## 1 Introduction  
With the characteristics of precise positioning, high accuracy, small storage capacity, and broad application, vector maps play a crucial and fundamental role in the construction of numerous fields, such as smart cities, national defense deployment, ecological protection, new agriculture, and disaster prevention and control, thereby establishing their extremely high commercial value [1, 2, 3]. Now more than ever, producers, legal owners, and consumers of vector maps are suffering enormous losses as a result of unlawful trafficking, dissemination, alteration, and unauthorized use of the data because of the ease with which electronic data may be transmitted. Therefore, there is a huge contradiction between the sharing and security of vector map data, and its security protection is highly valued by governments and scholars [4, 5, 6]. As the cutting-edge technique for copyright protection of digital products, digital watermarking provides technological support for the security protection of vector mapsand the healthy growth of the geographic information industry; thus, it has been a subject of considerable interest [7; 8; 9; 10]. Traditional digital watermarking conceals copyright information in electronic data invisibly using an algorithm that does not compromise data quality or availability. In addition, the invisible nature of the watermark ensures that other data consumers are unaware of its presence, hence enhancing the security of watermarking. The traditional watermarking for vector maps is commonly accomplished by altering the coordinate values. The way is typically inapplicable to vector maps, which have extremely low redundancy and require very high accuracy in the use of data, particularly for certain high-precision data where even the tiniest deviation is undesirable. Therefore, it is crucial to building a lossless watermarking technique for vector map copyright protection. Zero-watermarking extracts the feature information of cover data to generate watermark keys as the basis for copyright verification, and its non-embedding mechanism has garnered significant interest [11; 12; 13]. The robustness and security of the algorithm depend heavily on the stability of extracted feature information. In vector map zero-watermarking, therefore, mining as much stable feature information as possible has been an essential study direction. In this study, the benefits and drawbacks of spatial and frequency domain watermarking for vector maps are analyzed, and an attempt is made to extract the deep inherent feature information of vector maps by merging the spatial and frequency domains. On this basis, a robust zero-watermarking algorithm for vector maps is proposed. The contributions of this paper include the following:  
(1) Analysis of the existing works from the perspectives of the spatial and frequency domains, respectively, leads to the conclusion that the relevant study is characterized by unicity and completeness in feature information mining, with relatively one-sided robustness. (2) A vector map zero-watermarking combining spatial and frequency domains is proposed. In the spatial domain, a constrained Delaunay triangulation network (CDTN) is constructed after feature points and data blocking, and the angle values in CDTN are extracted as the initial feature information. In the frequency domain, the discrete Fourier transform (DFT) is used to further transform and mine sequences of angle values. The sequence of DFT phase coefficients in each sub-block is used as the final feature information for the generation of zero watermarks. (3) Under the above operation, a zero-watermarking scheme for vector maps is built. Experimental results and analysis demonstrate that watermarking has a comprehensive robustness to common geometric attacks, coordinate point attacks, and clipping attacks, among others. The rest of this paper is organized as follows: The related work is introduced in Section 2. The methodology containing basic algorithms and a detailed watermarking process is proposed in Section 3.