## 3 Methodology
### Spatial Domain Statistical Information Mining Based on CDTN  
The CDTN is made up of a series of connected but non-overlapping triangles and the outer circles of these triangles do not contain any other points in the domain, which is known as the empty outer circle property of CDTN ([PERSON] et al., 2009). Another property of CDTN is that the minimum angles of all triangles are maximized as much as possible, which indicates that all the triangles in CDTN are closest to regularization [PERSON] et al. (2009); [PERSON] et al. (2009). The above properties ensure that the construction of CDTN will form unique and reasonable triangles. Furthermore, adding, deleting, or moving some vertices affects only the adjacent triangle and has no effect on the construction of other triangles. As a result, CDTN has good global stability, which can serve as the foundation for the robust performance of watermarking. In the proposed watermarking scheme, angle values in the CDTN are extracted as stable spatial statistics of vector maps and used for constructing feature information. The main steps are as follows:  
(1) Construction of CDTN based on feature points. After the Douglas-Peuker simplification of the original vector map, obtain the discrete points set \(V=\{v_{0},v_{1},v_{2}\ \ldots\ v_{i},\ \ldots\ v_{n}\}\), and randomly sort each point. If the point \(v_{i}\) is inside a triangle, connect it to the triangle's three vertices and divide the triangle into three triangles. If \(v_{i}\) is on the side of a triangle,connect the disjoint points of the two triangles that share a side with \(v_{i}\) and divide the two triangles that share a side into four triangles. The CDTN is formed after any vertex of the initial triangle is eliminated by regularizing the triangular dissection using the edge-swapping principle. (2) Extraction of angle values. Generating triangle networks based on each block's feature points, extracting angle values with the inverse trigonometric function (Equation (2)) by calculating the distance between each triangle (Equation (1)), and composing a sequence of angle values \(A_{r}\) (Equation (3)), ensuring that the length of the angle sequence \(A_{r}\) within each block is greater than the length of the watermark sequence \(W_{inf}\). \[\left\{ \begin{aligned} d_{12}&=\sqrt{\left(v_{1}^{x}-v _{2}^{y}\right)^{2}+\left(v_{1}^{y}-v_{2}^{y}\right)^{2}}\\ d_{23}&=\sqrt{\left(v_{2}^{x}-v_{3}^{y}\right)^{2}+ \left(v_{2}^{x}-v_{3}^{y}\right)^{2}}\\ d_{31}&=\sqrt{\left(v_{3}^{x}-v_{1}^{x}\right)^{2}+ \left(v_{3}^{y}-v_{1}^{y}\right)^{2}}\end{aligned} \right. \tag{1}\]  
\[\left\{ \begin{aligned} \alpha_{1}=arccos\,\left[\left(d_{12}^{2}+d_{31}^ {2}-d_{23}^{2}\right)/\left(2\cdot d_{12}\cdot d_{31}\right)\right]\\ \alpha_{2}=arccos\,\left[\left(d_{23}^{2}+d_{12}^{2}-d_{31}^{2} \right)/\left(2\cdot d_{23}\cdot d_{12}\right)\right]\\ \alpha_{3}=arccos\,\left[\left(D_{23}^{2}+D_{31}^{2}-D_{12}^{2} \right)/\left(2\cdot D_{23}\cdot D_{31}\right)\right]\end{aligned} \right. \tag{2}\]  
\[A_{r}=\{\alpha_{1},\alpha_{2},\alpha_{3},\cdots\alpha_{n}\},\,\,\,r=1,2, \ldots,m \tag{3}\]  
In Equation (1), \(v_{1}^{x}\), \(v_{2}^{x}\), and \(v_{3}^{y}\) are the horizontal coordinate values of the first, second, and third points in Figure 1, respectively; \(v_{1}^{y}\), \(v_{2}^{y}\), and \(v_{3}^{y}\) are the vertical coordinate values of the first, second, and third points in Figure 1, respectively; and \(d_{12}\), \(d_{23}\), and \(d_{31}\) are the distances between different vertices, respectively. In Equation (2), \(\alpha_{1}\),\(\alpha_{2}\), and \(\alpha_{3}\) are the three interior angle values within a triangle in Figure 1. The \(A_{r}\) denotes the set of angle values of the \(r\)-th block, \(m\) demotes the total number of sub-blocks, and \(n\) denotes the total number of angles in Equation (3). The angle sequence \(A_{r}\) is based on CDTN's invariance in the face of geometric attacks such as rotation, translation, and scaling.