## 3 Methodology
### Adaptive Kernel Combination  
Three different feature kernels are expressed based on the above feature information 3D space. From feature tensors information \(F_{faa}^{flag}\), where \(flag\) is a member of \(\{G,A,W\}\), a set of feature pixels is randomly selected as training samples, which is denoted as \(\left\{F_{1}^{faa,flag},F_{2}^{faa,flag},\cdots,F_{n}^{faa,flag}\right\}\), where \(n\) is the number of the training samples. Spatial pixels from \(F_{mean}^{flag}\) and \(F_{spa}^{flag}\) are extracted, respectively, corresponding to the position indexes of selected feature pixels, which are denoted as \(S_{1}^{mean,flag},\cdots,S_{n}^{mean,flag},S_{1}^{spa,flag},\cdots,S_{n}^{ spa,flag}\), respectively. After that, three different kernels use the radial basis function as kernel function, and are created based on the three kinds of training samples, that is  
\[K_{fea}^{flag}\left(F_{i}^{fea,flag},F_{j}^{fea,flag}\right)=\exp\Bigl{(}- \left\|F_{i}^{fea,flag}-F_{j}^{fea,flag}\right\|/2\delta^{2}\Bigr{)}, \tag{16}\]  
\[K_{mean}^{flag}\left(S_{i}^{mean,flag},S_{j}^{mean,flag}\right)=\exp\Bigl{(} -\left\|S_{i}^{mean,flag}-S_{j}^{mean,flag}\right\|/2\delta^{2}\Bigr{)}, \tag{17}\]  
\[K_{spa}^{flag}\left(S_{i}^{spa,flag},S_{j}^{spa,flag}\right)=\exp\Bigl{(}- \left\|S_{i}^{spa,flag}-S_{j}^{spa,flag}\right\|/2\delta^{2}\Bigr{)}. \tag{18}\]  
By adaptive kernel combination of the above three kernels, the composite kernel can be constructed. To determine the dynamic weight, principal component analysis [32] is used on \(F_{fea}^{flag}\), \(F_{mean}^{flag}\), and \(F_{spa}^{flag}\) respectively to obtain the first three principal components (PCs) in each feature space, which can reduce the computational cost. Canny filter is used to detect the edge on PCs to obtain the binary image of the contour. The image entropy [25; 26] is used to measure the amount of structure information in the binary image. If image entropy is high, this means the structural information contained in the image is rich. The image entropies are calculated as follows:  
\[\begin{split} E_{fea}^{flag}=-\sum_{i=0}^{1}P_{i}^{faa,flag} \log P_{i}^{faa,flag},E_{mean}^{flag}=-\sum_{i=0}^{1}P_{i}^{mean,flag}\log P _{i}^{mean,flag},\\ E_{spa}^{flag}=-\sum_{i=0}^{1}P_{i}^{spa,flag}\log P_{i}^{spa, flag}.\end{split} \tag{19}\]  
\(P_{i}^{faa,flag}\) is the probability that the pixel gray value is \(i\) in the binary image of \(F_{fea}^{flag}\), \(i\) is equal to 0 or 1. Similarly, \(P_{i}^{mean,flag}\) and \(P_{i}^{spa,flag}\) are the probability in the different features \(F_{mean}^{flag}\) and \(F_{spa}^{flag}\), respectively. \(E_{fea}^{flag}\), \(E_{mean}^{flag}\), and \(E_{spa}^{flag}\) are image entropy. The obtained image entropy ratios are set as the dynamic weight of the corresponding kernels, and the composite kernel is defined as  
\[\begin{split} W_{fea}^{flag}=\frac{E_{fea}^{flag}}{E_{fea}^{flag }},W_{mean}^{flag}=\frac{E_{fea}^{flag}}{E_{fea}^{flag}},W_{spa}^{flag}=\frac {E_{spa}^{flag}}{E_{fea}^{flag}},E_{sum}^{flag}=E_{fea}^{flag}+E_{mean}^{flag }+E_{spa}^{flag},\\ K_{count}^{flag}\left(F_{i}^{com,flag},F_{i}^{com,flag}\right)\\ =W_{fea}^{flag}K_{fea}^{flag}\left(F_{i}^{fea,flag},F_{j}^{fea, flag}\right)+W_{mean}^{flag}K_{mean}^{flag}\left(S_{i}^{mean,flag},S_{j}^{ mean,flag}\right)+W_{spa}^{flag}K_{spa}^{flag}\left(S_{i}^{spa,flag},S_{j}^{ spa,flag}\right).\end{split} \tag{20}\]  
The adaptive kernel combination obtains composite kernels, which are incorporated into the SVM classifier to implement the SAR classification. Each group of feature information generates a composite kernel and obtains a SAR classification result. The below three classification maps are fused to obtain SAR image final classification.