## 2 Methodology
### Feature Compression
#### 2.3.1 Auto-Encoder  
An auto-encoder (AE) is a neural network which is an unsupervised learning for data representation and its aim is to set the output values approximately equal to the inputs. The basic structure of a single-layer AE neural network consists of three parts: encoder, activation and decoder, which are shown in Figure 2, where the input layer (\(x\)), hidden layer (\(y\)) and output layer (\(z\)) have, respectively, \(n\) neurons, \(m\) neurons, and \(n\) neurons. The hidden layer is commonly used to implement the encoding for the input data, while the output layer is for the decoding operation. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Feature Extraction Schemes** & **Features** & **Dimension** \\ \hline \multicolumn{3}{c}{Polarization intensities} \\ \(|S_{HH}|\), \(|S_{HV}|\), \(|S_{VV}|\) & 3 \\ \hline \multicolumn{3}{c}{Amplitude of HH-VV correlation} \\ \(\left|S_{VV}S_{HV}^{*}/\sqrt{|S_{HH}|^{2}|\cdot|S_{VV}|^{2}}\right|\) & 1 \\ \hline \multicolumn{3}{c}{Phase difference of HH-VV} \\ \(a\text{tan}(\text{Im}(S_{HH}S_{VV}^{*})/\text{Re}(S_{HH}S_{VV}^{*}))\) & 1 \\ \hline \multicolumn{3}{c}{Co-polarized ratio} \\ \(10\log_{10}(2|S_{HV}|^{2}|/|S_{VV}|^{2})\) & 1 \\ \hline \multicolumn{3}{c}{Cross-polarized ratio} \\ \(10\log_{10}(|S_{VV}|^{2}|/|S_{HH}|^{2})\) & 1 \\ \hline \multicolumn{3}{c}{Co-polarization ratio} \\ \(10\log_{10}(2|S_{HV}|^{2}/|S_{HH}|^{2})\) & 1 \\ \hline \multicolumn{3}{c}{Degrees of polarization} \\ \(|S_{VV}|^{2}/|S_{HH}|^{2},2|S_{HV}|^{2}/(|S_{HH}|^{2}+|S_{VV}|^{2})\) & 2 \\ \hline \multirow{3}{*}{Incoherent decomposition} & [PERSON] decomposition & 5 \\ \cline{1-1}  & [PERSON] decomposition & 7 \\ \cline{1-1}  & [PERSON] decomposition & 3 \\ \cline{1-1}  & [PERSON] decomposition & 9 \\ \hline Other decomposition & Null angle parameters & 2 \\ \hline \multicolumn{3}{c}{Sum} & 36 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The 36-dimensional decomposition features of single-temporal PolSAR data. The weighted input \(a\) of each neuron in encoder is defined as  
\[a_{j}^{(1)}=\sum_{i=1}^{n}w_{ii}^{(1)}x_{i}+b_{j}^{(1)},j=1,\ldots,m, \tag{8}\]  
where \(w_{ji}^{(1)}\) represents the encoder weight coefficient, \(b_{j}^{(1)}\) is the bias of neuron \(j\) Then, the encoder output \(y\) can be written as the nonlinear activation of weighted input \(a\) as follows  
\[y_{j}=f(a_{j}^{(1)}),j=1,\ldots,m, \tag{9}\]  
where \(f(\cdot)\) is a sigmoid function, which is usually chosen as the logsig function:  
\[f(x)=\frac{1}{1+e^{-x}}. \tag{10}\]  
If \(m<n\), the output \(y\) can be viewed as the compressed representation of input \(x\), then the encoder usually plays the role of data compression. Whereas the decoder is a reverse process of reconstructing the compressed data \(y\), which achieves the restoration of the original data, i.e., output \(z\) represents the estimate of input \(x\). The weighted input of the decoder is defined as  
\[a_{i}^{(2)}=\sum_{j=1}^{m}w_{ij}^{(2)}y_{j}+b_{i}^{(2)},i=1,\ldots,n, \tag{11}\]  
where \(w_{ij}^{(2)}\) is the decoding weight coefficient, and \(b_{i}^{(2)}\) is the bias of neuron \(i\). The decoder output reads  
\[z_{i}=g(a_{i}^{(2)}). \tag{12}\]  
Here, \(g(\cdot)\) is the sigmoid function for decoder neurons, which is commonly chosen the same as \(f(\cdot)\). The training process of AE is based on the optimization of the cost function and obtained the optimal parameters of weight coefficients and bias. The cost function measures the error between the input \(x\) and its reconstruction at the output \(z\), which can be written as  
\[J_{mse}=\frac{1}{2Q}\sum_{q=1}^{Q}\sum_{i=1}^{n}[x_{i}(q)-z_{i}(q)]^{2}, \tag{13}\]  
Figure 2: Single-layer AE neural networks structure. where \(Q\) is the number of samples. Furthermore, a restriction term of weight decay is usually incorporated into the cost function to regulate the degree of the weight attenuation, which helps to effectively avoid overfitting and remarkably improve the generalization capacity for the network.