## 4 Experiments and Results
### Algorithm Robustness  
Robustness refers to the ability of multiple watermarking techniques to detect watermarks in the cover data, even after being subjected to various attacks. For vector geographic data, the most common attacks are randomly deleting vertices, cropping, compression, or randomly adding vertices. The proposed multiple watermarking algorithm was evaluated for its robustness via experiments conducted on diverse digital vector geographic maps in the ShapeFile format. Figure 11 shows three example maps of experimental data, which (a) are organized as points, (b) are organized as polylines and (c) are organized as polygons. The same experimental procedures were repeated for different vector geographic datasets, numbered from 1 to 30. Nos. 1 to 10 are organized as points, 11 to 20 are organized as polylines, and 21 to 30 are organized as polygons, sorted in ascending order based on the number of coordinate vertices in the cover data. A multiple watermarking algorithm with good robustness against data addition and deletion attacks is presented in [35]; therefore, it is chosen for the comparative experiments. In [35], the vector geographic data are divided into logic domains according to the number of watermarks, and then the watermarksare embedded into the cover data. This algorithm exhibits good performance in terms of robustness against the deletion of vertices and cropping attacks. The specific steps of the experiment were followed meticulously. Four watermarks were randomly generated. The experimental cover data successively embed Watermark 1, Watermark 2, Watermark 3, and Watermark 4 using both the proposed algorithm in Section 3 and the algorithm proposed in [35]. Then, the random deletion of vertices, cropping, compression, and the random addition of vertices are carried out. Finally, all watermarks are detected in the cover data being attacked. The detection process does not involve the original vector map or the original watermark, i.e., the blind watermarking algorithm. We conducted 30 experiments to repeat the above experimental procedure. The corresponding experimental results for the experimental maps are presented in Tables 3-5. In the above tables, \(\surd\) denotes the presence of a watermark in the data after an attack, while \(\times\) indicates its absence. The content in parentheses represents the number of experimental maps that can detect a specific watermark. The experimental results demonstrate that the proposed algorithm exhibits robustness against common attacks such as vertex deletion, feature deletion, cropping, data compression attacks, and vertex addition attacks. The algorithm's robustness remains effective even without reliance on the original watermark. In terms of determining whether a watermark exists in detected data using our proposed algorithm, there is no requirement for the presence of the original watermark when dealing with maps containing numerous vertices. Since the numbering of all experimental data increases with the increase in the data size, the number of remaining coordinate vertices in experimental maps with higher numbers is larger than the maps with lower numbers when the attack intensity increases. Therefore, data in large volumes demonstrate better robustness in experiments. The algorithm detailed in [35] has strong robustness against cropping and deletion attacks, but the proposed algorithm is better in this respect. The algorithm in [35] adopts a block-based approach, where the experimental data are first divided into blocks, and multiple watermarks are then embedded into the different blocks. By adopting a logical domain approach, the robustness of the watermark algorithm against the addition and deletion of vertices is enhanced. However, when the intensity of the addition or deletion increases, it is possible that the blocks containing a certain watermark may be deleted, resulting in an undetectable watermark. The proposed algorithm differs from this approach.