## 2 ROI Extraction
### The Proposed Saliency Map Detection Method
The spatial correlation of the region x to the domain X\({}^{\prime}\) for the pixel \((\mathrm{i,j})\) is defined as follows:  
\[\mathrm{Correlation}(i,j)=\frac{\mathrm{cov(x,x^{\prime})}}{\sigma_{\mathrm{x }}*\sigma_{\mathrm{x^{\prime}}}}=\frac{\mathrm{E(x\odot x^{\prime})-E(x)E(x^{ \prime})}}{\sqrt{E(x)-E^{2}(x)}\sqrt{E(x^{\prime})-E^{2}(x^{\prime})}} \tag{13}\]where \(x\) and \(X^{\prime}\) are the square regions centered on the pixel \(p(i,j)\) and \(p(i,j+N)\), respectively. I and \(j\) are the row and column coordinates of the image. N is set to 5 in this paper. \(x\) and \(X^{\prime}\) are areas represented by the black and blue borders in Figure 4. The sizes of them are both \(k\times k\) pixels.\(\alpha_{x}\) and \(\sigma_{x^{\prime}}\) are standard deviations of the corresponding areas. \(E(x)\) and \(E(x^{\prime})\) are the mean value of the region \(x\) and \(X^{\prime}\), respectively. \(x\odot x^{\prime}\) represents the dot product of the gray value in the corresponding position of the two regions, and the size of \(x\odot x^{\prime}\) is \(k\times k\) pixels. \(E(x\odot x^{\prime})\) is the mean value of the \(x\odot x^{\prime}\). We then calculate the unified LBP [17] to reflect texture properties of points, line corners, and flat areas in the image. \[ \begin{split} LBP_{N,R}^{tria2}=\left\{\begin{array}{ll}\sum \limits_{n=0}^{N-1}s(g_{n}-g_{c})&if(U(LBP_{N,R})<=2)\\ {N+1}&otherwise\end{array}\right.\end{split}  \tag{14}\]  
where \(s(. )\) is a binary symbol, \(s(x)=1,x>0\), and \(s(x)=0,x\leq 0\). \(U(LBP_{N,R})\) represents the number of exchanges between 0 and 1 on the N-bit binary number [17], \(g_{c}\) and \(g_{n}\) are the gray values of the central and the domain pixel, respectively. It was found that the LBP value of the background was higher than the target. The LBP map is processed as follows. \[ \begin{split} LBP(i,j)_{coeff}=\left|\frac{\overline{LBP}- LBP(i,j)}{\overline{LBP}}\right|\end{split}  \tag{15}\]  
where \(\overline{LBP}\) is the mean value of the LBP map, and \(|.|\) represents the absolute value. The final fusion feature is obtained by multiplying the correlation value \(Correlation(i,j)\) and the coefficient in the LBP\({}_{coeff}(i,j)\) in the corresponding position:  
\[ \begin{split} Fusion_{mup}=Correlation(i,j)\odot LBP_{coeff}(i, j)\end{split}  \tag{16}\]  
Through the above steps, we completed the construction of the feature maps. \(f_{0}\)- \(f_{3}\) in Equation (1) can be assigned: \(f_{0}\) is the original image, \(f_{1}\) is a contrast map by Equation (11), \(f_{2}\) is a Gabor transform map based on Equation (12), and \(f_{3}\) is a blend feature map combining spatial correlation with the texture feature based on Equation (16). Therefore, the quaternion is \(q(x,y)=f_{0}+f_{1}\mu_{1}+f_{2}\mu_{2}+f_{3}\mu_{3}\). \(Q[u,v]\) is obtained by performing the hyper-complex Fourier transform in Equation (6). The amplitude spectrum \(A(\ u,v\ )\) and phase spectrum \(P(\ u,v\ )\) of Fourier transform is calculated by Equations (7) and (8). At the same time, in order to enhance the significant part, the Gaussian kernel function with different scales is used to obtain the scale space of the amplitude spectrum. \[\overline{A_{i}(u,v)}=g_{i}(u,v)*A(u,v) \tag{17}\]  
\[g_{i}(u,v)= \frac{1}{\sqrt{2\pi}2^{k-1}t_{0}}e^{-\frac{q^{2}+v^{2}}{2^{2k-1}t_{0}^{ 2}}} \tag{18}\]  
Figure 4: Schematic diagram of the spatial correlation regions. where k is a spatial scale parameter, k = 1,..., K, \(K=\left[\log_{2}\min\{H,W\}\right]+1\), and H and W are the height and width of the image, respectively, \(t_{0}=0.5\). We built a series of saliency maps at different scales as follows:  
\[S_{\mathrm{k}}=g*\left\|Q^{-1}\left\{\overline{A}_{i}(u,v)e^{XP(u,v)}\right\} \right\|^{2} \tag{19}\]  
The best saliency map is selected by methods according to the theory of minimum information entropy, and other saliency maps are discarded [26]. However, the discarded saliency maps also contained significant information at different scales for differently sized ships.