# Water Extraction in PolSAR Image Based on Superpixel and Graph Convolutional Network
## 2 Materials and Methods
### Superpixel Segmentation and Feature Extraction
#### 2.1.2 Feature Extraction  
After superpixel segmentation, the features of each node were extracted, including:  
1. [label=(0)]
2. Scattering matrix and Statistical features  
Usually, the electromagnetic scattering characteristics of radar targets in the far field are linear. If the scattering space coordinate system and the corresponding polarization base are selected, there is a polarization component between the radar illumination wave and the target scattered wave. Therefore, the variable polarization effect of the target can be expressed in the form of a complex two-dimensional matrix called the scattering matrix,  
Figure 1: The flowchart of the proposed method. which represents the full polarization information of the target at a specific attitude and observation frequency. \[S= \begin{bmatrix}S_{HH}&S_{HV}\\ S_{VH}&S_{VV}\end{bmatrix}  \tag{1}\]  
where \(H\) stands for horizontal polarization and \(V\) means vertical polarization. \(S_{HH}\) and \(S_{VV}\) denote the co-polarization components of the scattering matrix. \(S_{HV}\) and \(S_{VH}\) are the cross-polarization components, equal for a reciprocal medium. All matrix elements are directly acquired from the complex values of multi-polarization single-look images. For dual-polarization data, only one co-polarization and one cross-polarization element were kept. These components tend to play different roles in extracting water. For instance, \(S_{HH}/S_{VV}\) images usually have a better signal-to-noise ratio than \(S_{HV}/S_{VH}\) in various targets, especially calm surface water bodies, due to the specular reflection mechanism; \(S_{HH}\) scarcely differs from \(S_{VV}\); and \(S_{HV}/S_{VH}\) is more sensitive to rough water surfaces, such as flowing rivers or floods. A scattering matrix is acquired for each pixel, and to extract the features representing each superpixel, all pixel-level matrix elements need to be translated into superpixel-level statistics values. For example, the mean, median, standard deviation, maximum, and minimum of each scattering matrix element should be calculated for all pixels belonging to the same superpixel. Among these features, maximum and minimum values are easily affected by the speckle noise, and the median can measure the central tendency better than the mean. Texture features and covariance matrix  
Texture analysis is important for SAR segmentation and has already been investigated in some literature [36]. Textures can be described by first-order metrics, such as the mean, variance, or entropy, as well as second-order metrics, such as gamma distribution features [7]. They can be calculated in the superpixel space and serve as a direct indicator of 'disorder' within them. All the above are based on single-polarization images; although they can be calculated for each polarization channel, a more sensible method is to obtain a mixed texture value from multi-polarization data. Considering that heterogeneous regions are common in high-resolution SAR images, the spherically invariant random vector (SIRV) product model can be adopted [37]. In this model, the maximum likelihood estimator of the texture value is given by:  
\[\tau=\frac{k^{H}M^{-1}k}{p} \tag{2}\]  
where \(k\) is the target vector in the linear bases as \(\left[S_{HH},\ \sqrt{2}S_{HV},S_{VV}\right]^{T}\) for quad-polarization data. \(M\) refers to the normalized covariance matrix (\(k\times k^{T}\)). \(H\) denotes the conjugate transpose operator and \(p\) equals 2 (dual-polarization) or 3 (quad-polarization). It is important to note that the covariance matrix is the mean of all pixels in the superpixels rather than a single pixel. It contains all the polarization information and is usually taken as the basis of PolSAR processing. Thus, the upper triangular elements of \(M\) (symmetric matrices) could be used as features. The elements of the main diagonal are highly related to the statistical mean values of the \(S\) matrix, and they refer to the mean intensity and magnitude value, respectively.