# Interactive Visualization of Geographic Vector Big Data Based on Viewport Generalization Model
## 3 Materials and Methods  
### Viewport Generalization Visualization Model  
The key ideas of the viewport generalization visualization model (VGVM) of geographic vector data are introduced in this section. Traditional geographic vector data visualization methods generally adopt a data-driven model--that is, starting from each geographic vector feature, the visualization results under a given resolution viewport are analyzed and calculated; then, the visualization results of all geographic vector features are superimposed, combined, and outputted. By summarizing the process of traditional methods, we propose a general processing model of traditional geographic vector data visualization methods, which can be expressed by Equation (1). In the equation, \(d_{i}\) is a vector feature of the geographic vector dataset \(D\) within the viewport, \(N\) is the number of features, \(V(D)\) is the result of the visualization processing of \(D\), \(M(. )\) is a plotting function of vector features, and \(F_{1}\otimes F_{2}\otimes\cdots\otimes F_{k}\) represents the K-connected analytical processing functions. The meaning of the equation is as follows: For each vector feature in the dataset, K analysis processing operations are executed in sequence and associated to obtain the analysis processing result of the feature, and then the results are plotted. Finally, the plotted results of all vector features are aggregated to obtain the final visualization result \(V(D)\). \[V(D)=\sum_{i=1}^{N}M[F_{1}\otimes F_{2}\otimes\cdots\otimes F_{k}(d_{i})],D=(d_ {i}|i=(1,\ldots,N)) \tag{1}\]  
Assuming that the average cost of analyzing each vector feature is \(A\) and the average number of drawn pixels for visualizing the analysis results of each vector feature is \(P\), the computational complexity of Equation (1) can be expressed as Equation (2). \[O[N\times(A\times K+P)] \tag{2}\]  
We observe that the visualization of geographic vector data is achieved by calculating the display value of each pixel within the viewport. According to Equation (1), when a pixel is hit by multiple geographic vector features, its display value may be computed several times; thus, a large number of redundant calculations are generated. Therefore, this paper designs a visualization model from the viewport perspective, whereby the display values of image pixels are quickly calculated by combining multiple geographic vector features within the viewport pixel range, as shown in Equation (3). In the equation, \(D\) is the geographic vector data, \(p_{i}\) is the pixel with the viewport, \(C\) is the total number of viewport pixels, and \(R(p_{i})\) is a general calculation function of geographic vector features that can be designed in different forms. In Equation (4), a binarization design of \(R(p_{i})\) is shown, the core of which is the comprehensive calculation of the geographic vector features in the spatial range corresponding to image pixel \(p_{i}\); when there are no geographic vector features in the spatial range corresponding to image pixel \(p_{i}\), the pixel value is 0. Conversely, when there are geographic vector features in the spatial range, the pixel value is 1. At the same time, as in Equation (5), \(R(p_{i})\) can also be designed as a function of the number of vector features in the corresponding spatial range of \(p_{i}\)--that is, a function related to the density distribution of geographic vector data. \[V(D)=\sum_{i=1}^{C}R(p_{i}) \tag{3}\]  
\[R(p_{i})= \begin{cases}0&S(p_{i})\cap D=\phi\\ 1&S(p_{i})\cap D\
eq\phi\end{cases}  \tag{4}\]  
\[R(p_{i})=F(|S(p_{i})\cap D|) \tag{5}\]  
As can be seen from the above equation, the visualization result \(V(D)\) of the dataset \(D\) has been converted from relying on the plotted value of each vector feature \(d_{i}\) (Equation (1)) to relying on the intersection result of the corresponding spatial range of the image pixel \(p_{i}\) with \(D\) (Equation (3)). Meanwhile, the computational complexity of Equation (3) mainly comes from the computational complexity of \(R(p_{i})\).