## 2 Methodology
### Zero-Watermark Based on Singular Value Decomposition  
Singular Value Decomposition (SVD) is an orthogonal transformation of matrix without requirements for the matrix size. Watermarking algorithms based on SVD are robust against perturbation as SVD transformation produces matrix that are stable to the operation of transpose, displacement and rotation [53]. The singular values reflect intrinsic map features that represent relations among matrix elements and therefore zero-watermark constructed based on non-feature vertices and SVD is robust to common geometric attacks.  
Figure 1: Construction of Delaunay triangulation mesh based on feature vertices. DTM: Delaunay Triangulation Mesh.  
Taking the horizontal direction as an example, where the operation to the vertical direction is analogous, steps to implement SVD to construct zero-watermark are follows:  
Based on non-feature vertices in all polylines and polygons, a fixed-size real matrix \(C=\left(C_{ij}\right)\in R^{M\times N}\) is constructed for each non-feature vertex set P of every polyline and polygon. The matrix of C is decomposed by applying the SVD algorithm (6).  
\[C=U\!\left[\begin{array}{cc}\Sigma&0\\ 0&0\end{array}\right]\!V^{T}=\left[u_{1},u_{2},u_{3}\cdots,u_{k}\right]\! \left[\begin{array}{cccc}\sigma_{1}&0&&&0\\ &\sigma_{2}&&&\\ &&&\ddots&&\\ &&&\sigma_{k}&\\ 0&&&0\end{array}\right]\!\left[\
u_{1},\
u_{2},\
u_{3}\cdots,\
u_{k}\right]^{T} \tag{6}\]  
where \(\left[u_{1},u_{2},u_{3}\cdots,u_{k}\right]\) and \(\left[\
u_{1},\
u_{2},\
u_{3}\cdots,\
u_{k}\right]\) denote their left and right eigenvectors, respectively; the diagonal matrix \(\sum=diag(\sigma_{1},\sigma_{2}\cdots\sigma_{k})\) satisfies \(\sigma_{1}\geq\sigma_{2}\geq\cdots\geq\sigma_{r}\geq\sigma_{r+1}\cdots\geq \sigma_{k}=0\); r denotes the rank of \(\sum\) and equals to the number of non-zero singular values; \(\sigma_{i}\) is a unique singular value of the matrix C as determined by decomposition.  
Because the first singular value \(\sigma_{1}\) contains the most information of the matrix elements, meaning that it is more robust than other singular values, the first singular value \(\sigma_{1}\) of the matrix C, let \(S=\left\{\sigma_{1}^{f}\right\}\), \(f=1,2,\cdots,m\) (m is the number of polylines and polygons, \(S\) is the set of the first singular values from all polylines);  
Let \(\overline{\sigma_{1}}=\frac{1}{m}\sigma_{1}^{f}\), the mean of all the first singular values \(\overline{\sigma_{1}}\) can be obtained, and subsequently compare the size of \(\sigma_{1}^{f}\) and \(\overline{\sigma_{1}}\) to get a binary table \(B_{f}\), as follows:  
\[B_{f}=\left\{\begin{array}{ll}1,\ \sigma_{1}^{f}>\overline{\sigma_{1}}\\ 0,\ \mbox{otherwise}\end{array}\right.\ (f=1,2,\cdots,m) \tag{7}\]  
Through the XOR operation between the binary table \(B_{f}\) and the binary sequence of \(W_{n}\), a zero-watermark \(Z_{f}\), which is consistent with the size of the watermark image, is obtained. If the number of the first singular values is less than the number of watermark bits, the first singular values would be reused orderly to construct zero-watermark. Similarly, a legally validated timestamp is added to the zero-watermark \(Z_{f}\) and is saved to the IPR center.  
\[Z_{f}=B_{f}\oplus\mbox{Bit}(W_{n}),f=1,2,\cdots,m. \tag{8}\]