CONSISTENT LINE SIMPLIFICATION BASED ON CONSTRAINT POINTS

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1 CONSISTENT LINE SIMPLIFICATION BASED ON CONSTRAINT POINTS Shen Ying, Lin Li and Yuanyu Zhang School of Resource and Environment Science, Wuhan University, China, ABSTRACT Line generalization, one of the most important fields of the cartographic generalization, still has a robust life with many years development. That has two reasons: one is the importance of polylines in map representation, the other is the imperfectability and immaturity of the theory of cartographic generalization. Making a comprehensive view of the past algorithms and papers, we find that almost all of them are inclusively concerned with polyline s geometric characters, and care little about cartographic contexture which map features deal with. But the constraints and the operations play an important role in line simplification. They guide and govern map generalization. Topological consistency is a primary criterion of the estimation of cartographic generalization, especially in structural cartographic generalization based on geographic environment. During line simplification we should keep spatial relations consistent according to the geographic characters adjacent to linear features. This paper develops some algorithms to simplify polylines consistently based on the constraint points, whether they are on lines or not. Keywords: map Generalization; line simplification; Binary-Line-Generalization Tree; Constraint Point INTRODUCTION Line simplification plays an important role in cartographic generalization because most map features are represented by polylines and polygons composed of polylines. Line simplification, known as a method of reducing the polyline details, not only make map more legible and clearly but improves efficiency and visualization of cartography. With several decades development, there are several approaches in polyline simplification such as Douglas-Peucker (DP) algorithm (Douglas D H, Peucker T K., 1973), Reumann-Witkam algorithm (1974), Strip-tree (Buttenfield B P. 1986; 1989;1991) and BLG-tree (Peter van Oosterom, Vincent Schenkelaars, 1995; Zhan F, Mark D M., 1993; Saalfeld A., 1999; Cromley, R. G.,1991;1992). We can come to one point according to these algorithms that they are all based on the simplification of feature individuals. They simplify the polylines simply through selection of geometric characteristic points on polylines of each map individual object. However, geographic structural/context characteristics is another crucial factor in cartographic generalization and the research about simplification based on them, especially on the spatial structural characteristic constraints, becomes a hotspot and more and more popular. There are two kinds of characteristic constraint points: natural characteristic points and cultural characteristic points. The natural characteristic points are mainly defined as the characteristic points of landform and geometry like inflexions of landform, geometric inflexions and extreme value points, while the cultural characteristic points are characteristic points formed by human beings during the process of conquest and reconstruction of nature varying from cities, residences to bridges, boundary positions and so on. Of course, many polygon features have been abstracted into points in the foregoing simplification. We always assume there are no self-intersection and topological conflicts to shy away these problems, let the solution methods be alone. Actually these happens Because of improper geometric threshold, non-synchronous simplification or ignorance of spatial structural characteristics, which all influence data accuracy and the efficiency of spatial database Let us begin by illustrating some things that can go wrong with simplification algorithm. For example, Figure 1 below is a simplified North European map from which we can see many topological character or sidedness of some points altered as a result of DP algorithm: the land Canterbry lies in the sea, Maastricht has been changed its national boundary and there also exists self-intersection in the simplified map which is opposite to the true situation of the original map. The reasons for these inaccurate results are the polyline simplification algorithm has not taken the spatial relation of polyline and cultural characteristic constraint points into account, which changed the positions of map objects essentially. Proceedings of the 21 st International Cartographic Conference (ICC) Durban, South Africa, August 2003 Cartographic Renaissance Hosted by The International Cartographic Association (ICA) ISBN: Produced by: Document Transformation Technologies

2 Figure 1. Inconsistent Simplification Using DP Algorithm (Mark de Berg, et al, 1998) The paper applies DP algorithm and BLG-tree which are commonplace in map generalization integrated with relationships among points, polylines and polygons to carry out consistent line simplification with constraint points. The methods in this paper have small complexity of computation and practicability. The paper is organized as follows. Section 2 analyzes the theory and relation between geographic context characteristics and geometric graph; section 3 develops three methods to solve consistent line simplification in illusion to characteristic points on polylines and section 4 aim at constraint points out of polylines; section 5 gives an example and some concluding remarks. 2. ANALYSIS In the course of spatial analysis and decision-making process, geographic spatial contextual characteristics are often in a higher level while geometric ones in a lower level, between which the relationship is similar to that of decision-making level and implemental level. The objects of Cartographic generalization are geographic structural characteristics objects not geometric detail characteristics of specific features, and the aim of generalization is to make decisions to stand out the main characteristics of geographic environment on the basis of structural analysis of geographic context. While the implementation of map generalization is carried out by a series of geometric operators and the geometric graphs express and present the process and result of map generalization. These form the mutual conditional and restricted relationship between geographic spatial characteristics and geometric characteristics. The relationships among spatial objects, as the primary method to describe the geographic characteristics, can be interpret as integrity constraints of spatial database; topological relations, as the essential representation of spatial relationships, can be considered as primary factor in map generalization. Above all, during line simplification based on geographic structural characteristics, the paper focuses on characteristic constraint points, topological relation is in the first status and geometric precise second. Actually in most cases, the precise of characteristics constraints is more important than that of polylines because of for their locational, cultural and military significances. Hence, no replacement among constraint points is constantly required in the following analysis. Generally speaking, generalized polylines after line simplification should have less internal nodes, geometric simplification conditions satisfied and no self-intersection. Considering restrict conditions of constraint points, the relations between points in the plane and polylines should keep consistent both before and after simplification. That is to say we shouldn t change the structure of characteristic points distribution in the plane with line simplification. So, if line simplification has no self-intersection, satisfy geometric threshold and maintain topological relationship, we can define it as consistent line simplification. Many researches have be studied in the field of consistent polyline simplification: research of intersection and selfintersection in contour generalization (SHEN Ying, 2001), comparisons among the relations of points, lines and polygons (Qingsheng Guo, 1993), a cognitive study about constraint conflicts of simplification (Zhan F, Mark, 1993), topologically correct subdivision simplification using the bandwidth criterion (Mark de Berg et al., 1998), consistent line simplification with the Douglas-Peucker Algorithm(Saalfeld A, 1999) and about frameworks for generalization constraints and operations based on object-oriented data structure in database generalization(yaolin Liu, et al.,2001). 3. CONSISTENT SIMPLIFICATION BASED ON CONSTRAINT POINTS ON THE POLYLINE For the constraint points on the polyline, such as a residential point on a road, a bridge on a river, a boundary point on a bourn and so on, they are not allowed to have any displacement for their locational significances. So we should make

3 sure the simplified polyline passes or contains these cultural characteristics points as they do before line generalization. Three methods are introduced to deal with this problem in this section. 3.1 Constraint point segmentation If we segment a polyline according to the constraint point, the point will become a starting-point or end-point of new polylines. Then we simplify the new polylines with DP algorithm or any other algorithms as to reserve starting-points and end-points frequently in the polyline simplification. In Figure 2, for example, we hypothesize that P 3 and P 7 are constraint points. Figure 2a shows DP algorithm in the entire polyline while Figure 2b is the result of segmenting the line on P 3 and P 7 and corresponding simplification separately. Obviously, because of the difference of begin-point, the chosen polyline s characteristic points and the ultimate graphs are correspondingly different. That also indicates the drawbacks of DP algorithm like that for its excessive dependence on starting-points. The disadvantage of the segmentation method is that: it adds certain the complexity of spatial database since segmentation inevitably will need more identified information when polylines are to be stored. 3.2 BLG-tree simplification The algorithm of BLG-tree to construct a hierarchical structure about points on line is based on the Douglas-Peucker algorithm (e.g. Figure 3b). BLG-tree simplification can not only reduce the repeatedly calls for DP algorithm to speed the data retrieve, but also simplify on the fly in different scale with the help of reducing or adding the whole or part of polyline points tree level according to scales or details requirement. Figure 2. Constraint Points Segmentation To simplify with constraint points using BLG-tree, first we should construct a BLG-tree of a polyline and do pay much attention that constraint points are the polyline interior nodes all the same but they need special labels. During line simplification, we trace throughout BLG-tree, select nodes from the top to bottom of the tree and examine whether they meet the requirements of simplification, that s geometric threshold. If there are still some constraint points left to be chosen and then we need go on selecting nodes in the part of polyline until it meet all the constraint conditions, especially the topological constraints. For example, in Figure 2a, given P 7 is a constraint point that has not been chosen within DP algorithm. Figure 3a is the BLG-tree for figure 2a in the graph. We can see the simplified result with BLGtree choosing points by tracing throughout the sub-tree from P 4 to P 8 and end up on P 7 which is a constraint point. BLGtree simplification can not only guarantee the whole coordination, symmetry and perfection of graph structure, but also keep consistent topological relation constraint between points and polyline unchanged. Figure 3. BLG-Tree Simplification 3.3 Points displacement In addition, we can calculate the nearest point to constraint point P on DP simplified line and displace that point to the constraint one. But it is merely an post-process method for the later modifications and disposal of a few points.

4 Figure 4. Points Displacement 4. CONSISTENT SIMPLIFICATION BASED ON CONSTRAINT POINTS OUT OF POLYLINES The constraint points on the line can be retained compulsive as characteristic points, while constraint points out of polyline will be more complicated to handle with and we should examine the topological relation between constraint points and polyline. In order to reduce the volume of data, we can cut down the field of researched rang with constraint points. There are often two means to achieve it:! Just test characteristic points on the contiguity relevant to polyline (always two);! Just test constraint points in the convex hull determined by polyline. In essence, topologically consistent simplification require the topological relation between constraint points and polyline (prior to or after simplification) should be unchanged, which means the area bounded by the source polyline and the target one should contain no constraint points. Then spatial relations among geographic structural characteristics objects will not change. We can tell whether topological inconsistency happens or not by examine if a constraint point lies inside the polygon formed by the source and simplified polylines. From the viewpoints of plane division, the domain of these polygons is smaller than both areas acquired by former two approaches. Through the analyses, we can infer that polylines of source and target make up of several polygons and each polygon is composed of the segment of simplified polyline and a series of original polyline segment. For an instance, polygon P 4 P 5 P 6 P 7 P 8 in Figure 5 is a complicated one that consists of original segment P 4 P 5 P 5 P 6 P 6 P 7 P 7 P 8 and simplified line P 4 P 8. Topological inconsistency can be identified by means of examining if there are constraint points in the polygon. Two approaches are described in the following in the opinion of constraint point out of polylines. 4.1 Consistent simplification with shortcut What we called shortcut is that it connects the starting-point with the point followed to build a new straight line. It makes threshold or bandwidth etc. a geometric condition and the topological relation between the new polygon, formed by the shortcut besides original polylines, and constraint points a topological condition. If the result meets these two conditions, the shortcut will be a generalized straight line, and the starting-point will alter to the next one and iterate that. It can be divided into two steps as below. Figure 5. Consistent Simplification with Shortcut The first step: Supposing we start from point P i (i [1,n-2], n is total node number), connect P i with P i+j (j [n+2,n], n is total node number), and then we can get a polygon P i P i+1 P i+j formed by shortcut PiPi+j and original polyline segments P i P i+1, P i+1 P i+2 P i+j-1 P i+j.if the conditions satisfy geometric constraints (threshold and bandwidth). And no constraint point is in the polygon, the connected point travel forward, e.g. P i with P i+j+1 ; if there is a constraint point in the polygon P i P i+1 P i+j P i+j+1, the shortcut P i P i+j will be the straight line of simplification of polyline P i P i+1 P i+j. In Figure 5, given P 4 is the starting point, shortcut P 4 P 7 and P 4 P 8 meet geometric simplification standards, but P 4 P 8 has caused topological inconsistency. As a consequence, P 4 P 7 is selected as a simplified line.

5 The second step: Continue to start from P i+j, connect P i+j+2,p i+j+3 and repeat the first step until come to the last point of the polyline. In a short word, the polylines characteristic points are chosen as critical points when it meets geometric precise and topological relation happens to change according to the above method. The approach can be used to examine constraint points adjacent to the polyline; it can effectively reduce the test range of constraint points and consequently reduce the computed data volume and computation complexity. 4.2 BLG-tree algorithm Simplified segments and original partly polyline may form one or more than one polygons (e.g. P 1 P 4 and P 4 P 8, P, P are constraint points in Figure 6). To guarantee the topological relation of the constraint points and polyline to be consistent after simplification is to make sure these points are not in these polygons. In order to avoid using DP algorithm repeatedly, we use the foregoing BLG-tree: create BLG-tree for the polyline or only for the research area, then we can inquire the sub-tree in which the research area is included and examine the topological relation between the constraint points and newly formed polygons. We can choose nodes on the sub-trees level by level until topological consistency is kept. Figure 6. Using BLG-Tree to Simply polylines with constraint points which are out of polylines (Point P and P are constraint points). The constraint point P in Figure 6 lies in polygon P 4 P 5 P 6 P 7 P 8, and topological inconsistency takes place in the left subtree of point P 8 from the BLG-tree. So we pick up P 7 in the sub-tree and examine whether P lies in the polygon Q composed of P 4 P 5 P 6 P 7.If not, P 7 will be selected or else continue to examine P 6. We can deal with constraint point P in polygon P 1 P 2 P 3 P 4 with the same approach in Figure 6a. The result of the topological consistent simplification result of Figure 6a is showed in Figure 6c. In the meanwhile, this approach is different from shortcut method because it lessens the test field by choosing point from BLG-tree. If the constraint point lies in the abandoned polygon, it indicates that the point is not in the constraint area, which is complementary concept of the above approach. As a matter of fact, holistic and contextual simplification requirement are considered in the approach to keep coordination of the graph. 5. CONCLUSION It should be noticed that we only need to deal with the sub-tree where the constraint point lies in and as more points are selected, the sub-tree will become smaller and smaller, which is of great importance for accelerating system data processing. Additionally, if a constraint point comes up in a minute sub-tree or in the details, we can artificially exaggerate it appropriately. Figure 7 is an example for consistent line simplification with BLG-tree based on the constraint points (cross line) on or not on the polyline comparing to DP algorithm. Figure 7. An Example for consistent simplification (the crosses are constraint points)

6 For the diversity of spatial objects themselves and the complexity of the spatial relation among geographic objects, it is more complicated and difficult to hold multi-feature spatial analysis. And for cartographic generalization, which is a decision-making process on the basis of spatial analysis, further research is needed to solve the spatial constraints and conflicts among geographic structural characteristics objects. This paper only solves these complex problems with simple methods based on constraints of spatial topological relation among geographic structural characteristics objects and did some abstraction and geometric work. This is a hot spot of cartographic generalization up to now. To describe spatial relations among spatial objects, especially among different kinds of spatial features, and how to handle spatial relations are more complicated and difficult in GIS spatial database design and practical implementation. 6. REFERENCES [1] Douglas D H, Peucker T K. 1973, Algorithms for the Reduction of the Number of Points Required to Represent a Line or its Caricature. The Canadian Cartographer, 10(2): [2] Buttenfield B P Digital Definitions of Scale-Dependent Structure. Auto-Carto, 1, [3] Buttenfield, B. P A Rule for Describing Line Feature Geometry. In Map generalization: Making Rules for Knowledge Representation, B. P. B. a. R. B. McMaster, (ed) Essex: Longman Scientific & Technical, pp [4] Buttenfield, B.P Scale-Dependence and Self-Similarity in Cartographic Lines. Cartographica, 26(1), [5] Cromley, R. G Hierarchical Methods of Line Simplification. Cartography and Geographic Information Systems. 18(2): [6] Peter van Oosterom, Vincent Schenkelaars. 1995, Development of an Interactive Multi-scale GIS, International Journal of Geographical Information Systems, 9(5): [7] Zhan F, Mark D M. 1993, Conflict Resolution in Map Generalization: a Cognitive Study. Auto-Carto. No [8] Saalfeld A. 1999,Topologically Consistent Line Simplification with the Douglas-Peucker Algorithm. CGIS, 26(1): [9] Cromley,R. G. and Campbell,G. M Integrating Quantitative and Qualitative Aspects of Digital Line Simplification. The Cartographic Journal. 29(1): [10] Mark de Berg, Marc ban Kreveld, Stefan Schirra, 1998, Topologically Correct Subdivision Simplification Using the Bandwidth Criterion. CGIS. 25(4): [11] Ying Shen, Guo Renzhong, 2001,Yan Haowen, Lin Henggui. An Algorithm for Detecting and eliminating intersection of Polylines in Cartographic Generalization. Science of Surveying and Mapping. 26(4) :39-41 [12] Guo QingShen, 1993,Relationships of point, linear and area features, Journal Of Wuhan Technical University Of Surveying And Mapping(Wtusm),vol18. [13] Liu yaolin, et al. 2001,Frameworks for Generalization Constraints and Operations Based on Object-Oriented Data Structure in Database Generalization, Geo-spatial information science, Vol.4,No.3.