## 2 Materials and Methods
### Linear Extractions
#### 2.2.2 Improved Line Segment Tracking Method for Lineament Extraction and Automatic Connection  
This lineament extraction is to determine whether the obtained line element is a ridge by judging the continuity of the pixel brightness value (DN value) in the remote sensing image. The continuous direction of the pixel can be connected to obtain a lineaments diagram by a matlab program. The specific steps are as follows.  
Continuous Direction Judgment of the Line Element  
To improve the search accuracy of lineaments and reduce the computation time, we choose the 11 \(\times\) 11 line element as the center, set the search window, and set 16 radial search directions with equal angle intervals in the range of 180\({}^{\circ}\) around the center, as shown in Figure 9.  
Figure 7: Image data after Gaussian low-pass filtering.  
Then, the deviation \(\varepsilon(x)\) is used to evaluate the continuity of the DN value in the straight line direction:  
\[\varepsilon(x)=\int_{-a}^{a}w(x+h)\{z(x)-z(x+h)\}^{2}dh\]  
where \(zx\) is the \(DN\) value of image pixel \(x\). \(zx+h\) is the \(DN\) value of another pixel passing through the pixel \(x\), and the angle is \(\theta\), which is a straight line from the pixel \(x\) to the other pixel with a distance of \(h\). \(W(x)\) is the weight coefficient corresponding to the pixel \(x\), \(w(x+h)\) is the weight coefficient of the pixel with \(h\) distance from the line element \(x\), and the weight coefficient is designed to strengthen the effect of the line element near the pixel \(x\). Then, we distinguished the \(DN\) value difference between each pixel. When the deviation \(\varepsilon(x)\) is the smallest, it indicates that the direction has the best continuity for the DN value of this pixel \(x\).  
Figure 8: Lineament interpretation results.  
Figure 9: The 16 directions of the data search.  
Line Element Determination  
After determining the direction with the best pixel continuity, the basic pixels can be constructed linearly in the continuation direction. Certain thresholds were set to determine the pixels. Meanwhile, the difference between different locations in the image should be eliminated. The difference can be defined as follows:  
\[\lambda=\frac{\left\{d^{2}z(x)/d\,x^{2}\right\}^{2}}{z(x)}\]  
where \(\lambda\) is the change rate of the pixel value \(zx\) to eliminate the pixel extraction gap in different regions of the image.  
Then, the quadratic integral function can be obtained in the direction perpendicular to the best direction of pixel continuity.  
\[\wedge=\sum_{i=-4}^{4}\lambda_{i}=\sum_{i=-4}^{4}\frac{\left(z_{i+1}-2z_{i}+z_ {i-1}\right)^{2}}{z_{i}}\]  
The threshold \(T\) is seen as the limit threshold of the central pixel. If the change rate of the central pixel \(\lambda_{i}\) is greater than the threshold \(T\), the line element can be retained. This pixel is defined as \(p\).  
\[T=v+\mu\sigma\]  
\[\mu=\frac{\phi_{1}}{1+(\eta-1)sin\theta}\]  
where \(v\) and \(\sigma\) represent the mean and standard deviation of \(\wedge\), respectively, and \(\eta\) can be determined by the variogram \(\gamma(h)\) of the pixel value. If there is sunlight, the variograms parallel and perpendicular to the incident direction of sunlight can be denoted as \(\gamma_{p}(h)\) and \(\gamma_{c}(h)\). The value of \(\eta\) can be defined by the ratio of the two different directions on the adjacent pixel \((h=1)\).  
\[\eta=\gamma_{p}(1)/\gamma_{c}(1)\]  
The \(\gamma(h)\) is the variant function that characterize the relationship between the variance of the change value of the data and the distance of the data. The formula of \(\gamma(h)\) is defined as:  
\[\gamma(h)=E\Big{[}\Big{\{}z(x+h)-z(x)\}^{2}\Big{]}\ /2\]  
where \(E\) is the mathematical expectation.