A Rapid Algorithm for Topology Construction from a Set of Line
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A Rapid Algorithm for Topology Construction from a Set of Line Segments SEBASTIAN KRIVOGRAD, MLADEN TRLEP, BORUT ŽALIK Faculty of Electrical Engineering and Computer Science University of Maribor Smetanova ulica 17, SI - 2000 Maribor SLOVENIA [email protected], [email protected], [email protected]Abstract: - An efficient algorithm for topology construction from a set of line segments is considered in the paper. The algorithm works in two steps: input analyzing, and topology construction. In the first step, inconsistencies in input are solved. The second step consists of two parts: the envelope construction, and determination of neighbouring relations between the obtained neighbouring The time complexity of the first part highly depends on the input data distribution, and the second part, which presents our original work, runs in O(n), what is confirmed by experiments using artificial input and real data from geographic database. Key-Words: - Geographic Information System (GIS), Topology Construction, Uniform Plane 1 Introduction In the past, land-maps were drawn manually on large sheets of papers. These traditional maps have been gradually replaced by digital formats as computers took- over the storage and maintenance of the data with the graphical content. The traditional maps have started to be digitalised, filtered, and vectorized [1, 2]. In this way, more and less complete and correct set of line segments is obtained. However, although these maps could look perfectly correct, the obtained set of line segments does not contain topological information, which is necessary for various spatial analyses and a geometric reasoning of land-maps [3, 4]. However, the lack of the topology information is not related only with the scanned old cadastre maps, but it also arises during actual activities. For example, spatial data are frequently obtained in a pure digital form (i.e. satellite images, orthophoto), surveyors measure visible points and not closed parcels. Therefore, solutions, which automatically construct the topological information, are more than welcome [3]. Some actual GIS systems include program solutions for the problem being considered here, but we have not found any hints for implementation. Recently, Žalik [5] proposed an algorithm for topology construction. We have analyzed his solution, identified bottlenecks, optimized the algorithm, and partially employed it in our solution. As a result, a very fast, numerically stable, and easy to implement solution has been obtained, and it is presented in the paper. The paper is organized in five sections. After the introductory one, detection and removal of inconsistencies in the input data are briefly considered in Section 2. Section 3 presents the algorithm for topology construction. In Section 4, analysis of the algorithm time complexity, and practical results are given. At the end, the results are emphasized and the work is evaluated. 2 Detection and Removal of Inconsistencies Input set of line segments frequently contains inconsistencies, which have to be detected and removed before the topology construction algorithm is used. Fig. 1 shows some of the characteristic inconsistencies: • Line segments intersect but the intersection point is not marked as a vertex (case (a) in Fig. 1). • line segment crosses for a small distance another line segment instead of ending on it (case (b) in Fig. 1). • Similarly to the previous case, a line segment should touch another line segment in a vertex (case (c) in Fig. 1). • A vertex is isolated (case (d) in Fig. 1). • Vertices should be joined (case (e) in Fig. 1). • In a vertex, at least two line segments should meet each other (case (f) in Fig. 1). • A line segment is isolated (case (g) in Fig. 1). A detailed description how the inconsistencies are detected and removed is described by Žalik [5]. We give just a brief summary here. The following tasks are implemented: 1. All input polylines are split into line segments. 2. The presented line segments are checked for intersections. The approach described by Žalik [5] uses a uniform plane subdivision for this task. This approach includes computationally expensive rasterisation of the line segments, and relatively complicated way of marking the line segments being already tested for intersection. It turns out as the most critical part concerning the spent CPU time. Instead of the uniform plane subdivision, the sweep- line approach can be applied to accelerate Proceedings of the 5th WSEAS International Conference on Telecommunications and Informatics, Istanbul, Turkey, May 27-29, 2006 (pp133-138)
