## 4 Results and Discussion
### Robustness Evaluation
#### 4.2.3 Points Attacks  
The coordinate point attack consists of three parts: adding or removing points at random and data compression. As the experiment progresses, we subject the experimental data to a range of data adding, deletions, and compression, all the while trying to extract the optimal watermark images and recording the resulting BER values. The points-adding attack is to add random coordinate points to the line elements of the experimental data, with the intensity of the incremental points increasing gradually by 10% of the number of vertices in the original experimental data, up to double the original data. The experimental outcomes are illustrated in Figure 6. At 20% strength of the points-adding attack, watermark images with low BER values can typically be retrieved from the data, and we believe that copyright identification is attained. When the attack intensity exceeds 20% (data (c) is larger than 30%), the BER values of the retrieved watermark image are typically greater than 15%, making it difficult to recognize legitimate copyright content and impossible to validate copyright. The points-adding attack affects feature point extraction, which in turn contributes to CDTN's shaky building blocks. Experimental results show that the proposed zero-watermarking approach is still usable under the incremental point attack with an intensity level of up to 20%. The points-adding attack influences feature point extraction, which in turn leads to inconsistencies between the original and rebuilt CDTN. Experiments show that the proposed zero-watermarking scheme is reliable even when facing point-adding attacks with an intensity of 20% or less.  
The points-deletion attack is to randomly delete the vertices in the line elements, and the intensity of the deletion increases in the order of 5% of the number of vertices in the original data, up to 50% of the total number of vertices. Figure 7 displays the statistical outcomes of the points-deletion attacks. When the attack intensity is less than 50%, the BER values of the restored watermark images in different data are generally less than 10%. For example, in data (a) under a 45% density of points-deletion attack, the BER value of the extracted watermark image is 9.63%, and the NC value is 0.9425, allowing for clear identification of the copyright content. The results of the experiments show that the proposed watermarking scheme is highly resistant to the points-deletion attack. This is primarily because of the consistency of the feature points and the fact that local changes to the CDTN do not affect the whole. Furthermore, the majority of the deleted points are non-feature points, which have less of an effect on the reconstruction of the CDTN within some sub-blocks.  
Figure 6: Result of points adding attack.  
Data compression is a unique form of point attack that simplifies the vector map's line elements. Figure 8 displays the BER values of the ideal watermark pictures that may be retrieved from three sets of experimental data at varying degrees of Douglas-Peucker compression. Evidently, it is always possible to extract watermark pictures with a BER of 0 as long as the compression ratio during data preprocessing is not exceeded. Additionally, even if the compression attack surpasses its original compression ratio by a small proportion, it can still be resisted well and remain robust. As depicted in Figure 8, when the compression ratio hits 80%, which slightly exceeds the predetermined compression ratio of 75% for data (a) and data (b), watermark images with high recognition can still be spotted. Since the proposed zero-watermarking is constructed based on feature points and the compression ratio is set to a high value, it retains a high level of robustness despite a relatively heavy compression attack.  
Figure 8: Result of compression attacks.  
Figure 7: Result of points deletion attack.