A Skeleton-based Density Partition Method for
the Evaluation of Road Network Selection
Fuquan Xiong*, Chang Ren*, Rui Wang*, Jing Tian* **
* School of Resource and Environment Science, Wuhan University
** Key Laboratory of Geographic Information System, Ministry of Education, PR China
Abstract. Road density is a widely used index in the analysis of ecological
effects, road network planning, delimitation of urban areas and road network selection. To evaluate results of road network selection, many indexes
are being used, such as using similarity to evaluate the length similarity of
road networks before and after selection, using connectivity to measure the
connectivity of retained road networks, and using visual inspection to visually check whether the road network pattern is preserved. However, little
attention has been paid to use density to evaluate results of selection. Although mesh-density based approach and the recently proposed combination of stroke-mesh approach can retain the road network density difference,
these methods did not use density to evaluate results of selection. This paper proposes a road density partition method based on skeleton partition of
road network and the idea of spatial autocorrelation. Firstly, constrained
Delaunay triangulation is applied to the road segments and skeletons are
generated. Then, approximated Voronoi cells of each road segment are
formed based on these skeletons. Secondly, Getis-ord Gi* is used to identify
statistically significant spatial clusters of high and low values of cellular
area. Finally, the neighboring cells are aggregated based on the statistically
significant spatial clusters of high and low values. The road network data of
Hong Kong Island was obtained from OpenStreetMap, and Google Maps
was referenced to select roads at different scales for the evaluation of road
network selection. The result shows that the densities in the experimental
partitions by and large match the road network density in reality, and well
reflect the road density contrast before and after selection. Additionally, the
result shows that this method is better compared with grid density method.
Keywords: road network selection, density, skeleton partition, spatial autocorrelation, Getis-Ord’s Gi*
1.
Introduction
The level of detail varies on maps of different scales because the available
space for delineating objects is limited. To avoid congestion and conflict
between cartographic objects when the map scale is reduced, objects have
to be selectively omitted, simplified and displayed. That is exactly the job of
cartographic generalization, a process that requires that the main characteristics and the logical pattern of those objects are preserved. Density is
one of those main characteristics. Although it is unpractical to preserve all
the nuances in local densities, the density contrast should be preserved;
that is, after generalization, regions with a relatively higher density should
still be the dense regions, not the converse.
Roads are one of the most important cartographic objects. Existing automated road selection methods can be classified into four categories, according to the difference between the methods of network simplification. The
first category is road intersection selection/aggregation, which involves
selecting roads by selecting or merging road intersections (Bjørke 2003).
The second is road meshes merging, which starts from the place where the
constraint conditions are not met, and calculates the importance degree of
the segments that constitute meshes or calculates the influence after merging meshes and then selectively removes less important road segments to
make new meshes (Peng & Muller 1996, Edwardes & Mackaness 2000,
Chen et al. 2009, Touya 2010). The third is road omission/selection, which
involves eliminating those road units that are less important and preserving
the road units that can reflect the thematic function of networks
(Mackaness 1995, Richardson & Thomson 1996, Thomson & Richardson
1999, Jiang & Claramunt 2004, Jiang & Harrie 2004, Zhang 2005, Tomko
et al. 2008, Liu et al. 2010, Yang et al. 2011). The fourth is the recently proposed stroke-density combination algorithm (Li & Zhou 2012, Weiss &
Weibel 2014). For the indexes to evaluate road network generalization
methods, there exist three measures: Similarity, Connectivity and Visual
Inspection. Similarity evaluates the similarity between road networks before and after generalization; Connectivity measures the retained road network's connectivity, and Visual Inspection visually checks whether the road
network pattern is preserved or not (Li & Zhou 2012). Since density is a
main characteristic of cartographic objects, another evaluation index is
whether the generalization result has preserved the density contrast and
how well it has done so. Some pioneer works proposed the concepts of
mesh density (Chen et al. 2009) and skeleton density (Liu et al. 2009) in
road selection. However, these were used to identify the regions where spatial conflicts lie and determine the importance of road segments, rather
than to evaluate the selection result.
As a kind of high order information, road density is not only used in cartographic generalization but also widely applied in other domains, such as in
the analysis of ecological effects (Hawbaker et al. 2005, Cai et al. 2013),
road network planning (Yang et al. 1996), and delimitation of urban zones
(Borruso 2003). Existing methods for calculating road density include traditional density (Hawbaker 2004), grid density (Borruso 2003), kernel
density (Borruso 2003, Cai et al. 2013), mesh density (Chen et al. 2009)
and skeleton density (Liu et al 2009, Liu et al. 2010), based on three statistical units: road intersections, road segments and road meshes. The aforementioned computation methods for road density can be primarily divided
into two categories. One is to first divide the whole area into several regions
and then, in each region, calculate the ratio between the road length and the
area of the region, consistent with the traditional density definition. The
other is to calculate the density of one road segment or one mesh. The division in the former category is arbitrary and some important parameters
such as the grid resolution are not easy to determine. Additionally, the divisions usually do not reflect the density distribution. The latter does not represent the density of an area since what it calculates is the density of one
segment or mesh, so it differs from human perception of dense areas and
sparse areas. Skeleton density, proposed by Liu et al. (2009), can reveal the
overall road distribution, yet the road network was not partitioned explicitly. This paper proposes a density partition method based on skeleton partition of road networks and applies it to the evaluation of road network selection results.
The rest of paper is structured as follows. Section 2 describes the method
for density partition based on skeleton partition and Getis-Ord’s Gi*. Section 3 presents the experiment and the application in evaluation of road
network selection. Conclusions are drawn in section 4.
2.
Methods
Inspired by the extraction of the “natural city” based on the spatial autocorrelation of the cells made by a road network (Jiang & Jia 2011), this article
proposed a density partition method based on spatial autocorrelation. In
this method, the skeleton of the road network partitions the whole studied
area into polygons, which are similar to Voronoi cells. Visually denser regions of the road network correspond to clusters of small Voronoi cells,
whereas sparse regions correspond to clusters of large Voronoi cells. The
method consists of two steps. First, the approximate Voronoi diagram of the
road segment is generated based on the skeleton, to partition the entire
area. Then, the high-value and low-value clusters that are statistically sig-
nificant are extracted based on the index of local spatial autocorrelation,
Getis-Ord’s Gi*. Since it is the area of Voronoi cells that is considered, the
extracted low-value clusters correspond to denser road network regions,
and the extracted high-value clusters correspond to the sparse regions. The
Voronoi cells corresponding to neighboring low-value areas and neighboring high-value areas are merged respectively, thus finalizing the road network partition.
2.1. Skeleton partition of road network
First, constrained Delaunay triangulation is applied to the road segments,
and then skeletons are generated. Following this, approximate Voronoi cells
can be formed based on these skeletons (Liu et al. 2009, Liu et al. 2010). As
an example, part of the road network of Hong Kong and the corresponding
skeletons are shown in Figure 1.
2.2. Density partition based on Getis-Ord Gi*
The high-value cluster areas of road segments (small-area Voronoi cell clusters) correspond to areas with dense road distribution, whereas the lowvalue cluster areas (large-area Voronoi cell clusters) correspond to areas
with sparse road distribution. This indicates local spatial autocorrelation.
Getis-Ord Gi* is a common statistic to describe local spatial autocorrelation.
It tells where statistically significant features with either high or low values
cluster spatially (Getis & Ord 1992, Ord & Getis 1995, Esri 2013). The Gi*
statistic returns a z-score and a p-value for each feature (each cell in this
work). The z-scores are equal to the ratios between the deviation from the
Figure 1. Part of Hong Kong road network and corresponding skeletons.
observed value to the expected value and the standard deviation; the pvalues are probabilities that a random process created the observed pattern.
When the absolute value of the z-score is large and the p-value is small,
there exists a high value spatial clustering. Otherwise, when the value of the
z-score is small and negative and the p-value is large, there exists a low value spatial clustering. The larger (or the smaller) the z-score, the greater the
degree of clustering. If the value of the z-score is close to 0, there is no obvious spatial clustering (Getis & Ord 1992). In general, a 95% confidence level
is chosen for statistically significance test. The detailed calculation is as follows:
Gi 
*

n
j 1
wi , j x j  X  j 1 wi , j
n


2
n
n

2
n
w

w
 j 1 i, j 
  j 1 i , j
S
n 1

X
S

n
j 1
n
n
j 1
xj
n
x j2
X2
(1)
(2)
(3)
in which Gi* is actually a z-score (Esri 2013); xj is the area of the jth Voronoi
cell; X is the mean value of the areas of all the Voronoi cells; wij is the
weight of the Voronoi cells i and j. The weight used in this paper is based on
a constant distance threshold; i.e. the weights between one Voronoi cell and
other cells within a certain distance are all 1, while the weights for cells beyond this distance are all 0. n is the number of cells and is also the number
of road segments.
A key problem is determining the distance threshold. According to ArcGIS
Help (Esri 2013) and some successful cases (Truong & Somenahalli 2011),
the criteria to choose an appropriate distance threshold is as follows: (1)
Make sure that each element has a neighbor; and (2) If the area of Voronoi
cells is not normally distributed, make sure that each element has eight
neighbors. According to the above criteria, in our experiment, the smallest
distance to ensure that every element has a neighbor is 600m.
After acquiring the statistically significant clusters with high and low values
based on Getis-Ord’s Gi*, the study utilized terminology in the textbook
(Wang et al. 1993) and merged those neighboring cells which corresponded
to statistically significant low value zones (Gi*<-1.96) at a confidence level of
95%; these cells are dense regions. Then, the study merged those neighboring cells that corresponded to regions that did not have statistical significance (-1.96≤Gi*≤1.96) at a confidence level of 95%; these cells are medium
density regions. In the end, the study merged those neighboring cells that
corresponded to statistically significant high value zones (Gi*>1.96) at a
confidence level of 95%. These cells are sparse regions.
3.
Application of the Proposed Method in Evaluating
Road Selection
3.1. Data and experiment methods
The purpose of the experiment presented in this section is to validate
whether the proposed density partitioning method correctly reflects the
variation of road density, before and after road selection. The experiment
also assesses whether the proposed method can be used to evaluate the
quality of road selections. This experiment used road network data at multiple scales. The road network data of Hong Kong Island was obtained from
OpenStreetMap (OSM), and Google Maps was referenced to select roads at
different scales (Fig. 2).
The experiment was performed by the following procedure:
(1) Perform density partitioning to the data with the most detailed scale,
using the proposed method.
(2) After partitioning, calculate the road density of each partition, based on
the ratio of the total length of roads in this partition to the total area of this
partition. Check if the density of each partition is in proper order, i.e. dense,
medium and sparse regions have a highest, medium and lowest road densi-
(a)
(b)
(c)
Figure 2. Road network of Hong Kong Island. (a) level 14 (b) level 13 (c) level 12
ty, respectively.
(3) Partition the multi-scale data based on the partitioning result of step 1,
and calculate the road density of each partition.
(4) Determine whether the density variations of the multi-scale data are in
accordance with the principles of map generalization. The principles, in
terms of the density contrast plot before and after road selection, include
the following: a) for the same area, density decreases; b) for denser areas,
the decrease is more significant; c) there is no major reverse of the density
curve.
3.2. Results and comparison
(a)
(b)
Figure 3. Partition result. (a) before merging (b) after merging
Figure 4. Road density contrast before and after road selection using the proposed method.
Figure 3 shows the partitioning result of the level 14 road network. The
small-area regions produced because of boundary effects are marked by
ellipses (Fig. 3a). To eliminate the impact of boundary effects, the smallarea regions were merged to their neighboring regions; this helps to filter
clutter in the partition result. Fig. 3b shows the final partition result. As
seen in this figure, the resultant dense and sparse regions are generally consistent with visual perception.
Figure 4 shows the density variation comparison among the road networks,
represented in three scales, partitioned according to Fig. 3b. As can be seen,
for the same region, the density value decreases (the level 12 density curve
is below the level 13 density curve, and they both are below the level 14 density curve). Only one of the regions has a reversed density after road selection (the three density curves have similar shapes and fluctuations, except
for Region 6 in level 12). The density of the denser regions decreased more
significantly than the medium density regions, and both decreased more
(a)
(b)
(c)
(d)
Figure 5. Road density contrast before and after road selection using the grid density. (a) 3*3 grid; (b) 4*4 grid; (c) road density contrast using 3*3 grid; (d) road
density contrast using 4*4 grid
than the density of sparse areas. Although a certain level of evening is produced, the density decrements are acceptable; this can be seen through
comparison with Fig. 2. Overall, the density regions partitioned with the
proposed method well reflect the density variations of the road network at
multiple scales.
The proposed method was compared with the grid density method. The
reason to choose the grid density method was that it produces explicit partitions, which is not a feature of the kernel density, mesh density or skeleton
density methods. The grid density partitioning result is shown in Fig. 5; two
resolutions were chosen, one close to and one greater than the number of
partitions of our method. As seen in Fig. 5, the reversed densities of certain
regions occurred in a 4-by-4 grid (Regions 7 and 8; 13 and 14), and the density difference was truncated in a 3-by-3 grid (Regions 1 and 4; 6 and 7). In
addition, the partitions produced by the grid density method do not show a
clear difference among dense and sparse regions. When the resolution is
close to the number of partitions of the proposed method, the trends of the
three density curves are more similar. Therefore, the proposed method is
superior to the grid density method, while simultaneously overcoming the
latter’s problem of sticky resolution determination.
4.
Conclusion
This paper proposes a method for road density partitioning. The method
first generates skeletons for road segments and then extracts high value
clusters and low value clusters of approximate Voronoi cells using GetisOrd’s Gi* based on the idea of local spatial autocorrelation; finally, associated with a 95% confidence level, it merges those cells that correspond to the
high value and low value regions to implement the partition.
This method does not use regular meshes or administrative divisions but
the characteristics of the road itself to implement the partition, and the
shape of each zone is irregular. Experiments on Hong Kong Island level 12,
13 and 14 road network data were carried out. Density calculation and comparison analysis indicated that the method correctly reflected the road density contrast before and after generalization.
The proposed method has boundary effects. The boundaries of the data
used in the experiments are assumed to be rectangles, which are often not
natural boundaries. Thus, further studies should focus on the boundary
revision.
Acknowledgment
We thank OpenStreetMap contributors for providing the impressive road
network data. This work was supported by National Science Foundation for
Fostering Talents in Basic Research of the National Natural Science Foundation of China under Grant No. J1103409.
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