# Change Detection of Water Resources via Remote Sensing: An L-V-NSCT Approach
## 2 Directional Logarithmic Variogram and the VSSIM Model  
At present, variogram analysis [14; 15] is typically used to study the texture structure and spatial correlation of an entire image. This paper mainly uses a variation function to consider the structural characteristics of surface features in different period's images and highlights the edge contour structure of target features such as rivers and lakes in images, as well as the variation characteristics of the main edges in the detection area. Based on the classic mathematical model of the variogram, a logarithmic variogram model with horizontal (0\({}^{\circ}\)), vertical (90\({}^{\circ}\)), and diagonal (45\({}^{\circ}\), 135\({}^{\circ}\)) directions was proposed. The model can quickly locate and analyze the main edge contour information of image objects. Set \(f(x,y)\) as the size of the collected image M \(\times\) N; the sliding window with the designed width \(L=2d+1\) was designed to analyze and process different types of ground object data in the image source. The center pixel of the ground object in the window can be set as \((i_{0},j_{0})\), the coordinates of the ground object pixels in other positions can be set as \((i,j)\), and the edge analysis model (log-vari) of the directional logarithmic variation function can be set as follows:  
\[{r_{h}}^{0}(i_{0},j_{0})=\frac{1}{2N_{0}(h)}\sum\limits_{i-i_{0}-d}^{i_{0}+d} \sum\limits_{j-j_{0}-d}^{j_{0}+d-h}|\text{ln}[f(i,j)]-\text{ln}[f(i,j+h)]| \tag{1}\]  
\[{r_{h}}^{45}(i_{0},j_{0})=\frac{1}{2N_{45}(h)}\sum\limits_{i-i_{0}-d}^{i_{0}+d -h\,j_{0}+d-h}|\text{ln}[f(i,j+h)]-\text{ln}[f(i+h,j)]| \tag{2}\]  
\[{r_{h}}^{90}(i_{0},j_{0})=\frac{1}{2N_{90}(h)}\sum\limits_{i-i_{0}-d}^{i_{0}+d -h}\sum\limits_{j-j_{0}-d}^{j_{0}+d}|\text{ln}[f(i,j)]-\text{ln}[f(i+h,j)]| \tag{3}\]  
\[{r_{h}}^{135}(i_{0},j_{0})=\frac{1}{2N_{135}(h)}\sum\limits_{i-i_{0}-d}^{i_{0} +d-h\,j_{0}+d-h}|\text{ln}[f(i,j)]-\text{ln}[f(i+h,j+h)]| \tag{4}\]  
where \(f(i,j+h)\) and \(f(i,j)\) are two coordinate values with distance h in the area. The values of the directional value \(\theta\) are {0\({}^{\circ}\), 45\({}^{\circ}\), 90\({}^{\circ}\), 135\({}^{\circ}\)}. \(N_{\theta}(h)\) is the number of all two-coordinate points in the region with a distance of h, and the final output mathematical matrix sequence \(r^{\theta}(x,y)\) is the texture coefficient matrix in the direction of \(\theta\). In the process of feature similarity analysis, the structure similarity model (SSIM) can be constructed from the brightness, contrast, and structure factor of the image [16]. The expression is as follows:  
\[SSIM(x,y)=[l(x,y)]^{\alpha}.[c(x,y)]^{\beta}.[s(x,y)]^{\gamma} \tag{5}\]  
where \(l(x,y)\) is the luminance coefficient matrix, \(c(x,y)\) is the contrast parameter, and \(s(x,y)\) is the structural parameter. The parameters \(\alpha,\beta,\gamma\) are all less than 1. To improve the detection accuracy of texture details of different types of objects in seismic images and the clearer edge structure of lakes and rivers, we used the edge texture coefficient matrix obtained in the upper section to improve and optimize the SSIM model and propose the structure of directional logarithmic variogram model similarity (VSSIM). We selected two images of size M \(\times\) N, which are represented by X and Y. First, the average value of the edge texture \(r^{\theta}(x,y)\) of the two images was calculated by using the log-vari model in the direction of \(\theta\). The extracted edge texture feature coefficients are \(r_{X}(x,y)\) and \(r_{Y}(x,y)\). At the same time, the structural similarity algorithm \(v(x,y)\) of the logarithmic variation function was constructed to judge the change degree of edge features in the texture feature matrix.