International Journal of
Geo-Information Article
An Algorithm based on the Weighted Network Voronoi Diagram for Point Cluster Simplification Xiaomin Lu 1,2 , Haowen Yan 1,2, *, Wende Li 3 , Xiaojun Li 1,2 and Fang Wu 4 1 2 3 4
*
Faculty of Geomatics, Lanzhou Jiaotong University, Lanzhou 730070, China;
[email protected] (X.L.);
[email protected] (X.L.) Gansu Provincial Engineering Laboratory for National Geographic State Monitoring, Lanzhou 730070, China School of Resource and Environmental Sciences, Wuhan University, Wuhan 430079, China;
[email protected] Faculty of Geomatics; Information Engineering University, Zhengzhou 450000, China;
[email protected] Correspondence:
[email protected]; Tel.: +86-136-0931-0452
Received: 4 January 2019; Accepted: 23 February 2019; Published: 27 February 2019
Abstract: Points on maps that stand for geographic objects such as settlements are generally connected by road networks. However, in the existing algorithms for point cluster simplification, points are usually viewed as discrete objects or their distances are considered in Euclidean spaces, and therefore the point cluster generalization results obtained by these algorithms are sometimes unreasonable. To take roads into consideration so that point clusters can be simplified in appropriate ways, the network Voronoi diagram is used and a new algorithm is proposed in this paper. First, the weighted network Voronoi diagram is constructed taking into account the weights of the points and the properties of the related road segments. Second, the network Voronoi polygons are generated and two factors (i.e., the area of the network Voronoi polygon and the total length of the dilated road segments in the polygon) are considered as the basis for point simplification. Last, a Cartesian coordinate system is built based on the two factors and the point clusters are simplified by means of the “concentric quadrants”. Our experiments show that the algorithm can effectively and correctly transmit types of information in the process of point cluster simplification, and the results are more reasonable than that generated by the ordinary Voronoi-based algorithm and the weighted Voronoi-based algorithm. Keywords: map generalization; weighted network Voronoi diagram; point cluster simplification; network Voronoi polygon
1. Introduction Point clusters are widely used to visually represent geographic features on maps. For example, settlements, islands, and control points can be presented using point symbols on certain scale maps [1]. When the scale of a map containing point symbols becomes smaller, the point clusters generally become crowded and the map becomes illegible. Under such a circumstance, the point clusters need to be simplified so that the smaller scale map can be read easily. In the process of point cluster simplification, some more important points are retained and the other less important ones are deleted. Automated simplification of point clusters depends on algorithms. By far many achievements have been made at this aspect. As shown in Table 1, existing algorithms can be divided into two categories: the ones that consider weights and the other ones that do not.
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Table 1. Two types of algorithms for point cluster simplification. Algorithm
Weights
Values of the Weights
Settlement-spacing ratio-based Gravity modeling-based Distribution-coefficient Circle-growth MWVD-based
Yes Yes Yes Yes Yes
Hierarchical Voronoi-based
Yes
Convex hull-based On spatial distribution properties preservation Genetic Algorithm Dot maps simplification Circle characters-based Kohonen-based Ordinary Voronoi-based
No
Given based on the importance of the settlements Given based on the importance of the settlements Given based on the importance of the settlements Given based on the characteristics of the points Given according to experience of experts The distance between the current point and other points /
No
/
No No No No No
/ / / / /
Note: ‘/’ means the algorithm does not take the weights into account.
The first type includes five algorithms. Three of them are proposed by Langran & Poiker (1986), i.e., the settlement-spacing ratio algorithm, the distribution-coefficient algorithm and the gravity model. In these algorithms, the simplification is done based on the importance of the settlements and the two-dimensional Euclidean distances between settlements [2]. In addition, van Kreveld et al. (1995) proposed an algorithm based on the idea of circle growth, in which the weights of the points were represented as the size of circles, and the simplification is done based on the size of corresponding circles [3]. The fourth one, the algorithm based on a multiplicatively weighted Voronoi diagram (Yan & Wang, 2013), sets the weights of the points according to the experience of experts, and the weighted Voronoi diagram is employed in point cluster simplification [4]. The last one is proposed by Li et al. (2014), which is based on the hierarchical Voronoi diagram. In the algorithm, the cluster analysis and simplification are done based on the linear distance between current point and other points [5]. The second type includes seven algorithms. The first one is the algorithm based on the convex hull (Wu, 1997), it simplifies point clusters by merging the convex hulls and simplifying the polygons. It can transmit the distribution characteristics of the original point cluster well [6]. The second one is the algorithm for spatial distribution properties preservation (Ai & Liu, 2002), which simplifies the outer points and the internal points respectively to maintain the geometric characteristics of the point cluster [7]. The third one is the algorithm based on the genetic algorithm (Deng et al., 2003). It simplifies point clusters by the adaptive algorithm and the generic algorithm, and it can transmit the distribution scope and local density well [8]. The fourth is the dot map simplification algorithm (de Berg et al., 2004), which simplifies points by ε-approximation and clustering algorithms, and the distribution density and clustering characteristics of the original point cluster can be maintained well through the simplification [9]. The fifth algorithm is based on circle characters (Qian et al., 2006), which simplifies point clusters by repetitive clustering and simplifying operations. It can maintain the distribution center and scope correctly after simplification [10]. The sixth one, a Kohonen net-based algorithm (Cai et al., 2007), can well and effectively preserve the local density and structure characteristics of the original point clusters [11], and the last one, a Voronoi diagram-based algorithm (Yan et al., 2008) can maintain the geometric and topological features after point cluster simplification [12]. It can be concluded that the second type of algorithm focuses on the preservation of geometric structure and topological features of the original point clusters in the simplification. Nevertheless, the weight of the point is ignored in this type of algorithm. In the first type of algorithm, although the weight of the point is considered in the simplification, two problems still exist: (1) The world is abstracted as an infinitely homogeneous and isotropic space, and the distance between two events or facilities is measured by the Euclidean distance [13,14]. However, in the real geographic space, the connection or the distance between points is usually constrained by the road network.
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(1) The world is abstracted as an infinitely homogeneous and isotropic space, and the distance (1) The world is abstracted as an infinitely homogeneous and isotropic space, and the distance between two events or facilities is measured by the Euclidean distance [13,14]. However, in the real between two events or facilities is measured by the Euclidean distance [13,14]. However, in the real ISPRS Int. J. Geo-Inf. 2019, 8, 105 3 of 16 geographic space, the connection or the distance between points is usually constrained by the road geographic space, the connection or the distance between points is usually constrained by the road network. network. (2) Road networks are not taken into account in these algorithms, which can generate the same (2) Road networks are not taken into account in these algorithms, which can generate the same (2) Road networks are not taken into account in these algorithms, which can generate the same results if the same point cluster is put into two different road networks and simplified by the same results if the same point cluster is put into two different road networks and simplified by the same results if the same point cluster is put into two different road networks and simplified by the same algorithm. This is obviously unreasonable, because the relations among the points are not pure algorithm. algorithm. This This is is obviously obviously unreasonable, unreasonable, because because the the relations relations among among the the points points are are not not pure pure geometric relations but are affected by the road network [15]. geometric relations but are affected by the road network [15]. geometric relations but are affected by the road network [15]. To settle down the two problems, the “network Voronoi diagram” is introduced into the new To settle down the two problems, the “network Voronoi diagram” is introduced into the new To settle down the two problems, the “network Voronoi diagram” is introduced into the new algorithm. It is a type of special Voronoi diagram which establishes links between facilities by algorithm. algorithm. It Itis is a a type type of of special special Voronoi Voronoi diagram diagram which which establishes establishes links links between between facilities facilities by by considering the road network related to the facilities rather than only the Euclidean distances among considering the road network related to the facilities rather than only the Euclidean distances among considering the road network related to the facilities rather than only the Euclidean distances among them. To be specific, it takes into account the properties of the roads related to the facilities. Compared them. To be specific, it takes into account the properties of the roads related to the facilities. Compared them. To be specific, it takes into account the properties of the roads related to the facilities. Compared with the ordinary Voronoi diagram, the network Voronoi diagram is more appropriate for analyzing with the ordinary Voronoi diagram, the network Voronoi diagram is more appropriate for analyzing with the ordinary Voronoi diagram, the network Voronoi diagram is more appropriate for analyzing the spatial phenomena and activities that are constrained by networks. For example, it has been the are constrained constrained by by networks. networks. For the spatial spatial phenomena phenomena and and activities activities that that are For example, example, it it has has been been applied in analyzing traffic accidents and crime distribution and describing service regions [15‐18]. applied in analyzing traffic accidents and crime distribution and describing service regions [15‐18]. applied in analyzing traffic accidents and crime distribution and describing service regions [15–18]. Thus, it should be a natural thought to use network Voronoi diagrams to simplify the point clusters Thus, it should be a natural thought to use network Voronoi diagrams to simplify the point clusters Thus, it should be a natural thought to use network Voronoi diagrams to simplify the point clusters that stand for geographic entities on the Earth and are closely related to the nearby road networks. that stand for geographic entities on the Earth and are closely related to the nearby road networks. that stand for geographic entities on the Earth and are closely related to the nearby road networks. The organization of this paper is as follows. After this introduction, the idea of the new algorithm The organization of this paper is as follows. After this introduction, the idea of the new algorithm The organization of this paper is as follows. After this introduction, the idea of the new algorithm is introduced in Section 2. The construction of a weighted network Voronoi diagram (Section 3) and is introduced in Section 2. The construction of a weighted network Voronoi diagram (Section 3) and is introduced in Section 2. The construction of a weighted network Voronoi diagram (Section 3) and the the generation of network Voronoi polygon (Section 4) are presented in detail; the procedure of the the generation of network Voronoi polygon (Section 4) are presented in detail; the procedure of the generation of network Voronoi polygon (Section 4) are presented in detail; the procedure of the deletion deletion of the point is illustrated in Section 5, and then the experiments will be illustrated and deletion of is the point is in illustrated Section 5, and then the will experiments will be discussed illustrated of the point illustrated Section 5,in and then the experiments be illustrated and toand test discussed to test the validity of the new algorithm (Section 6). Finally, some conclusions will be drawn discussed to test the validity of the new algorithm (Section 6). Finally, some conclusions will be drawn the validity of the new algorithm (Section 6). Finally, some conclusions will be drawn and a number of and a number of potential research topics will be given (Section 7). and a number of potential research topics will be given (Section 7). potential research topics will be given (Section 7). 2. Framework of the New Algorithm 2. Framework of the New Algorithm 2. Framework of the New Algorithm Suppose that a number of settlements in a block of a city are represented as point cluster on a Suppose that a number of settlements in a block of a city are represented as point cluster on a Suppose that a number of settlements in a block of a city are represented as point cluster on a map at scale S, and the information of the roads pass through the block is given (Figure 1). If the scale map at scale S, and the information of the roads pass through the block is given (Figure 1). If the scale map at scale S, and the information of the roads pass through the block is given (Figure 1). If the scale of the map becomes S/5, the points on the map become crowded and need to be simplified. Figure 2 of the map becomes S/5, the points on the map become crowded and need to be simplified. Figure 2 of the map becomes S/5, the points on the map become crowded and need to be simplified. Figure 2 shows the procedure of the point cluster simplification proposed in the new algorithm. shows the procedure of the point cluster simplification proposed in the new algorithm. shows the procedure of the point cluster simplification proposed in the new algorithm.
Figure 1. Original point cluster and the road network. Figure 1. Original point cluster and the road network. Figure 1. Original point cluster and the road network. Point cluster + road network Point cluster + road network
Point cluster + weighted network Point cluster + weighted network Voronoi diagram Voronoi diagram
Point cluster + network Voronoi Point cluster + network Voronoi polygon polygon
Construction of weighted Network Construction of weighted Network Voronoi diagram Voronoi diagram
Generation of network Generation of network Voronoi polygons Voronoi polygons
Deletion of points Deletion of points Generalized point cluster Generalized point cluster Figure 2. Flowchart of the new algorithm.
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ISPRS Int. J. Geo‐Inf. 2018, 7, x FOR PEER REVIEW Figure 2. Flowchart of the new algorithm.
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Figure 2. Flowchart of the new algorithm. This network Voronoi diagram‐based algorithm comprises three steps: (1) Firstly, the weighted ISPRS Int. J. Geo-Inf. 2019, 8, 105 4 of 16 network Voronoi diagram is constructed by taking into account the grades of the points, the traffic
This network Voronoi diagram‐based algorithm comprises three steps: (1) Firstly, the weighted capacity, and the directions of the related road segments. After that, the different roads are network Voronoi diagram is constructed by taking into account the grades of the points, the traffic distributed to different points (marked in different color in Figure 3b). (2) Then, the network Voronoi This network diagram-based algorithm comprisesAfter three steps: (1) Firstly, the roads weighted capacity, and the Voronoi directions of the related road segments. that, the different are polygons are constructed (Figure 3c), where the area of the network Voronoi polygon and the total network Voronoi diagram is constructed by taking into account the grades of the points, the traffic distributed to different points (marked in different color in Figure 3b). (2) Then, the network Voronoi length of dilated road segments in the polygon are proposed and used to calculated the selection capacity, and the directions of the related road segments. After that, the different roads are distributed polygons are constructed (Figure 3c), where the area of the network Voronoi polygon and the total probability of the points. (3) Lastly, the point deletion is carried out based on the selection probability to different pointsroad (marked in different color in Figure 3b). (2) Then,used the network Voronoi polygons length of dilated segments in the polygon are proposed and to calculated the selection of the points. are constructed (Figure 3c), where the area of the network Voronoi polygon and the total length of probability of the points. (3) Lastly, the point deletion is carried out based on the selection probability dilated road segments in the polygon are proposed and used to calculated the selection probability of of the points. the points. (3) Lastly, the point deletion is carried out based on the selection probability of the points.
(b)
(a)
(c)
(b) (c) (a) Figure 3. Construction of network Voronoi diagram and network Voronoi polygons: (a) point cluster
and the road network related to them; (b) the network Voronoi diagram of the points; and (c) the Figure 3. Construction of network Voronoi diagram and network Voronoi polygons: (a) point cluster Figure 3. Construction of network Voronoi diagram and network Voronoi polygons: (a) point cluster network Voronoi polygons of the points. and the road network related to them; (b) the network Voronoi diagram of the points; and (c) the and the road network related to them; (b) the network Voronoi diagram of the points; and (c) the network Voronoi polygons of the points. network Voronoi polygons of the points. 3. Construction of Weighted Network Voronoi Diagrams
3. Construction of Weighted Network Voronoi Diagrams To construct weighted network Voronoi diagrams, the stream flowing method, which has been 3. Construction of Weighted Network Voronoi Diagrams tested to be an effective method for constructing the network Voronoi diagrams, is employed here To construct weighted network Voronoi diagrams, the stream flowing method, which has been To construct weighted network Voronoi diagrams, the stream flowing method, which has been [18]. It is described step by step in the following subsections. tested to be an effective method for constructing the network Voronoi diagrams, is employed here [18]. tested to be an effective method for constructing the network Voronoi diagrams, is employed here It is described step by step in the following subsections. [18]. It is described step by step in the following subsections. 3.1. Network Tessellation 3.1. Network Tessellation The roads are tessellated to obtain a discrete representation, which is similar to the rasterization 3.1. Network Tessellation The roads are tessellated to obtain a discrete representation, which is similar to the rasterization of the planar space. This can be achieved by the following way: First, the road network is broken into The roads are tessellated to obtain a discrete representation, which is similar to the rasterization of planar space. This can beconnecting achieved by theneighboring following way: the road (e.g., network is broken a the set of consecutive segments two road First, intersections AB and BC in of the planar space. This can be achieved by the following way: First, the road network is broken into into a set of consecutive segments connecting two neighboring road intersections (e.g., AB and BC in Figure 4). Second, each segment is subdivided into a number of equal‐length linear units, which are a referred to as lixels. The lixels can be interpreted as the line pixel in an image [19]. It should be noted set of segments two into neighboring intersections (e.g., AB and BC are in Figure 4).consecutive Second, each segmentconnecting is subdivided a numberroad of equal-length linear units, which Figure 4). Second, each segment is subdivided into a number of equal‐length linear units, which are referred to as lixels. The lixels can be interpreted as the line pixel in an image [19]. It should be noted that the length of the linear unit is set according to the practical application. For example, compared referred to as lixels. The lixels can be interpreted as the line pixel in an image [19]. It should be noted that the length of the linear unit is set according to the practical application. For example, compared with the bus stop, the service region of a train station is much greater and a greater lixel length can that the length of the linear unit is set according to the practical application. For example, compared with the bus stop, the service region of a train station is much greater and a greater lixel length can be be set for the network tessellation. with the bus stop, the service region of a train station is much greater and a greater lixel length can set for the network tessellation. be set for the network tessellation. A S1
lixel
A
lixel S2 S1 S3 P1’ S4 P1 S2 P1’’ S3 P1’ S4 P1 P1’’
P4 P2
P4 P3
P6 P5
P6
B B
P2 P5 P3 Figure 4. An example of network tessellation. Figure 4. An example of network tessellation.
C C
Once the road network isFigure 4. An example of network tessellation. tessellated, points are projected onto the nearest lixels. The thick linear Once the road network is tessellated, points are projected onto the nearest lixels. The thick linear units in Figure 4 represent the projected lixels. units in Figure 4 represent the projected lixels. Once the road network is tessellated, points are projected onto the nearest lixels. The thick linear According to the role the lixels play in the dilation operation, the lixels may be classified into units in Figure 4 represent the projected lixels. three types: (1) unoccupied lixel, (2) expanding lixel, and (3) occupied lixel. An unoccupied lixel refers to the one that has not been occupied by any of the generators (e.g., S1 in Figure 4). An expanding lixel is the one that has been occupied by a generator but has at least one unoccupied neighboring lixel. It
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According to the role the lixels play in the dilation operation, the lixels may be classified into three types: (1) unoccupied lixel, (2) expanding lixel, and (3) occupied lixel. An unoccupied lixel refers ISPRS Int. J. Geo-Inf. 2019, 8, 105 5 of 16 to the one that has not been occupied by any of the generators (e.g., S1 in Figure 4). An expanding lixel is the one that has been occupied by a generator but has at least one unoccupied neighboring acts as an active head in dilation operation (e.g., S2 , S4 in Figure 4). An occupied lixel refers to the one lixel. It acts as an active head in dilation operation (e.g., S 2, S4 in Figure 4). An occupied lixel refers to itself and its neighbors have been occupied (e.g., S3 in Figure 4). the one itself and its neighbors have been occupied (e.g., S 3 in Figure 4). 3.2. Expansion Operator 3.2. Expansion Operator The The idea idea of of stream stream flowing flowing which which is is similar similar to to the the expansion expansion principle principle in in mathematical mathematical morphology is employed in expansion operations here [18]. In the process of expansion, the projected morphology is employed in expansion operations here [18]. In the process of expansion, the projected lixels of the points are viewed as the source of the generator starting from which the flows expand lixels of the points are viewed as the source of the generator starting from which the flows expand with a step along with related road road segments simultaneously. The flows expand different with a certain certain step along with related segments simultaneously. The flows along expand along tributaries before they reach the intersections of the road network, and the expansion does not stop different tributaries before they reach the intersections of the road network, and the expansion does until the flows encounter each other or reach the end of the road network (Figure 5). As a result, the not stop until the flows encounter each other or reach the end of the road network (Figure 5). As a routes taken by different generators form the network Voronoi diagrams, which partition the impacting result, the routes taken by different generators form the network Voronoi diagrams, which partition area in the network space. the impacting area in the network space.
P4
P1
P4
P1
P6 P2
P5
P3 (a)
P6 P2
P5
P3 (b)
Figure 5. Process of dilation operation: (a) result after the first dilation operation and (b) result after Figure 5. Process of dilation operation: (a) result after the first dilation operation and (b) result after the dilation operation is finished. the dilation operation is finished.
As shown in Figure 5, the stream flowing assumes that the streets are two-way and in As shown in Figure 5, the stream flowing assumes that the streets are two‐way and in homogeneous spaces, and the facility features have equal importance. However, in the real geographic homogeneous spaces, and the facility features have equal importance. However, in the real world, some roads are one-way and/or less important, and some others might be two-way and/or geographic world, some roads are one‐way and/or less important, and some others might be two‐ more important, and the point features themselves have different importance. Obviously, all of the way and/or more important, and the point features themselves have different importance. Obviously, factors take effect in computing the influential regions of the facilities. Therefore, these factors should all of the factors take effect in computing the influential regions of the facilities. Therefore, these be considered when the expansion speed is calculated. These factors are listed as follows: factors should be considered when the expansion speed is calculated. These factors are listed as • Direction of the roads follows: The flow can expand along the road if its expanding direction is the same as the direction of the Direction of the roads road. Otherwise, the flow cannot expand along the road, e.g., if the road extends from west to east, the The flow can expand along the road if its expanding direction is the same as the direction of the flow that is expanding from east to west has to stop on this road. This can be expressed by Equation (1): road. Otherwise, the flow cannot expand along the road, e.g., if the road extends from west to east, the flow that is expanding from east to Lwest has to stop on this road. This can be expressed (1) by p1 = k × F ( P1 , P2 ) Equation (1): where, LP1 is the expansion unit, k is the standard length Lp1 k F ( P1 , of P2 )a lixel, and F(P1 , P2 ) is a direction function. (1) If the flow is the same as the direction of the road, F equals 1; otherwise, F equals 0. •where, L TrafficP1capacity of the roads is the expansion unit, k is the standard length of a lixel, and F(P 1, P2) is a direction Because different categories of roads have different traffic capacity, the new algorithm set different function. If the flow is the same as the direction of the road, F equals 1; otherwise, F equals 0. weights for different categories of roads, e.g., assign 2 to arterial roads and 1 to secondary roads. The Traffic capacity of the roads function that traffic capacity plays in expansion operationtraffic can becapacity, expressed by:new algorithm set Because different categories of the roads have different the different weights for different categories of roads, e.g. assign 2 to arterial roads and 1 to secondary L p2 = k × W ( x ) (2) roads. The function that traffic capacity plays in the expansion operation can be expressed by:
Lp2 k the W(traffic x) capacity of the roads. where, LP2 is the expansion unit and W(x) denotes (2) • Weight of the points where, LP2 is the expansion unit and W(x) denotes the traffic capacity of the roads. The weight of the point depends on the scale and power of the corresponding facility. The higher Weight of the points the weight of the point, the larger its influence area is, which can be represented in expansion operation by Equation (3): L p3 = k × W ( p) (3)
The weight of the point depends on the scale and power of the corresponding facility. The higher the weight of the point, the larger its influence area is, which can be represented in expansion The weight of the point depends on the scale and power of the corresponding facility. The higher operation by Equation (3): the weight of the point, the larger its influence area is, which can be represented in expansion operation by Equation (3):
Lp3 k W ( p) Lp3 k W ( p)
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(3) where, LP3 is the expansion unit and W(p) denotes the weight of the point. If all the above factors are considered, the expansion speed can be defined as Equation (4) where, LP3 is the expansion unit and W(p) denotes the weight of the point. where, L P3 is the expansion unit and W(p) denotes the weight of the point. If all the above factors are considered, the expansion speed can be defined as Equation (4) If all the above factors are considered, the expansion speed can be defined as Equation (4) Lp k W ( x) W( p) F(P1, P2 ) (4) k WW((xx))× W F(FP(1P ,P ) (4) Lp L =p k× W((pp))× (4) 12, P2 ) Figure 6 shows the expansion result when all of the three factors are considered, where routes that are generated from different points are marked in different colors. Figure 6 shows the expansion result when all of the three factors are considered, where routes Figure 6 shows the expansion result when all of the three factors are considered, where routes that are generated from different points are marked in different colors. that are generated from different points are marked in different colors. Point weight is 2 Point weight is 1 Point weight is 2 Arterial roads Point weight is 1 Secondary roads Arterial roads Road direction Secondary roads P1
P4
P1
P4
P2
P 3
P2
P 3
Road direction P6 P5 P6
P1
P4
P1
P4
P 5
(a)
P2
P3
P2
P3
P6 P6
P5 (b)
P5
(a) (b) Figure 6. Construction of weighted network Voronoi diagram: (a) initial point cluster and road Figure 6. weighted network Voronoi diagram: (a) initial point cluster and roadand network network and (b) weighted Voronoi diagram of point cluster. Figure 6. Construction Construction ofof weighted network Voronoi diagram: (a) initial point cluster road and (b) weighted Voronoi diagram of point cluster. network and (b) weighted Voronoi diagram of point cluster.
4. Construction of Network Voronoi Polygons 4. Construction of Network Voronoi Polygons 4. Construction of Network Voronoi Polygons In the weighted network Voronoi diagram, the road network and its related space are assigned In the weighted network Voronoi diagram, the road network and its related space are assigned to different points. The road segments generated from the same point (marked in the same color in In the weighted network Voronoi diagram, the road network and its related space are assigned to different points. The road segments generated from the same point (marked in the same color in Figure 6) may constitute a polygon region. The polygon region which can be named network Voronoi to different points. The road segments generated from the same point (marked in the same color in Figure 6) may constitute a polygon region. The polygon region which can be named network Voronoi polygon can represent: (1) the importance of the point, which is the basis of the point simplification; Figure 6) may constitute a polygon region. The polygon region which can be named network Voronoi polygon can represent: (1) the importance of the point, which is the basis of the point simplification; and and (2) the distribution characteristics of the point cluster, which should be considered in the polygon can represent: (1) the importance of the point, which is the basis of the point simplification; (2) the distribution characteristics of the point cluster, which should be considered in the simplification simplification of the point cluster. Therefore, network polygon important in in the the and (2) the distribution characteristics of the the point cluster, Voronoi which should be is considered of the point cluster. Therefore, the network Voronoi polygon is important in the simplification of a simplification of a point cluster. Its construction is described as follows: simplification of the point cluster. Therefore, the network Voronoi polygon is important in the point cluster. Its construction is described as follows: Step 1: Construct the constrained Delaunay triangulation using the nodes of the road segments simplification of a point cluster. Its construction is described as follows: Step 1: Construct the constrained Delaunay triangulation using the nodes of the road segments (points P 1‐P12 in Figure 7a) [20]. Step 1: Construct the constrained Delaunay triangulation using the nodes of the road segments (points P1 -P12 in Figure 7a) [20]. (points P1‐P12 in Figure 7a) [20].
P10
P12 P12 P11 P11 P1 P1
P9
P8 T3 T1 T3 T2 T1 P3 T2 P3 P2 (a) P2
P10
P12
P9 P8
P10
P10
P10
P12 P6 TP 5 6
P7
P11
P7
P11
T 5 P1 P4 P4
P1
P5
P9 P8
P9
P8 PL PL PR
P5
P10
P12 P12
P6 P6
P7 P7
P11 P11 P1
PR P3 P2
P3
P4
P2
(b)
P4
P1
P9 P8
P9
P8 P5 P5
P6
P7
P6
P7
P3 P2
P3
P4
P2
(c)
P4
(c) (a) (b) Figure Figure 7. 7. Construction Construction of of constraint constraint Delaunay Delaunay triangulation triangulation of of road road segments: segments: (a) (a) Delaunay Delaunay triangulation based onon thethe nodes of theof road segments; (b) influence polygons polygons (PL, PR) are(PL, constructed triangulation based nodes the road segments; (b) influence PR) are Figure 7. Construction of constraint Delaunay triangulation of road segments: (a) Delaunay by deleting triangles (T -T5triangles ) that intersect the road segments; and (c) Delaunay triangulation of the 5) road that intersect segments; and (PL, (c) Delaunay constructed by deleting (T1‐T triangulation based on 1 the nodes of the segments; the (b) road influence polygons PR) are influence polygons. triangulation of the influence polygons. constructed by deleting triangles (T1‐T5) that intersect the road segments; and (c) Delaunay triangulation of the influence polygons. Step 2: Calculate the boundary polygon by the method of dynamic threshold ‘stripping’, which
may be described below: (1) Set threshold d = k × Avelength, where k is the grade of ‘stripping’ (here, k = 2) [21], Avelength is the average value of all the triangle edges in the Delaunay triangulation, and d is dynamically updated after every “stripping” [22].
Step 2: Calculate the boundary polygon by the method of dynamic threshold ‘stripping’, which Step 2: Calculate the boundary polygon by the method of dynamic threshold ‘stripping’, which may be described below: may be described below: d d kkAv eAle g th (1) (1) Set Set threshold , where k is k the grade of ‘stripping’ (here, k = k 2) v enle n g th threshold , where is the grade of ‘stripping’ (here, = [21], 2) [21], Avelength is the average of all the the triangle edges in the Delaunay triangulation, and d 16 is Avelength is the average value of all triangle edges in the Delaunay triangulation, and d is ISPRS Int. J. Geo-Inf. 2019, 8, 105 value 7 of dynamically updated after every “stripping” [22]. dynamically updated after every “stripping” [22]. (2) Compare every outside non‐feature edge (the edge which has only one adjacent triangle and (2) Compare every outside non‐feature edge (the edge which has only one adjacent triangle and (2) Compare every outside non-feature edge (the edge which has only one adjacent triangle and is not the the feature edge of the linear object group) with threshold d. If it than d, then is not feature edge of the linear object group) with threshold d. If is it greater is greater than d, is then not the feature edge of the linear object group) with threshold d. If it is greater than d, then determine determine if the other two edges of the current triangle can form a triangle with any other edge after determine if the other two edges of the current triangle can form a triangle with any other edge after if the other two edges of the current triangle can form a triangle with any other edge after deleting the deleting the current edge. If the result is positive, delete this edge and turn to 3), otherwise the edge deleting the current edge. If the result is positive, delete this edge and turn to 3), otherwise the edge current edge. If the result is positive, delete this edge and turn to 3), otherwise the edge should be should be retained; repeat this step. should be retained; repeat this step. retained; repeat this step. (3) (3) Set the the other two edges of the triangle that the the deleted edge belongs to as edges, Set other two edges of the triangle that deleted edge belongs to outside as outside edges, (3) Set the other two edges of the triangle that the deleted edge belongs to as outside edges, recalculate the value of Avelength and update the value of d. Strip inward layer by layer until the recalculate the value of Avelength and update the value of d. Strip inward layer by layer until the recalculate the value of Avelength and update the value of d. Strip inward layer by layer until the length of all the outside non‐feature edges is less than the current threshold d. length of all the outside non‐feature edges is less than the current threshold d. length ofThe ‘stripping’ result of the linear object group in Figure 7c is shown in Figure 8a. all the outside non-feature edges is less than the current threshold d. The ‘stripping’ result of the linear object group in Figure 7c is shown in Figure 8a. The ‘stripping’ result of the linear object group in Figure 7c is shown in Figure 8a. (4) Connect the outside edges end to end to form the final boundary polygon (Figure 8b) (4) Connect the outside edges end to end to form the final boundary polygon (Figure 8b)
(b) (b)
(a) (a)
Figure 8. Construction of boundary polygon: (a) result of stripping outside triangles of the road Figure 8. Construction of boundary polygon: (a) (a) result of stripping outside triangles of the road Figure 8. Construction of boundary polygon: result of stripping outside triangles of the road segments and (b) the final boundary polygon. segments and (b) the final boundary polygon. segments and (b) the final boundary polygon.
(4) Connect the outside edges end to end to form the final boundary polygon (Figure 8b). By doing this, the network Voronoi polygons are constructed (i.e. S 1 in Figure 9), by which the By doing this, the network Voronoi polygons are constructed (i.e. S 1 in Figure 9), by which the By doing this, the network Voronoi polygons are constructed (i.e., S1 in Figure 9), by which the weights of the points can be calculated and the point cluster simplification can be done . . weights of the points can be calculated and the point cluster simplification can be done weights of the points can be calculated and the point cluster simplification can be done. S1: network Voronoi polygon of point P 1 S1: network Voronoi polygon of point P 1
P1 P1
P4 P4 P2 P2 P3 P3
P6 P6 P5 P5
Figure 9. Network Voronoi polygon. Figure 9. Network Voronoi polygon. Figure 9. Network Voronoi polygon.
5. Deletion of Points 5. Deletion of Points 5. Deletion of Points 5.1. Strategies Used in Information Transmission 5.1. Strategies Used in Information Transmission 5.1. Strategies Used in Information Transmission One of the main goals of the algorithm for point cluster generalization is to correctly transmit the information contained in the original point cluster. Therefore, some strategies are used in the One of the main goals of the algorithm for point cluster generalization is to correctly transmit One of the main goals of the algorithm for point cluster generalization is to correctly transmit proposed algorithm to transmit thethe four different informations on point maps, i.e., statistical, thematic, the the information contained in the original point cluster. Therefore, some strategies are are used in the information contained in original point cluster. Therefore, some strategies used in the topological, and metric information [23]. proposed algorithm to transmit the four different informations on point maps, i.e., statistical, proposed algorithm to transmit the four different informations on point maps, i.e., statistical, • Statistical information thematic, topological, and metric information [23]. thematic, topological, and metric information [23]. The Law is an extensively applied method for calculating the number of objects that • • Radical Statistical information Statistical information should appear on a target scale map in map simplification. Thus, the number of points thatobjects should bethat The Radical Law is an extensively applied method for calculating the the number of that The Radical Law is an extensively applied method for calculating number of objects retained on the target map can be determined by Equation (5) [24]. should appear on a target scale map in map simplification. Thus, the number of points that should should appear on a target scale map in map simplification. Thus, the number of points that should be retained on the target map can be determined by Equation (5) [24]. be retained on the target map can be determined by Equation (5) [24]. q N = N0 · So /S f (5) where, N is the number of points on the target map; N0 is the number of points on the original map; So is the denominator of the original map scale, and Sf is the denominator of the target map scale. • Thematic information
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N N 0 S0 Sf
(5)
0 is the number of points on the original ISPRSWhere, N is the number of points on the target map; N Int. J. Geo-Inf. 2019, 8, 105 8 of 16
map; So is the denominator of the original map scale, and Sf is the denominator of the target map scale. • Thematic information The importance of the point is considered as the basis for point cluster simplification, which is The importance of the point is considered as the basis for point cluster simplification, which is decided by both the weight of the point and the properties of the related road segments. In the new decided by both the weight of the point and the properties of the related road segments. In the new algorithm weighted network Voronoi diagram, it can be seen that the of theof point algorithm based based on on the the weighted network Voronoi diagram, it can be seen that weight the weight the and the properties of the road segments can be represented by the network Voronoi polygons. point and the properties of the road segments can be represented by the network Voronoi polygons. For the network Voronoi polygon, firstly, the region of it will be larger if the point and its related For the network Voronoi polygon, firstly, the region of it will be larger if the point and its related road segments are more important; secondly, it has a lot to do with the density of the road network. The road segments are more important; secondly, it has a lot to do with the density of the road network. resulting network Voronoi polygons can be quite different, although the flows expand with equal speed The resulting network Voronoi polygons can be quite different, although the flows expand with equal at equal times. For example, the network Voronoi polygon of P1 is larger than that of P2 (Figure 10), speed at equal times. For example, the network Voronoi polygon of P 1 is larger than that of P 2 (Figure but the total length of the expanded routes (road segments in polygon) of P is smaller than that of P2 . 1 10), but the total length of the expanded routes (road segments in polygon) of P1 is smaller than that Such results were produced because of the different local densities. Thus, the regions of the network of P2. Such results were produced because of the different local densities. Thus, the regions of the Voronoi polygons as well as the total length of dilated road segments in the polygon are treated as network Voronoi polygons as well as the total length of dilated road segments in the polygon are the basis for point simplification. In the following simplification, a basic rule is generally abided by: treated as the basis for point simplification. In the following simplification, a basic rule is generally ‘the larger the network Voronoi polygon and the longer the total length of dilated road segments in abided by: ‘the larger the network Voronoi polygon and the longer the total length of dilated road the polygon, the more probable the point can be retained on the resulting map’. This rule obviously segments in the polygon, the more probable the point can be retained on the resulting map’. This rule may ensure that the points of great importance and points next to the important roads have a higher obviously may ensure that the points of great importance and points next to the important roads have possibility to be shown on the generalized map. a higher possibility to be shown on the generalized map.
P1
P2
(a)
(b)
Figure 10. Measurement of points’ weights. (a) Network Voronoi polygon of P1 and (b) network Figure 10. Measurement of points’ weights. (a) Network Voronoi polygon of P1 and (b) network Voronoi polygon of P2. Voronoi polygon of P2.
In the new algorithm, the relative area of the network Voronoi polygon (Pi1 ) and relative total In the new algorithm, the relative area of the network Voronoi polygon (Pi1) and relative total length of road segments in polygons (Pi2 ) are computed by Equation (6) and Equation (7). After that, length of road segments in polygons (Pi2) are computed by Equation (6) and Equation (7). After that, Pi1 and Pi2 together determine the probability of a point to be retained. Pi1 and Pi2 together determine the probability of a point to be retained. n
n Pi1Pi1= AAii/ ∑ AAi i
ii= 11
(6) (6)
th point. where, A i is the area of the network Voronoi polygon of the i where, Ai is the area of the network Voronoi polygon of the ith point.
Pi 2 li
n
n l i i
1 l Pi2 = li / ∑ i
i =1
(7) (7)
where, li is the total length of road segments in the network Voronoi polygon of the ith point. where, li is the total length of road segments in the network Voronoi polygon of the ith point. There may be a special case in which the road segments generated from a point cannot form a Theree.g. may be ais special caseroad in which the road segments generated a point cannotonly form a polygon, there only one segment generated from the point. from In this situation, the polygon, e.g., there is only one road segment generated from the point. In this situation, only the length of the road segment is calculated for thematic information. length road segment is calculated for thematic information. • of the Topological information •Although it is impossible to protect topological relations among the points in the process of point Topological information Although it is impossible to protect topological relations among the points in the process of point cluster simplification, it is still a principle to try and minimize damaging their topological relations. cluster simplification, it is still a principle to try and minimize damaging their topological relations. Thus, the rule ‘do not delete any two neighboring points’ needs to be observed [25], i.e., any two Thus, the rule ‘do not delete any two neighboring points’ needs to be observed [25], i.e., any two points points whose network Voronoi polygons are neighbors cannot be deleted simultaneously in the same whose network Voronoi polygons are neighbors cannot be deleted simultaneously in the same round round of point deletion. of point deletion. In the simplification, each point may be in one of the three statuses: ‘fixed’, ‘deleted’, or ‘free’. In the beginning, all the original points are marked as ‘free’. If a point is marked as ‘deleted’, it means that this point is a candidate that will be deleted but not all at once. To ensure the efficient transmission
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the beginning, all the original points are marked as ‘free’. If a point is marked as ‘deleted’, it means that this point is a candidate that will be deleted but not all at once. To ensure the efficient of the topological information, if a point is marked as ‘deleted’, its neighbors should be marked as transmission of the topological information, if a point is marked as ‘deleted’, its neighbors should be ‘fixed’. Fixed points cannot be marked as ‘deleted’ in the same round of point deletion, ensuring that marked as ‘fixed’. Fixed points cannot be marked as ‘deleted’ in the same round of point deletion, no adjacent points are deleted simultaneously. This step is repeated until no points can be marked as ensuring that no adjacent points are deleted simultaneously. This step is repeated until no points can ‘deleted’. In this procedure, the points marked as ‘fixed’ belong to type I, which will be retained; the be marked as ‘deleted’. In this procedure, the points marked as ‘fixed’ belong to typeⅠ, which will points which are marked as ‘deleted’ belong to type II and will be deleted; other points belong to type be retained; the points which are marked as ‘deleted’ belong to type II and will be deleted; other III and will compete to decide whether they will be retained or deleted [26]. points belong to type III and will compete to decide whether they will be retained or deleted [26]. • Metric information • Metric information In the new algorithm, the relative area of the network Voronoi polygon that reflects the local In the new algorithm, the relative area of the network Voronoi polygon that reflects the local relative density is employed as a metric measure. It works together with the rule ‘do not delete any relative density is employed as a metric measure. It works together with the rule ‘do not delete any two neighboring points’ and ensures that the metric information can be clearly transmitted. two neighboring points’ and ensures that the metric information can be clearly transmitted. 5.2. Process of Point Deletion 5.2. Process of Point Deletion In the process of point deletion, it is easy to select the points of type II because if a point will In the process of point deletion, it is easy to select the points of type II because if a point will be be deleted in the simplification, it is very likely that its relative area of the network Voronoi polygon deleted in the simplification, it is very likely that its relative area of the network Voronoi polygon (Pi1 ) and the relative total length of dilated road segments in the polygon (Pi2 ) are both small. If the (Pi1) and the relative total length of dilated road segments in the polygon (P i2) are both small. If the corresponding points are marked in the Cartesian coordinate system, where the area of the network corresponding points are marked in the Cartesian coordinate system, where the area of the network Voronoi polygon is the abscissa and the total length of the road segments in the polygon is the ordinate, Voronoi the be abscissa and total of the road on segments the deletion polygon the the pointspolygon of type IIis will nearer to thethe origin oflength the system. Based this, the in point is is done ordinate, the points of type II will be nearer to the origin of the system. Based on this, the point as follows: deletion is done as follows: Step 1: The number of points to be deleted is determined by Equation (8): Step 1: The number of points to be deleted is determined by Equation (8): nn = int((N int N0 −NN) )
(8) (8)
O
where, n is the number of points to be deleted; N0 is the0 is the number of points on the original map, number of points on the original map, and N where, n is the number of points to be deleted; N is the number of points on the target map which is calculated by the Radical Law. and N is the number of points on the target map which is calculated by the Radical Law. Step 2: The values of Pi1i1 and P and Pi2i2 are marked as weighted points in the Cartesian coordinate system are marked as weighted points in the Cartesian coordinate system Step 2: The values of P with Pi1i1 as its abscissa and P as its abscissa and Pi2i2 as its ordinate. The weighted points of the point cluster in the example as its ordinate. The weighted points of the point cluster in the example with P are shown in Figure 11a. are shown in Figure 11a.
(a)
(b)
.
Figure 11. Deletion of points: (a) normalized weighted point and (b) deletion method of Figure 11. Deletion of points: (a) normalized weighted point and (b) deletion method of concentric concentric quadrants. quadrants.
Step 3: Concentric quadrants are drawn starting from a quadrant whose center is the origin and Step 3: Concentric quadrants are drawn starting from a quadrant whose center is the origin and the radius is the distance between the origin and the nearest weighted point to the origin. the radius is the distance between the origin and the nearest weighted point to the origin. Step 4: Concentric quadrants are drawn recurrently at one step intervals of the minimum distance Step the 4: weighted Concentric quadrants drawn recurrently at one intervals of the and minimum between points (Figureare 11b). The weighted points on step the current quadrant in the distance between the weighted points (Figure 11b). The weighted points on the current quadrant and stripe between it and the prior quadrants will be selected after each round, and their corrsponding in the instripe between it whose and the prior are quadrants will be neighbor selected points after each their points the point cluster, statuses ‘free’ and their have round, not beenand marked
corrsponding points in the point cluster, whose statuses are ‘free’ and their neighbor points have not been marked as ‘deleted’, will be marked as ‘deleted’. Meanwhile, their neighbor points will be marked as ‘fixed’. After this round of marking, the number of points marked as ‘deleted’ will be compared with value n. If it is smaller than n, then Step 4 will be repeated; if it is greater than n, turn ISPRS Int. J. Geo-Inf. 2019, 8, 105 10 of 16 to Step 5; otherwise, turn to Step 6. Step 5: The points that have been marked as ‘deleted’ in the prior stripe are marked as “free” and they will compete with each other to decide which one should be deleted. The area of the network as ‘deleted’, will be marked as ‘deleted’. Meanwhile, their neighbor points will be marked as ‘fixed’. Voronoi polygon is treated as the main basis for deletion in this situation. The corresponding points After this round of marking, the number of points marked as ‘deleted’ will be compared with value n. are sorted by ascending order of the area of their network Voronoi polygons, and the points in the If it is smaller than n, then Step 4 will be repeated; if it is greater than n, turn to Step 5; otherwise, turn tail will be marked as ‘deleted’. Turn to Step 6 until the number of the points marked as ‘deleted’ to Step 6. equals n. The points that have been marked as ‘deleted’ in the prior stripe are marked as “free” and Step 5: Step 6: The points marked as to ‘deleted’ are deleted from the point cluster and of the rest points they will compete with each other decide which one should be deleted. The area the network constitute the target point cluster. Voronoi polygon is treated as the main basis for deletion in this situation. The corresponding points are sorted by ascending order of the area of their network Voronoi polygons, and the points in the tail 6. Experimental Studies and Discussion will be marked as ‘deleted’. Turn to Step 6 until the number of the points marked as ‘deleted’ equals n. Step 6: The points marked as ‘deleted’ are deleted from the point cluster and the rest points 6.1. Experiments constitute the target point cluster. The new algorithm has been implemented by the authors in Matlab (R2013a) on Microsoft 6. Experimental Studies and Discussion Windows 7. A number of datasets have been used to test the validity of the algorithm, and two of them are shown here. The data used in experiment 1 is a block of the Lanzhou City, China which 6.1. Experiments contains 10 hospitals and 337 road segments, as shown in Figure 12a. The hospitals are of the same The new algorithm has been implemented by the authors in Matlab (R2013a) on Microsoft weight, and the road segments are of the same category. The original map scale is 1:10K and the target Windows 7. A number of datasets have been used to test the validity of the algorithm, and two of them map scale is 1:25K. The dataset used in experiment 2 shows a much more complicated city block (it are shown here. The data used in experiment 1 isin a block of the Lanzhou City, contains China which contains 10 is Shenzhen city, China) compared with that experiment 1. This map 96 educational hospitals and 337 road segments, as shown in Figure 12a. The hospitals are of the same weight, and institutions (63 public educational institutions and 33 private educational institutions) and 4652 road the road segments are of the same category.1765 The secondary original map scale is 1:10K and target map scale is segments (2217 arterial road segments, road segments, and the 670 pedestrian road 1:25K. The dataset used in experiment 2 shows a much more complicated city block (it is Shenzhen segments), as shown in Figure 14a. The original map scale is 1:50K and the target map scale is 1:100K. city, China) compared with that in experiment 1. This map contains 96 educational institutions (63 To demonstrate that the results generated by the new algorithm are more reasonable than the public educational institutions and 33 private educational institutions) and 4652 road segments (2217 algorithm based on the ordinary Voronoi diagram, the same point clusters shown in Figure 12a and arterial road segments, 1765 secondary road segments, and 670 pedestrian road segments), as shown Figure 14a are used to test them. The generalized results are shown in Figure 13b and Figure 15b, in Figure 13a. The original map scale is 1:50K and the target map scale is 1:100K. respectively.
P5 P1
P7
P3 P2 P4 P6
P8
P9
P10
(a)
P5 P1
P7
P3 P2 P4 P6
(b)
P7 P8 P9
P1 P10
P2
P3 P4
P8 P10
(c)
.
Figure 12. Experiment 1 for testing the new algorithm: (a) source data at scale 1:10K; (b) network Voronoi diagram of the points; and (c) generalized point data at scale 1:25K. The maps are not shown exactly to scale.
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Public educational institution Private educational institution Arterial road segments Secondary road segments 2 Pedestrian road segments
(a)
Points deleted in the generalization
P1’ P2’
(c)
(b)
P3’
.
Figure 13. Experiment 2 for testing the new algorithm: (a) source data at scale 1:50K; (b) network Figure diagram 14. Experiment for testing new algorithm: source data 1:100K. at scale The 1:50K; (b) are network Voronoi of point 2 cluster; and the (c) generalized point(a) data at scale maps not Voronoi diagram of point cluster; and (c) generalized point data at scale 1:100K. The maps are not shown exactly to scale. ISPRS Int. J. Geo‐Inf. 2018, 7, x FOR PEER REVIEW 11 of 16 shown exactly to scale.
To demonstrate that the results generated by the new more1:10K; reasonable than the Figure 12. Experiment 1 for testing the new algorithm: (a) algorithm source data are at scale (b) network algorithm based on the ordinary Voronoi diagram, the same point clusters shown in Figures 12a and Voronoi diagram of the points; and (c) generalized point data at scale 1:25K. The maps are not shown 13a are used to test them. The generalized results are shown in Figures 14b and 15b, respectively. exactly to scale. Points deleted in the generalization
P5
P1
P
P3 P2
P4
P6
(a)
P5 P8 P 9
P10
P2
P3 P6
P7 P9
P10
(b)
(a) (b) Figure 14. Experiment 1 for testing the algorithm based on the ordinary Voronoi diagrams and . Figure 13. Experiment 1 for testing the algorithm based on the ordinary Voronoi diagrams and comparing it with the new algorithm: (a) ordinary Voronoi diagram of the points and (b) generalized comparing it with the new algorithm: (a) ordinary Voronoi diagram of the points and (b) generalized Figure Experiment 2 for testing the shown algorithm based on the weighted Voronoi diagrams and point data15. at scale 1:25K. The maps are not exactly to scale. point data at scale 1:25K. The maps are not shown exactly to scale. comparing it with the new algorithm: (a) weighted Voronoi diagram of point cluster and (b) generalized point cluster. The maps are not shown exactly to scale.
It can be concluded from Figure 12 and Figure 13: (1) In Figure 12b, the flows generated from different points are of the same expansion speed, because all points have the same weight and so do The points marked in red in Figure 14b and Figure 15a are the points to be deleted in the process the roads. (2) In the simplification based on the proposed algorithm, the effect of the road network of simplification. From Figure 14 and Figure 15, it can be concluded that: (1) The points of greater on the influence region of the points is taken into account. For example, it can be seen that P importance have higher possibilities to be retained in the process of simplification if 9 is deleted the data is and P8 is retained in Figure 12c, while P9 is retained and P8 is deleted in Figure 13b. This is because in the new proposed algorithm, P8 has a larger influence region because it is nearer to the intersection of the road network than P9 is. But in the algorithm based on the ordinary Voronoi diagram, P8 is
(c)
(b)
P3’
.
Figure 14. Experiment 2 for testing the new algorithm: (a) source data at scale 1:50K; (b) network ISPRS Int.Voronoi diagram of point cluster; and (c) generalized point data at scale 1:100K. The maps are not J. Geo-Inf. 2019, 8, 105 12 of 16
shown exactly to scale.
Points deleted in the generalization
(a)
(b)
.
Figure 15. Experiment 2 for testing the algorithm based on the weighted Voronoi diagrams and Figure 15. Experiment 2 for testing the algorithm based on the weighted Voronoi diagrams and comparing it with the new algorithm: (a) weighted Voronoi diagram of point cluster and (b) generalized comparing it with the new algorithm: (a) weighted Voronoi diagram of point cluster and (b) point cluster. The maps are not shown exactly to scale. generalized point cluster. The maps are not shown exactly to scale.
It can be concluded from Figures 12 and 14: (1) In Figure 12b, the flows generated from different The points marked in red in Figure 14b and Figure 15a are the points to be deleted in the process points are of the same expansion speed, because all points have the same weight and so do the of simplification. From Figure 14 and Figure 15, it can be concluded that: (1) The points of greater roads. (2) In the simplification based on the proposed algorithm, the effect of the road network on the importance have higher possibilities to be retained in the process of simplification if the data is influence region of the points is taken into account. For example, it can be seen that P9 is deleted and P8 is retained in Figure 12c, while P9 is retained and P8 is deleted in Figure 14b. This is because in the new proposed algorithm, P8 has a larger influence region because it is nearer to the intersection of the road network than P9 is. But in the algorithm based on the ordinary Voronoi diagram, P8 is deleted because of a smaller Voronoi polygon than that of P9 . The points marked in red in Figures 13b and 15a are the points to be deleted in the process of simplification. From Figures 13 and 15, it can be concluded that: (1) The points of greater importance have higher possibilities to be retained in the process of simplification if the data is generalized by the new algorithm (91.9% public educational institution are retained and 32.3% of private educational institutions are retained); (2) the points nearer to the intersections of roads are more likely to be retained in the simplification than the farther ones if the data is generalized by the new algorithm (i.e., P1 ’ , P2 ’ in Figure 13c); (3) the points whose related roads have greater weights have greater possibilities to be retained after generalization if the data is generalized by the proposed algorithm (for example, P3 ’ in Figure 13c). 6.2. Algorithm Evaluation As mentioned in the previous sections, the generalization algorithm should transmit the four types of information well. Therefore, the following standards are defined to evaluate the new algorithm quantitatively: (1) The transmission rate of the statistical information (Ds ) is measured by the deviation between the number of generalized points (Ng ) and the theoretical number of points (No ) that should be retained (Formula (9)). Ds = Ng − No
(9)
The smaller the value of Ds , the better is the transmitted statistical information. (2) The transmission rate of the thematic information (Dth ) is measured by the deviation between the average weight value of the original points (Wo ) and that of the generalized points (Wg ) (Equation (10)).
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Dth = Wg − W0
(10)
The greater Dth is, the more probable it is that points with greater weights are retained. Therefore, the greater Dth is, the better the thematic information is transmitted. (3) The transmission rate of the topological information (Dtp ) is measured by the difference between the mean number of the neighbors of the original network Voronoi polygon of the points on the generalized map (d g ) and that of the original map (do ) (Equation (11)). Dtp = d0 − d g
(11)
The smaller Dtp is, the better the topological information is transmitted. (4) The transmission rate of the thematic information (Dm ) is measured by the change of the area of the range polygon (Dm ), which can be evaluated by Equation (12). Po − Pg (12) Dm = Po where, Po is the area of the original range polygon, and Pg is the area of the range polygon on the generalized map. The smaller Dm is, the better the metric information is transmitted. The results of the indices calculated using the experiments are listed in Table 2. Table 2. Used in the experiments for evaluating the new algorithm. Experiment
Experiment 1
Experiment 2
Algorithm
Ds
Dth
Dtp
Dm
Algorithm based on the ordinary Voronoi diagram
0.68
/
0.59
2.85%
New algorithm based on the weighted network Voronoi diagram
0.68
/
0.34
0.628%
1
0.056
1.563
1.324%
0.21
0.154
1.412
1.383%
Algorithm based on the weighted Voronoi diagram New algorithm based on the weighted network Voronoi diagram
Table 2 indicates that the new algorithm transmits the information of the original points correctly after simplification. A number of insights can be gained from Table 2: (1) The number of the points on the generalized map obtained using the proposed algorithm is approximately equal to the theoretical number of the points that should be retained. (2) The deviation in the mean normalized weights (Dth ) increases during the simplification, which means the points with greater weights are retained and those with less weights are deleted. (3) The average change of the number of the neighbors (Dtp ) is small, which means the topological relations are transmitted well. (4) The change of the range polygon (Dm ) is not large, and the errors are acceptable. 6.3. Discussions From the experiments and evaluation, it can be seen that: (1) The network Voronoi diagram is different from the ordinary Voronoi diagram. The space divided by the ordinary Voronoi diagrams occupies the entire planar area, while the space divided by the network Voronoi diagrams only covers the space occupied by the road network. In addition, the boundary of an ordinary Voronoi polygon is a smooth line, while the boundary of a network Voronoi polygon is rather jagged. It can be imaged that the ordinary Voronoi diagrams and the network Voronoi diagrams can be the same if the road network is dense enough; and (2) the network Voronoi diagram performed better than the ordinary Voronoi diagram in point cluster simplification because roads are taken into account which adds more
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necessary and useful information to the construction of the weighted network Voronoi diagrams and, therefore, makes the process of point cluster simplification reasonable. Compared with the algorithm based on the ordinary Voronoi diagrams, the algorithm based on the weighted network Voronoi diagram has higher time complexity, because all road segments in the road network are traversed in the construction of weighted network Voronoi diagrams. Table 3 shows the time costs for the experiments with different datasets and different lixel unit lengths at 1m and 5m, respectively. From the table it can be seen that the smaller length of the lixel unit is, the longer the experiment time, and the time cost has approximately linear growth with the refining process of lixel length. Thus, the determination of lixel length is a key problem in the construction of the network Voronoi diagrams as well as in point simplification. In the new algorithm, the lixel unit length is given according to the following factors: (1) The average distance between points, i.e., the greater the average distance between the two points, the larger the linear unit can be; and (2) the influence region of the point, for example, the influence region of train stations is much greater than that of bus stops; when we construct the network Voronoi diagram for train stations, the lixel unit length can be set much greater. Table 3. Time costed in point simplification. Experiments
Lixel Length (m)
Number of Lixels in the Road Network
Total Time (s)
Experiment 1 (Figure 12)
1 5
1323 321
14.3 7.1
Experiment 2 (Figure 13)
1 5
32859 7278
19.4 9.1
7. Conclusions Point cluster simplification is an important part of map generalization. It also plays an important role in spatial analysis and urban planning. Because points that stand for geographic objects are generally connected and constrained by road networks, the network Voronoi diagram rather than the ordinary Voronoi diagram is used in the new algorithm. In addition, the importance of the points are affected by their connected and/or nearby road networks; therefore, in this new algorithm, the weighted network Voronoi diagram is employed as a tool to simplify point clusters and it is constructed by taking into account the weight of the points and the properties of the related road segments. To complement the point deletion, the network Voronoi polygons are generated and two factors (area of network Voronoi polygon and total length of dilated road segments in the polygon) are proposed and used to calculate the probability of the point to be retained. Based on these two factors, point simplification is done by the method of “concentric circle”. The classic method of point simplification is based on the ordinary Voronoi diagrams or the weighted Voronoi diagrams. Compared with them, the proposed new algorithm has the following advantages: (1) more effective transmission of information of the original point cluster and (2) more reasonable generalization results because of the integration of the road network information into point cluster simplification. Some pieces of progress made in this study can also be used in other similar domains. For example, the network Voronoi diagram as well as the network Voronoi polygon (Peterson et al., 2017) [27] can be applied to other research. The method of “concentric circle” proposed for point deletion can be used in other similar studies. The algorithm can also be extended to lines and polygons simplification. However, the algorithm may be further improved by introducing other possible factors such as human traffic and traffic flow into the construction of a network Voronoi diagram. We will apply the algorithm and improve it in the future. Compared with the other algorithms, the difference of time-consumption between the new algorithm and the other existing algorithms lies in the time spent in the construction of the network
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Voronoi diagram, which has approximately linear growth with the refining process of lixel length. In practical applications, the length of the lixel can be adjusted by the actual demand. For example, the length of lixel can be set as a small value if higher accuracy is required; on the other hand, a big value can be set to the length of lixel if a higher speed is demanded. To solve the problem of large amounts of data, the divide-and-conquer approach can be referred to, i.e., the construction of a weighted network Voronoi diagram and the simplification of a point cluster can be done based on a series of space slices. Author Contributions: All the authors contributed to the development of the proposed generalization algorithm and this manuscript. X.L. and H.Y. proposed the methodology. X.L. performed the experiments and wrote the draft of the manuscript. W.L. Visualized the weighted network Voronoi diagram. F.W. analyzed and evaluated the results. H.Y. and X.L. guided the research and revised the manuscript. Funding: This research was funded by the National Key R&D Program of China (No. 2017YFB0504203) and the National Nature Science Foundation of China (No. 41801395, 41671447 and 41761088). Acknowledgments: The authors are grateful to anonymous reviewers, whose comments and suggestions have helped us to improve the context and presentation of the article. Conflicts of Interest: The authors declare no conflicts of interest.
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