## 2 Morphological Reconstruction and Multi-feature Analysis for TIR Ship Images
### Ship Target Verification Based on Contour Description and Shape Constraint  
#### 2.4.1 Contour Description of TIR Ship Based on Eigenvalue Analysis of Structure Tensor  
For TIR remote sensing images, the pixel intensities of small ship targets tend to be evenly distributed, they have an appearance of high uniformity and will not show large gray scale changes. Therefore, after GMR-based saliency map fusion, the contour information of ship targets can be completely retained. Recall that the contours of ship target in the OGMR and CGMR preprocessed images, as shown in Figure 5d-f, are depicted as ridges, like a simplified level-set model. Meanwhile, inspired by the successful applications of closed contour extraction, such as Chan-Vese model in ship detection [20; 21], a novel contour measure based on GMR and eigenvalue analysis of structure tensor is presented to characterize candidate ship targets and eliminate residual clutter simultaneously. Structure tensor has become a powerful tool for edge analysis, and achieved some impressive results for its good perceptivity on the dominant direction at the local image [39; 40]. To investigate the local contour information of TIR ship targets, the structure tensor \(ST\) is firstly calculated as:  
\[ST(x,y)=G_{\sigma}*\left\{\
abla u(x,y)\times[\
abla u(x,y)]^{T}\right\}=G_{ \sigma}*\left\{ \begin{array}{cc}\left(\frac{\partial u}{\partial x}\right)^{ 2}&\frac{\partial^{2}u}{\partial x\partial y}\\ \frac{\partial^{2}u}{\partial x\partial y}&\left(\frac{\partial u}{\partial y }\right)\end{array} \right\}=\left[ \begin{array}{cc}ST_{11}&ST_{12}\\ ST_{21}&ST_{22}\end{array} \right], \tag{21}\]  
where \(G_{\sigma}\) is a Gaussian kernel with variance \(\sigma\) and \(*\) is a convolution operator, \(u\) denotes pre-processed infrared ship image after OGMR or CGMR operation. The Gaussian kernel can be considered as  
Figure 8: The final results of saliency map fusion. (**a**–**c**) are the final results of saliency map fusion of Figure 5a–**c**, (**d**–**f**) are their corresponding 3-D mesh plots, (**g**–**i**) the binary segmentation results of their final saliency maps. a window of \(ST\) to perceive the local information and \(\sigma\) determines the size of the window. The \(\sigma\) is empirically set to 2. Then the normalized large eigenvalue of matrix \(ST\) is obtained:  
\[\lambda_{\text{large}}(x,y)=\mathbf{N}\bigg{[}\frac{1}{2}\bigg{(}ST_{11}+ST_{2 2}+\sqrt{\left(ST_{11}-ST_{22}\right)^{2}+4 ST_{12}ST_{21}}\bigg{)}\bigg{]}, \tag{22}\]  
here, we use normalized large eigenvalue \(\lambda_{\text{large}}\) of structure tensor to delineate the edges of an infrared ship image. Because the large eigenvalue of structure tensor can indicate the predominate direction and the coherence degree of the gradient trend, the eigenvalues \(\lambda_{\text{large}}\) of the closely adjacent surroundings of infrared ship targets are almost the maximum values in the global eigenvalue image and present an approximately closed contour. Based on this cue, the average eigenvalue measure of structure tensor (STAEM) is proposed to characterize the contour feature of candidate TIR ship targets and eliminate residual clutter simultaneously, and the STAEM can be defined as:  
\[\lambda_{\text{azver}}=\frac{\sum\limits_{(m,n)\in D_{\text{can}}}\lambda_{ \text{large}}(m,n)}{\sum\limits_{(m,n)\in D_{\text{can}}}1}, \tag{23}\]  
where \(\lambda_{\text{azver}}\) denotes the STAEM value of the specified connected region. \(D_{\text{can}}\) is the domain of binary contour obtained by dilation of the connected region. On the right side of the formula, the numerator is the total value of normalized large eigenvalues in the domain \(D_{\text{can}}\), and the denominator indicates the total pixel number of the domain \(D_{\text{can}}\). Finally, any connected regions in the binarized final saliency map whose STAEM value is larger than a distinctive threshold \(Thr^{*}\) are retained, and all other pixels are set to 0.