# A Global Continuous Multi-Scale Terrain Contour Simplification Method Based on the Level Set
## 3 Experiments and Results
### Results and Analysis
#### 3.2.2 Comparison of Shape Similarity Change Trend  
In this subsection, the shape similarity-scale trend lines obtained by the three methods were compared. The shape of an object is an important visual feature, and the difference in shape similarity can be used to describe shape change [40; 41]. To avoid contingency, all initial contours in the study area were simplified by using the above three methods. All these methods generated 100 simplified results for each initial contour. The intermediate simplified contours between the initial and target contours were also represented by a serial number. The initial contour scale was taken as the source scale (i.e., 1:50,000), and the target contour scale was taken as the target scale. To further analyze the deformation of the simplified results, the target contour was the most simplified contour line containing two points (i.e., the starting and ending points of the initial contour), whereas the target contour scale was designated as a smaller scale (e.g., 1:20,000,000) or even approaching the infinitesimal scale (1:\(\infty\)). Equation (10) was used to calculate the similarity between the simplified contour and the corresponding initial contour.  
\[Sim(L_{initial},\ L_{i})=1-\delta_{dF}(L_{initial},\ L_{i})/Length(L_{initial}) \ (i=1,2,\ldots,100) \tag{10}\]  
where \(L_{initial}\) represents the initial contour, \(L_{i}\) represents the simplified contour, and \(i\) represents the corresponding sequence number of the simplified contour.  
Figure 10 shows the shape similarity-scale trend line corresponding to the three simplification methods. The horizontal axis represents the sequence number \((i=1,2,\ldots,100)\). The horizontal axis can also represent the scale because our setting parameters are continuously changing, and so should be the scale. The vertical axis represents the average shape similarity value of all initial contours at the corresponding sequence number \(i\). When \(i=1\), the scale of the contour line is: 1:50,000; When \(i=100\), the scale of the contour line is: 1:20,000,000) or even approaching the infinitesimal scale (1:\(\infty\)).  
Figure 9: Simplified contours generated by the DP algorithm, the WM algorithm, and the CCSM method. (**a**) The simplified contours generated by DP algorithm. (**b**) The simplified contours generated by WM algorithm. (**c**) The simplified contours generated by CCSM method.  
Overall, the average shape similarity value decreased along with the scale for all three simplification methods. As shown in Figure 10, both the blue and green curves jumped (denoted by red rectangles), which means that the shapes of the simplified contour lines generated by the DP and WM algorithms were not continuously changing. In other words, as the scale gradually changed, the changes in the shape of the simplified contours generated by the DP and WM algorithms were not continuous but abrupt, thereby leading to the loss of some intermediate scale details. In this case, the user demand for visual continuity is left unsatisfied. By contrast, the yellow curve in Figure 10 representing CCSM shows a monotonic and continuous trend that closely reflects the ideal relationship between similarity and scale at a continuous multi-scale (Figure 7). The average shape similarity gradually decreased along with the scale. Therefore, CCSM can obtain smooth and continuous multi-scale simplified contours.