## 3 Method of Recognition
### Pseudo-Target Criterion Based on Shape Prior Information
Since the gravitation in the local neighborhood is calculated, there is no significant difference in the distance between the data points and the neighbors, so Equation (12) can be simplified as  
\[\textbf{F}_{ij}=Gm_{i}m_{j}\textbf{D}_{ij} \tag{13}\]  
The Local Resultant Force (LRF) on \(\textbf{X}_{i}\) produced by its k nearest neighbor points can be simplified as:  
\[\textbf{LRF}(i,k)=\sum\limits_{j=1}^{k}\textbf{F}_{ij}=Gm_{i}\sum\limits_{j=1 }^{k}m_{j}\textbf{D}_{ij} \tag{14}\]  
where k represents the number of the nearest neighbor points of \(\textbf{X}_{i}\). According to Equation (14), the greater the mass of \(\textbf{X}_{i}\), the greater the influence on the neighborhood. Similarly, the smaller the mass, the more susceptible it is to the influence of the nearest neighbor points. Therefore, the **LRF** can be simplified as:  
\[\textbf{LRF}(i,k)=\frac{1}{m_{i}}\sum\limits_{j=1}^{k}\textbf{D}_{ij} \tag{15}\] \[\text{s.t. }m_{i}=1/\sum\limits_{j=1}^{k}d_{ij}\]  
On the basis of the above equations, the LGM clustering process includes the following: Firstly, the concentration of the data points is calculated using the LGM model, and the distance between each data point and the points with high concentration is determined based on the concentration. Secondly, the data points with high concentration and high distance values are selected as the clustering centers. Finally, the rest of the data points are allocated based on the principle that the concentration of the points inside the cluster is much higher than that of the boundary points, and the parameters are automatically adjusted by balancing the K-Nearest Neighbor. The area of the homogeneous region at a point is denoted as \(\textbf{A}_{i}\), which is supposed to satisfy the following conditions:  
\[T_{A}^{1}<A_{i}<T_{A}^{2} \tag{16}\]  
where \(\textbf{T}_{A}^{1}\) and \(\textbf{T}_{A}^{2}\) represent the lower and upper limits of the area threshold, respectively. Given a homogeneous region, the length of \(\textbf{w}_{1}\) and width of \(\textbf{w}_{2}\) of its minimum bounding rectangle can be obtained. Due to the impact of various factors, there may be some true target areas missing from the homogeneous regions, but the proportion of true extractions is supposed to exceed a certain limit of \(\textbf{T}_{\text{h}}\). \[\frac{A_{i}}{w_{1}\times w_{2}}>T_{h} \tag{17}\]  
Since the underwater targets, which are treated as the focus of this study are squares, the aspect ratio of the minimum bounding rectangle should be close to 1. However, due to the inability to extract the true target area completely, the threshold \(\textbf{T}_{\text{l}}\) of the aspect ratio can be appropriately relaxed. \[\frac{w_{1}}{w_{2}}<T_{l} \tag{18}\]Based on the prior experience, the value of \(\rm T_{A}^{1}\) and \(\rm T_{A}^{2}\) are set to 30 and 200, respectively; \(\rm T_{h}\) and \(\rm T_{l}\) are set to 0.4 and 1.6, respectively. When the conditions as set out in Equations (16)-(18) are not satisfied in the homogeneous region at a point, this point is treated as a pseudo-target point. The threshold values depend on the spatial resolution of the hyperspectral image. To make the threshold more applicable, we can apply the spatial resolution of the image used in this study as a reference, and the ratio of the resolution of new images to the reference image resolution can be used as a scaling factor for the threshold. Figure 4 lists the shape information of the candidate target points in the three situations: Situation 1 indicates that although the area of the homogeneous region of the candidate target point meets the requirements of Equation (16), its aspect ratio is higher than \(\rm T_{l}\), indicating that the candidate point is more inclined to linear features (such as cables), which is not consistent with the shape of the underwater target in our work.