# A Blockchain Copyright Protection Model Based on Vector Map Unique Identification
## 3 Basic Idea and Preliminaries
### Vector Map Unique Identification Based on Geometric and Topological Relationships
To facilitate understanding, we illustrate the distance measurement using Figure 4. Let us suppose that, in the dataset, the similar features to feature A are Points 1 and 2, while the dissimilar features are Polyline 1, Polyline 2, and Polygon 1. Among these, only Polyline 1 and Polygon 1 are disjointed and do not intersect with A. By calculating the Hausdorff distance between feature A and Polyline 1, as well as between feature A and Polygon 1, and then taking the minimum value, we obtain the nearest non-intersecting heterogeneous feature, \(NN(A)\). It is necessary to utilize the R-tree spatial index to find the nearest candidate features; this should be followed by an exact Hausdorff distance calculation for the candidate set. Assuming the candidate set for feature \(i\) is \(C_{i}\), the precise calculation process is as follows:  
\[\forall j\in C_{i},if\ d(i,j)=\underset{k\in C_{i}}{\text{min}}(i,k),\ then\ record\ d(i,j) \tag{5}\]  
For each feature \(I\), it is necessary to record the nearest non-intersecting heterogeneous feature \(d(i,NN(i))\) into the result sequence \(D\). The sequence is formatted as follows:  
\[D=\{d(i,NN(i))|i\in S\} \tag{6}\]After obtaining the distance sequence \(D\), we need to calculate the mean value of the sequence \(D\). Let \(D\) contain \(n\) distance values, denoted as \(D=\{d_{1},\,d_{2},\ldots\,d_{n}\}\). The formula for calculating the mean \(\overline{d}\) is as follows:  
\[\overline{d}=\frac{1}{n}\sum_{i=1}^{n}d_{i} \tag{7}\]  
In the formula, \(d_{i}\) is the \(i\)-th distance value in the sequence \(D\), and \(n\) is the length of the sequence \(D\) (i.e., the total number of features). Ultimately, by comparing the specific distance of each geographical feature with the global mean, the correlation between geographical features is quantified in a binary manner to obtain the topological feature parameters of the vector map dataset. \[T_{i}= \begin{cases}1&\quad\text{if}\ \frac{d_{i}}{\overline{d}}>1\\ 0&\quad\text{if}\ \frac{d_{i}}{\overline{d}}\leq 1\end{cases}  \tag{8}\]  
where \(T_{i}\) represents the topological feature parameter in the \(i\)-th position of the unique identifier. Finally, by using the geometric feature parameter \(H_{ijk}\) as the index and the topological feature parameter \(T_{i}\) as the feature value, the unique identifier of the vector map can be constructed. Continuing with the example of rotational transformation, we focus on analyzing the stability of geometric and topological feature parameters amidst geometric transformations like rotation, scaling, and translation (RST). For the geometric feature parameter \(H_{ijk}\), the calculation of the angle after rotation is independent of the rotation itself. According to Equation (9), the geometric feature parameter \(H_{ijk}\) remains unchanged after rotation. According to Equation (10), distance calculation only involves positional information and is unaffected by rotation. Therefore, the topological feature parameter \(T_{i}\) also remains unchanged after rotation transformation. \[H_{ijkr}=f(\theta_{1r},\theta_{2r},\ldots,\theta_{lenr})=f(\theta_{1},\theta_{ 2},\ldots\theta_{len})=H_{ijk} \tag{9}\]  
Figure 4: Schematic of nearest neighbor non-intersecting heterogeneous feature query based on Hausdorff distance. \[T_{i}\left(r\right)=d_{ir}/\overline{d_{r}}=d_{i}/\overline{d}=T_{i} \tag{10}\]  
where \(H_{ijkr}\) and \(T_{i}(r)\) represent the geometric and topological feature parameters, respectively, after rotation transformation. \(\theta_{1r}\), \(\theta_{2r}\), and \(\theta_{lenr}\) denote the angle values of each geographical feature in the vector map dataset after rotation, while \(d_{ir}\) indicates the distance between each geographical feature after rotation and the nearest dissimilar feature. \(d_{r}\) represents the global mean distance of the vector map dataset after rotation.