## 4 Results and Discussion
### Watermark Robustness  
#### 4.2.1 Resistance to Geometric Attacks  
The experimental results for both ZC and SZ in Table 4 show that under different levels of translation, rotation, and scaling attacks, both DTM-based and SVD-based schemes could detect the watermark images. Under the coordinate transformation attacks, the \(NC\) values for the optimal feature vertex-based watermark are 0.84 and 0.83 in ZC, respectively, and 0.85 and 0.84 in SZ, respectively, when transforming to the Xian80 and Beijing54 coordinate projection. Although the corresponding images are relatively vague, we can still identify the copyright information. The \(NC\) values of the non-feature vertex watermarks obtained from these two maps are 1 under the coordinate transformation attack. Under the compression attacks, the DTM-based watermarking scheme can extract a 100% similar watermark image at a compression ratio of 78%. The watermark quality decreases as the compression strength exceeds 78%. In the case of a 50% compression rate, the extracted SVD-based watermark can hardly recognize the copyright information, and when the compression rate reaches to 78%, it is impossible to construct non-feature vertex zero-watermark. DTM is constructed based on the stability of the topological relationship among feature vertices such as that the angle values in the DTM remain the same as the vector map does not distort under operations of translation, rotation, or scaling. Singular values of non-feature vertices could change, but the relative size remains unchanged. The projective transformation has a high intensity of vertex operations, but the \(NC\) values for the SVD-based watermarks are almost unaffected as the relative sizes of the vertices' values have little change. The \(NC\) values of the DTM-based watermarks however are affected greatly, where the extracted watermark images have the \(NC\) values ranging between 0.83 and 0.85. The copyright information can still be identified and the waterm  
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slightly more than that in SZ as the original projection coordinate system and the degrees of projection transformation differ. The number of feature vertices is not affected by changes in the compression strength when the compression strength does not exceed a threshold and decreases as the compression strength increases after exceeding the threshold. As non-feature vertices could not be extracted, the watermarks for non-feature vertices could not be constructed. In essence, watermarks based on DTM exhibit robust to the compression attacks. Table 5 shows the \(BER\) values of the optimal watermarks that can be detected in ZC and SZ in the proposed methods and in the methods from literatures of [1] and [24] under six cropping patterns, where there are results for two different embedding methods in [24]. Under the six cropping patterns, the watermark based on DTM could be used to extract the watermark image with a \(BER\) value of 0, meaning that the extracted watermark information is completely consistent with original watermark information, which indicates a strong resistance capability to the cropping attacks. Correspondingly, the \(BER\) values of the watermarks extracted from the developed dual zero-watermarking scheme is 0 under different cropping attacks. The non-feature vertex watermarks have very high \(BER\) values under different cropping attacks, indicating a low resistance capability to the cropping attacks. The \(BER\) values obtained under the method in literature [1] in the corresponding patterns are notably better than that for the non-feature vertex watermark. The methods in [24] provide \(BER\) values lower than those obtained from the method in [1] for all cropping attacks except both Pattern 1 and Pattern 4. Overall, watermarks constructed based on the proposed DTM method or the dual methods are far superior to the comparative methods in literatures in terms of resistance to the cropping attack.