## 3 Method of Recognition
### Local Peak Detection Algorithm for Target Points  
Underwater robots and water bodies exhibit different features in hyperspectral imaging, which means local peaks appear in some particular regions. Allowing for this, a local peak detection algorithm is applied to screen out the pixels that may belong to the underwater autonomous vehicle. Let \(C\in R^{d_{1}\times d_{2}\times 1}\) be the Cayley-Klein measurement matrix corresponding to the hyperspectral image \(I\in R^{d_{1}\times d_{2}\times d_{3}}\). For any point of \((x,y)\) in the measurement matrix, the measurement of \(C(x,y)\) can be decomposed into:  
\[C(x,y)=c_{t}(x,y)+c_{b}(x,y)+\varepsilon(x,y) \tag{7}\]  
where \(c_{t}\), \(c_{b}\), and \(\varepsilon\) represent the target component, background component and noise, respectively. Assuming that there is a spectral difference between the UUV and the water, and there is supposed to be a local peak in the corresponding Cayley-Klein measurement, which is similar to the features of the point spread function (SPF), as can be described by a two-dimensional Gaussian function. For the local bright spot-shaped targets, the gray values in the locality are expected to show a peak that can be roughly described by  
Figure 2: (**a**) The pseudo-color image corresponding to the hyperspectral image, where the red box indicates the underwater target region; (**b**–**d**) show the results of \(d_{E}^{global}\), \(d_{E}^{local}\), and \(d_{F}\), respectively; (**e**–**g**) show the 3D views of \(d_{E}^{global}\), \(d_{E}^{local}\), and \(d_{F}\), respectively. a two-dimensional Gaussian distribution. That is, the target component of \(\rm{c_{t}(x,y)}\) can be approximately expressed as:  
\[f_{t}(x,y)=\left\{ \begin{array}{ll}I_{i}\exp\{-\frac{1}{2}\left[\frac{(x-x_{ ai})^{2}}{\sigma_{ii}^{2}}+\begin{array}{cc}\frac{(y-y_{ai})^{2}}{\sigma_{2i}^{2}} \end{array}\right]\}&,(x,y)\in\Omega_{i}\\ 0&,(x,y)\
otin\Omega_{i}\end{array} \right. \tag{8}\]  
In the above equation, \((\Omega=\Omega_{1}\cup\Omega_{2}\cup\Omega_{3}\cup\cdots\cdots\cup\Omega_{n})\), where \(\Omega_{i}\) represents the \(i\)-th target area, n indicates the number of target areas, and \(\rm{I_{i}}\) denotes the Cayley-Klein measurement of the i-th target. For those small targets, their small size can be expressed as any \(\Delta x^{2}+\Delta y^{2}=\rm{w^{2}}\), satisfying \(\left(x_{oi}+\Delta x,y_{oi}+\Delta y\right)\
otin\Omega_{i}\), where w represents a slightly larger value than the target size. By taken account into Equation (6), it can be known that:  
\[F(x_{ai},y_{ai})-F(x_{ai}+\Delta x,y_{ai}+\Delta y)= \tag{9}\] \[f_{t}(x_{oi},y_{oi})+[f_{b}(x_{oi},y_{oi})-f_{b}(x_{oi}+\ \Delta x,y_{oi}+\Delta y)]+\] \[[n(x_{oi},y_{oi})-n(x_{oi}+\ \Delta x,y_{oi}+\Delta y)], \Delta x^{2}+\Delta y^{2}=w^{2}\]  
When the background changes at a relatively slow pace and the intensity of the noise is much lower than that of the target, the last two terms on the right-hand side of Equation (7) can be ignored, which means:  
\[F(x_{ai},y_{ai})-F(x_{ai}+\Delta x\cdot y_{ai}+\Delta y)=f_{t}(x_{oi},y_{oi}) =I_{i} \tag{10}\]  
Notably, the intensity level of noise must be lower compared with the target to satisfy the requirements of Equation (8). In case of salt-and-pepper noise, additional processing is required to apply the equation, which will be explained later. The target must have a certain intensity, that is, \(\rm{I_{i}}>T\), where T indicates a fixed threshold value. Therefore, the discriminant formula used for the selection of the peak features is expressed as:  
\[F(x_{oi},y_{oi})-F(x_{oi}+\Delta x,y_{oi}+\Delta y)>T \tag{11}\]  
In practice, noise is inevitably present in the Cayley-Klein measures, which causes the frequent deviations of local peaks. To reduce the impact of noise on local peak detection, it is proposed in this paper to denoise the Cayley-Klein measures through median filtering. Figure 3 shows the local peak detection algorithm intended for the target points.