Eliminating contour line artefacts by using constrained edges
Introduction
Contour lines are one of the most important cartographic tools. They enable height and slope behaviour of the terrain to be represented on two-dimensional (2D) maps. Most people understand such a map intuitively, without special training. Therefore, automatic computation of contour lines is a natural and necessary part of GIS programs. However, automatically computed contour lines usually contain some artefacts—areas that an experienced cartographer would draw differently. It is easy for an expert to recognize these problematic places and correct them manually, but it takes time. At present, it is not known how to make all the necessary corrections automatically.
To improve the contour lines, we can concentrate either on the original triangulation, which is used as the first step for computation of contour lines, or on the resulting contour lines. We chose the first approach, as identical contour lines can be produced from the triangulation at any time, without storing contour lines and their corrections.
Our approach is based on inserting constrained edges into the triangulation. These edges are inserted into the places where the contour lines have been problematic. After local or global recomputation, the contour lines on this revised triangulation are improved and artefacts are eliminated or totally removed. The constrained edges approach can be used in a semi-automatic regime where a cartographic expert places the constrained edges manually in an editor; then the triangulation is recomputed automatically. All kinds of problems can be improved in this way, but it is very time consuming.
Another possibility is to combine the constrained edges approach with an algorithmic detection of artefacts. We developed two algorithms for an algorithmic detection of the most disturbing artefact, which is an absence, or strange behaviour of, contour lines in areas with flat triangles. The automatic regime is less reliable – not all contour line problems are corrected and some new artefacts can appear – but it can still achieve a substantial improvement—to about one third of the originally detected problematic flat areas.
The method of constrained edges used in a semi-automatic regime was published in our older paper, Kolingerová et al. (2004), but we have included this material here to make this paper self-contained.
Section 2 gives the necessary background, definitions and state of the art in triangulation and in contour line computation. Section 3 shows the main categories of artefacts appearing in automatically computed contour lines. Section 4 discusses which triangulation is good for contour line computation; Section 5 describes elimination of contour line artefacts using constrained edges. Section 6 presents experiments and results and Section 7 concludes the paper.
Section snippets
Background and definitions
Let us first define a triangulation, a fundamental concept needed in the paper. Definition 1 (A triangulation) A triangulation T(P) of a set P of N points in the Euclidean plane is a maximum set of edges E such that No two edges in E intersect at a point not in P. The edges in E divide the convex hull of P into triangles.
If our goal is to compute contour lines with a small number of artefacts, the first question that should be answered is which triangulation is appropriate for this purpose. We chose two
Main artefacts in automatically computed contour lines
Contour lines may be incorrect due to the following factors:
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Unsatisfactory input data.
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Imprecise numerical computation.
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Improper contour line interpolation.
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Convex hull instead of domain boundaries.
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Locally or globally improper triangulation.
This paper mostly addresses artefacts appearing in the last factor, but let us first briefly describe how to combat all these factors.
Unsatisfactory input data: Input points cause problems if they are either extremely non-uniform (in a cartographic application
Which triangulation is good for contour line computation?
We found two main differences in contour lines produced by DT and GT. One of the main criteria when producing triangulations for contour lines is avoiding triangles which are too long and skinny, which may cause artefacts. Skinny triangles occur more often in GT than in DT. Fig. 5 shows a typical situation: contour lines on DT are without problems, while GT gives unpleasant waves to the contour lines which are caused by a skinny triangle.
Both DT and GT produce horizontal or nearly horizontal
Elimination of contour line artefacts by constrained edges in the triangulation
Insertion of a constrained edge means that the triangulation edges intersecting the newly inserted edge have to be deleted and the hole in the triangulation retriangulated. Then, in the case of DT, the newly obtained triangles have to be checked to see if they satisfy the empty circumcircle criterion; eventual changes may spread into the whole triangulation, although usually they are only local. After the triangulation changes, the contour lines on the changed triangles have to be recomputed as
Experiments and results
We implemented the constrained edge approach as an editor of contour lines in Delphi using the MS Windows operating system, as an assisting program for a cartographer to place constrained edges manually, see Fig. 22. Results obtained by manual placement of constrained edges are shown in Fig. 23, where artefacts from Figs. 4b and d have been removed. In the same way, we can eliminate artefacts shown in Figs. 4a and c.
These results confirm that local changes of the triangulation in the form of
Conclusion
We have surveyed the artefacts which appear in contour lines generated automatically from triangulations. We have presented a comparison of greedy and Delaunay triangulations as to the quality and problems in the contour lines generated when using them. We have shown how contour lines can be improved by incorporating constrained edges, either manually or automatically. The principle of contour line improvement using constrained edges was verified in an editor. Two methods developed for
Acknowledgements
This work has been supported by the Ministry of Education, Youth and Sport of the Czech Republic—the Project no. LC 06008.
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