Frequency-domain watermarking is highly effective in copyright authentication for various forms of multimedia data, including but not limited to images, videos, and audios. However, it faces vulnerability to vertex attacks, such as map cropping and vertex deletion, when applied to vector maps. To enhance the robustness of frequency-domain watermarking for vector maps, this study proposes a new scheme based on virtual vertices (VVs) and frequency coefficient ratios. Prior to watermark embedding, VVs are inserted between each pair of adjacent vertices. Next, the discrete wavelet transform is employed to decompose x- and y-coordinates sequences, which include both actual and virtual coordinates. The ratios between high frequency coefficients of the x- and y-coordinates sequences are computed and selected as the embedding domain. To minimize distortion from watermark embedding, the ratios are amplified and modulated to conceal the watermark information. Furthermore, a watermark correction method is devised to address VVs inconsistency, further enhancing robustness. Experimental results demonstrate that the proposed watermark correction method improves robustness performance by up to 35% compared to the uncorrected baseline. The proposed scheme also exhibits remarkable robustness against vertex deletion, vertex addition, and map cropping, achieving bit error rate ranging from 0% to 9.65%. Furthermore, the watermark’s visibility is well controlled, with error statistics approaching 0, thereby ensuring its imperceptibility.

Symmetries and symmetry-breaking play significant roles in data security. Digital watermarking is widely employed in information security fields such as copyright protection and traceability. With the continuous advancement of technology, the research into and application of digital watermarking face numerous challenges. To gain a comprehensive understanding of the current research status and trends in the development of digital watermarking, this paper conducts a bibliometric analysis using the CiteSpace software, focusing on 8621 publications related to digital watermarking (watermark/watermarking) from the Web of Science (WOS) Core Collection database, spanning from 2004 to 2024. This study explores the research landscape and future trends in digital watermarking from various perspectives, including annual publication volume, keyword co-occurrence and burst detection, leading authors, research institutions, and publishing countries or regions. The results reveal a regional concentration of research efforts, with early research being primarily dominated by the United States, Taiwan, and South Korea, while recent years have seen a rapid rise in research from China and India. However, global academic collaboration remains relatively fragmented and lacks a well-integrated international research network. Keyword analysis indicates that research hotspots have expanded from traditional copyright protection to data integrity verification, multimedia watermarking, and the incorporation of intelligent technologies. Notably, the introduction of deep learning has propelled watermarking algorithms toward greater sophistication and intelligence. Using CiteSpace, this study is the first to systematically illustrate the dynamic evolution of digital watermarking research over the past 20 years, focusing on thematic trends and regional distributions. Unlike previous reviews that rely mainly on qualitative analyses, this study offers a quantitative and visualized perspective. These findings provide concrete references for the future development of more targeted research efforts.