## 3 Methodology
### Feature Information Construction Based on DFT  
The DFT is a fundamental Fourier analysis technique that converts signals from the time domain to the frequency domain [38]. DFT plays an important role in vector map watermarking  
Figure 1: Extraction of angle values of CDTN.  
schemes as a common method for frequency domain watermarking. The positive and inverse transform equations of DFT are shown in Equations (4) and (5), respectively.  
\[Y(k)=\sum_{n=0}^{N-1}x(n)\Big{(}e^{-i2\pi/N}\Big{)}^{kn},\;k\in[0,N-1] \tag{4}\]  
\[x(n)=\frac{1}{N}\sum_{n=0}^{N-1}Y(k)\Big{(}e^{2\pi/N}\Big{)}^{kn},n\in[0,N-1] \tag{5}\]  
where \(e\) denotes the natural logarithmic base, and \(i\) denotes the imaginary unit. The DFT is used to transform the angular values in CDTN and obtain the phase sequence as feature information, and the specific steps are shown as follows: by following the steps in Section 3.2 for obtaining the Angle values in the CDTN, the maximum and minimum values of each triangle Angle value are chosen to form the Angle sequence set \(A=\{A_{1},A_{2},\ldots,A_{m}\}\).  
\[\begin{cases}\alpha_{i}^{min}=min\big{\{}a_{i}^{1},\alpha_{i}^{2},\alpha_{i}^{ 3}\big{\}}\\ \alpha_{i}^{max}=max\big{\{}a_{i}^{1},\alpha_{i}^{2},\alpha_{i}^{3}\big{\}} \end{cases} \tag{6}\]  
\[A_{r}=\Big{\{}\alpha_{1}^{min},\alpha_{1}^{max},\alpha_{2}^{min},\alpha_{2}^{ max}\cdots\alpha_{k}^{min},\alpha_{k}^{min}\Big{\}} \tag{7}\]  
where \(\alpha_{i}^{1},\alpha_{i}^{2}\), and \(\alpha_{i}^{3}\) are the three interior angles of the \(i\)-th triangle; \(\alpha_{i}^{min}\) and \(\alpha_{i}^{max}\) are the minimum and maximum values of these three angular values, respectively. \(A_{r}\) is the sequence of angle values of the \(r\)-th block in Equation (7), and \(k\) is the number of triangles of the \(r\)-th block. The complex number \(A_{r}^{u}\) of the angle value is constructed by Equation (5). According to Equation (8), the discrete Fourier transform of complex sequence \(\{A_{r}^{u}\}\) was performed, and the Fourier coefficient \(\{A_{r}^{v}\}\) was obtained (Equation (9)). According to the property of Fourier coefficients, the amplitude sequence \(R_{r}^{v}\) and phase value sequence \(P_{r}^{v}\) of the \(r\)-th block can be obtained through Equations (10) and (11), in which \(P_{r}^{v}\) is represented by angle value.  
\[A_{r}^{u}=(\alpha_{min})_{u}+i\cdot(\alpha_{max})_{u},(u=0,1,2,3\ldots k-1) \tag{8}\]  
\[A_{r}^{v}=\sum_{u=0}^{k-1}A_{r}^{u}\Big{(}e^{-i2\pi/k}\Big{)}^{uv},(v=0,1,2,3 \ldots k-1) \tag{9}\]  
\[R_{r}^{v}=|A_{r}^{u}|=\sqrt{[R(A_{r}^{v})]^{2}+[I(A_{r}^{v})]^{2}},R_{r}=\{R _{r}^{v}\} \tag{10}\]  
\[P_{r}^{v}=arctan\left(R(A_{r}^{v})/I(A_{r}^{v})\right),P_{r}^{v}\in[0,180],P_{ r}=\{P_{r}^{v}\} \tag{11}\]  
where \(u\) is the number of triangles in \(A_{r}\), and \(i\) is the imaginary unit of complex numbers in Equation (8); \(v\) is the DFT series in Equation (9); and \(R(A_{r}^{v})\) and \(I(A_{r}^{v})\) represent the real and imaginary parts of the Fourier coefficients \(A_{r}^{u}\), respectively. Then, each angle value in the \(P_{r}^{v}\) is rounded down and converted into 8-bit binary form to form the feature sequence set \(P_{r}^{binary}\). The lengths of each sequence in \(P_{r}^{binary}\) are compared with the watermark sequence \(W_{inf}\), and the supplementary bits strategy in Equation (12) is adopted to make \(P_{r}^{binary}\) equal to the length of the watermark information \(W_{inf}\). To improve watermarking security, \(W_{inf}\) is scrambled, yielding the encrypted watermark information sequence \(W_{inf}^{evy}\). Finally, Xor operations are performed on \(P_{r}^{binary}\) and \(W_{inf}^{ecy}\), and each block obtains a zero watermark \(W_{r}^{zero}\) (Equation (13)).  
\[\left\{\begin{array}{l}e=\frac{L\left(W_{inf}\right)}{L\left(P_{r}^{binary }\right)}\\ P_{r}^{binary}=P_{r}^{binary}\ll(e-1)P_{r}^{binary}\end{array}\right. \tag{12}\]\[W_{r}^{zero}=P_{r}^{binary}\oplus W_{inf}^{exp},r=1,2\ldots m \tag{13}\]