (9)). Moreover, considering that local gradient energy is also one of image sharpness indicators, which can reflect the target edge and other detailed features, average gradient [35] is adapted to the characterization of link strength. The average gradient \(\overline{d}(x,y)\) is defined as follows:  
\[\begin{split}&\overline{d}(x,y)=\frac{1}{\phi}\sum\limits_{p=-1 }^{1}\sum\limits_{q=-1}^{1}sqrt([g_{1}(x+p,y+q)+g_{2}((x+p,y+q))]/2)\\ &\left\{\begin{array}{l}g_{1}(x,y)=[d(x,y)-d(x+1,y)]^{2}\\ g_{2}(x,y)=[d(x,y)-d(x,y+1)]^{2}\\ \end{array}\right.\end{split} \tag{9}\]  
where \(d\) (\(x,y\)) is the high frequency sub-band coefficient. \(g_{1}\) (\(x,y\)) and \(g_{2}\) (\(x,y\)) represent the gradient difference in the horizontal and vertical directions, respectively.  
Figure 3: Improved DPCNN model.  
Moreover, a novel sum-modified Laplacian (NSML) is set as external excitation, and denotes as:  
\[\begin{array}{rl}nml(x,y)=&\left|2\times d(x,y)-d(x-1,y)-d(x+1,y)\right|\\ &+\left|2\times d(x,y)-d(x,y-1)-d(x,y-1)\right|\\ &+\sqrt{2}\times d(x,y)-\left(\sqrt{2}/2\right)\times d(x-1,y-1)-\left(\sqrt{ 2}/2\right)\times d(x+1,y+1)\\ &+\sqrt{2}\times d(x,y)-\left(\sqrt{2}/2\right)\times d(x-1,y+1)-\left(\sqrt{ 2}/2\right)\times d(x+1,y-1)\right|\\ &NSML(x,y)=\sum\limits_{p=-1}^{1}\sum\limits_{q=-1}^{1}nml(x+p,y+q)^{2}\end{array} \tag{10}\]