# A Global Continuous Multi-Scale Terrain Contour Simplification Method Based on the Level Set
## 3 Experiments and Results
### Results and Analysis
#### 3.2.3 Comparison of Global Structural Feature Retention  
We then compared the ability of retaining the global structural features of the simplified contours generated by the DP algorithm, WM algorithm, and CCSM. To ensure a fair comparison, the retention degree of the global structural features on the generated simplified contours was examined at the same scale level. When simplifying line elements in electronic maps, the product of the minimum visual discriminant distance (usually set to 0.1 mm) and the reciprocal of the scale were utilized as simplification parameters. If the distance between two linear elements is less than this parameter, then their scale is the same [42]. In this study, the improved Frechet distance between the simplified and initial contours was used as a proxy for the scale parameter. For example, when the initial contour scale was 1:50,000, the scale parameter was set to 5 m (0.1 mm \(\times\) 50,000 = 5 m). When the improved Frechet distance between two contour lines was less than 5 m, the two scales can be considered the same. According to Equation (10), when calculating the similarity between an initial contour and an intermediate simplified result whose Frechet distance was within 5 m, the minimum similarity value was approximately equal to 0.998, that is, the difference \(\Delta\)s \(\approx\) 0.002:(1 \(-\) 0.998). Therefore, 0.002 was considered the maximum similarity value for identifying whether two simplified contours are at the same scale. Three simplification methods were used to simplify 20 contour lines that were randomly selected from the dataset (as shown in the first column of Table 2). For each initial contour line, 100 simplified contours were obtained by each method. The scale of the initial contour scale was 1:50,000. The target contour was the most simplified contour line containing two points, namely, the starting and ending points of the initial contour. The similarities between all simplified contours and their initial contours were calculated by  
Figure 10: Average shape similarity trend of three simplification methods. using Equation (10). To avoid contingency in the experimental process, for each initial contour line, multiple groups of simplified contours were selected within the maximum similarity value (as shown in the second column of Table 2), and each group had three simplified contours generated by the three simplification methods. Therefore, each group had three similar values corresponding to the three methods. To ensure that the simplified results are compared at the same scale, a verification was performed as follows. First, in every group, each two of three similarity values should be subtracted from each other, and the absolute value should be obtained. Second, for each initial contour line, the absolute values of all groups were averaged (as shown in columns 3 to 5 of Table 2). Third, the maximum value of the three averages was selected and verified to check whether this value is less than 0.002. The max average absolute value (MAAV) of different initial contours was calculated as Equation (11). \[ \begin{array}{c}MAAV=MAX(S_{d-ID}(i,DP,CCSM),S_{d-ID}(i,DP,WM),S_{d-ID}(i,WM, CCSM))\\ S_{d-ID}\left(i,A,B\right)=mean_{-ID}\left|Sim\left(L_{i}^{A},L_{initial-ID} \right)-Sim\left(L_{i}^{B},L_{initial-ID}\right)\right|\\ \left(ID=1,2,\ldots,20;\ i=1,2,\ldots,m\right)\end{array}  \tag{11}\]  
where \(L_{i}^{A}\) represents the simplified contour generated by the A algorithm, \(L_{i}^{B}\) represents the simplified contour generated by the B algorithm, \(L_{initial-ID}\) represents the initial contour, and \(i\) represents the number of groups. By using the above calculation method, the MAAV of the 20 contours was calculated and listed in Table 2. As shown in Table 2, the MAAV was within 0.002 (as shown in the last column of Table 2), thereby guaranteeing that the simplified results produced by the three methods were compared on the same scale.