## 3 Method of Recognition  
### Hyperspectral Background Field Suppression Based on Cayley-Klein Measure  
Let \(\mathrm{I}\in\mathrm{R}^{\mathrm{d}_{1}\times\mathrm{d}_{2}\times\mathrm{d}_{3}}\) be a hyperspectral image, where \(\mathrm{I}\) refers to the original data, while \(\mathrm{d}_{1}\), \(\mathrm{d}_{2}\), and \(\mathrm{d}_{3}\) represent the height, width, and number of spectral bands of the hyperspectral image, respectively. Let \(\mathrm{x}\in\mathrm{R}^{\mathrm{d}_{3}}\) be the spectral vector corresponding to each pixel. A number (\(\mathrm{N}\)) of samples are randomly selected from the original hyperspectral image to comprise a dataset of \(\left\{\mathrm{x}^{\mathrm{k}}\right\}\subset\mathrm{I}\), where \(\mathrm{k}=1,2,\dots,\mathrm{N}\). Let \(\mathrm{m}\) and \(\sum\) represent the mean and covariance matrix of the dataset, respectively. In this study, given the extremely small proportion of the target of the UUV in the entire image, it can be approximated that \(\left\{\mathrm{x}^{\mathrm{k}}\right\}\) represents the background field dataset. To calculate the discriminability of any spectral vector and the background field, the Mahalanobis distance is taken as a measure of similarity [17]. The Mahalanobis distance is expressed as follows:  
\[d_{M}\big{(}\mathrm{x}_{o},\mathrm{x}_{p}\big{)}=\big{(}\mathrm{x}_{o}- \mathrm{x}_{p}\big{)}^{T}\sum\big{(}\mathrm{x}_{o}+\mathrm{x}_{p}\big{)} \tag{1}\]  
where \(\mathrm{d}_{M}\) represents the Mahalanobis distance measure, \(\mathrm{x}_{o}\) and \(\mathrm{x}_{p}\) are referred to as the two spectral vectors, and \(T\) denotes the transpose. The Mahalanobis distance plays a role in mitigating the impact of the background field, enhancing the discriminability between the target and the background field, and improving the accuracy of target detection. In addition, the Mahalanobis distance can be used to transform the high-dimensional space in an image into a low-dimensional space, thus improving the efficiency of data processing. The calculation of the Mahalanobis distance requires the inversion of the covariance matrix, which is in essence a linear measure. However, hyperspectral image data usually have high dimensionality, and the estimated covariance matrix is ill-conditioned in high-dimensional space, thus making its inversion unstable or impractical. Previous studies show that the Cayley-Klein metric matrix is capable to approximate the Mahalanobis distance in extreme cases [18]. Thus, this study introduces the Cayley-Klein measure to define the pixel similarity of the hyperspectral image. In order to obtain the Cayley-Klein measure, \(\mathrm{m}\) and \(\sum\) are used in this study to define two invertible symmetric matrices \(\mathrm{G}^{\pm}\) as follows:  
\[\mathbf{G}^{\pm}= \begin{pmatrix}\sum&-\sum\mathbf{m}\\ -\mathbf{m}^{T}\sum&\mathbf{m}^{T}\sum\mathbf{m}\pm k^{2}\end{pmatrix} (k>0) \tag{2}\]  
where \(\mathrm{G}^{+}\) and \(\mathrm{G}^{-}\) denote positive definite matrix and positive indefinite matrix, respectively. The bilinear form of \(\mathrm{G}^{\pm}\) is expressed as follows:  
\[\sigma^{\pm}\big{(}\mathrm{x}_{o},\mathrm{x}_{p}\big{)}=\big{(}\mathrm{x}_{o} ^{T},1\big{)}\mathbf{G}^{\pm} \begin{pmatrix}\mathrm{x}_{p}\\ 1\end{pmatrix}