# A Global Continuous Multi-Scale Terrain Contour Simplification Method Based on the Level Set
## 4 Discussion
### Ability of Arbitrary Scale Simplification  
We aim to generate a set of global continuous multi-scale simplified contours based on the initial contour. The advantage of CCSM lies in the fact that the contour lines of the desired scale can be obtained between fixed scales. Therefore, obtaining a simplified contour with the specified scale from the initial contour is critical. Given that CCSM shows a monotonous and continuous trend between scale and similarity as discussed in Section 3.2.2, the similarity between the simplified and initial contours is considered a proxy indicator for obtaining a simplified contour with a specified scale. In other words, similarity can serve as an intermediary to obtain the simplified contour corresponding to an arbitrary desired scale. According to [PERSON] [35], an exponential regression function can be used to establish an empirical relationship between the similarities and scales of contours. However, such a relationship may be unfit for other types of polylines. The exponential regression function is defined in Equation (14). \[f(x)=a*\exp(b*x)+c*\exp(d*x) \tag{14}\]  
where \(x\) represents the scale, \(f(x)\) represents the similarity value between two contour lines, and \(a,b,c,d\) represent coefficient values in exponential regression functions. The detailed process is described as follows. First, all contours in the study area were divided into five groups (as shown in Figure 17), and the shape and size of the contour line were used as criteria [16]. Second, one contour in each group was selected as reference data (boldface in Figure 17). Third, for the reference data in each group, the similarity values (Equation (9)) between the initial contour (1:50,000) and the contours of the three other scales (1:100,000, 1:250,000, and 1:500,000 downloaded from Google Earth) were then calculated. Table 4 shows the computation results of reference data in every group. Table 4 shows that the exponential regression function coefficients for each group can be fitted by using the values of the known scales and the similarities of the reference  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Scale**} & \multicolumn{5}{c}{**The Similarity of Different Groups**} \\ \cline{2-6}  & **Group1** & **Group2** & **Group3** & **Group4** & **Group5** \\ \hline
1:50,000 & 1 & 1 & 1 & 1 & 1 \\
1:100,000 & 0.998 & 0.988 & 0.995 & 0.989 & 0.997 \\
1:250,000 & 0.992 & 0.951 & 0.986 & 0.978 & 0.993 \\
1:500,000  
contours in each group. Table 5 shows the coefficient values for each equation in these five groups. As for CCSM, the shape similarity-scale trend is continuous and monotonous as shown in Figure 10. Therefore, the scale and similarity value of each simplified result have a one-to-one correspondence. In this case, we can generate simplified contours at any arbitrary scale by using CCSM. A simplified contour of an arbitrary scale can be obtained as follows. First, by using the exponential regression function in every group, the similarity value corresponding to the desired scale can be obtained. Second, by referring to Equation (9), the corresponding \(Z_{i}\) can be deduced from the similarity value. Third, the desired scale simplified contour can be selected based on the value of \(i\). This method provides a potential mechanism for achieving arbitrary scale simplification. Figure 18 shows the simplified contours with four specified scales (1:100,000, 1:200,000, 1:250,000, and 1:500,000) obtained by CCSM. For most of the simplified contours, the essential shape was unchanged during the progressive change process, thereby suggesting that the overall curved shape was well maintained. Individual contours with a small size disappeared as the scale decreased because their similarity values were filtered out according to the similarity threshold based on the small scale and exponential regression function. This finding is consistent with the rule of discarding small elements in cartographic generalization [21].