Terrain Reconstruction from Contour Lines

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1 Terrain Reconstruction from Contour Lines JAN VANEK 1, BRUNO JEZEK 1, EVA MILKOVA 2 1 Faculty of Military Health Sciences, University of Defence 2 Faculty of Science, University of Hradec Kralove Hradec Kralove CZECH REPUBLIC vanek.conf@gmail.com jezek@pmfhk.cz eva.milkova@uhk.cz Abstract: This paper presents an algorithm for elevation grid reconstruction from contours. The algorithm takes advantage of the properties of contour lines and creates more plausible terrains with tighter fit to the input data than the mainstream algorithms currently at use. It is also faster than most of the currently used algorithms. Key-Words: terrain reconstruction, elevation grid, TIN, contour lines, digital terrain models, GIS 1 Introduction The basis for landscape visualization, simulation and many other tasks is a digital terrain model (DTM). It is derived from terrain-describing data, which can be acquired by geodesic survey or remote sensing as a scattered point cloud or a georeferenced regular elevation grid. In practice, the data are often processed into contours. Contours are frequently the only source of topography information in cases when the terrain changed (due to erosion, volcanic processes, human activity etc.) or when remote sensing data is not yet available in desired quality. However, contours as such are not suitable for the mentioned tasks and a DTM must be constructed from the contours first. 2 Problem Formulation A DTM can have the form of a triangulated irregular network (TIN) or a regular elevation grid. This defines two classes of reconstruction algorithms, which, however, overlap in some respects. To construct a TIN from terrain contours a general algorithm for surface reconstruction from contours of parallel slices can be used, as proposed by [1], [2], [3], [4], [5] or [6]. However, these algorithms do not take into account specific properties of geographic contours and mostly are overly complex while not addressing some typical problems such as local terrain extrema. Therefore, many experts propose specialized algorithms of reconstruction from scattered points. These are commonly based on Delaunay triangulation, often modified to integrate contours or other linear data such as drainage network (see [7], [8], [9], [10]). The resulting TINs still can exhibit some artefacts caused by triangles with all vertices on one contour. [11] and [12] propose to extend the data with medial axes, which leads to satisfactory result. Most authors solve the reconstruction of a regular elevation grid from contours as a search for a function with known values at contour vertices and a subsequent interpolation of grid values from. To estimate, usually Thin Plate Spline, differential equations are used most often, as in [13], leading to large equation systems with the number of unknowns proportional to the number of grid values (e.g. [14], [15]). Hierarchical subdivision optimizations are possible, as shown in [16]. Thin Plate Spline interpolation suffers from similar artefacts as triangulations at places where a triangle would have all vertices on a single contour, namely local extrema and ridges. Authors of [17] partially solve the problems by supplementing contours with estimated gradient lines; [18] and [19] add medial axes to contours. Authors of [20] triangulate the contour vertices first and then reduce the interpolation problem to one dimensional Hermite interpolation along gradient curve. A rather different approach is used by authors of [21] who employ raster operations of mathematical morphology. The computational complexity of the most efficient triangulation algorithms is, where is the number of contour vertices. Interpolation algorithms are generally more expensive; many also include triangulation. Satisfactory results are achieved only by [14], [16] and [20], however, at great computational costs. The ISBN:

2 quality is assessed mainly visually, as analytical criteria tend to cover only local properties and ignore undesirable artefacts such as oscillation or terracing (see [17]). Since the regular elevation grid is currently the preferred DTM form for rendering and simulation and indeed, it has many advantages over TIN in terms of inherent topology, data coherency and storage and management efficiency, the problem the proposed algorithm solves is the interpolation of an elevation grid. Without any loss of generality, the grid is assumed to be axes-aligned in a coordinate set with the -axis defining the elevation. 3 Problem Solution Based on the information contained in contours, any terrain point is only guaranteed to have the -coordinate in the interval defined by the elevation of the contours relevant to the point, i.e. contours the point lies between in terms of the plane. Thus, the reconstructed terrain should meet two criteria: every point must have the -coordinate in the interval, and the terrain should look plausible. No other qualities are expected of the output of the proposed algorithm, as e.g. continuity of any class is neither a necessary nor a sufficient condition of an acceptable look. The criterion of plausibility is very vague and cannot be precisely defined; its assessment is subjective [17]. 3.1 Definitions In GIS, contours are usually stored as a set of polylines, piecewise linear planar curves represented as lists of vertices of linear segments. Every polyline is assumed to either be closed or have the first and last vertex on the boundary of the terrain rectangle being reconstructed. Every polyline is stored together with its elevation. The set of all polylines of the same elevation is a layer. Obviously, no two layers can have the same elevation. There must be at least two layers in the reconstructed terrain rectangle. The layers must be equidistant, i.e. when sorted, all the neighbouring layers must have the same elevation difference. The distance from a point to a segment is either the distance from the point to the line the segment is a subset of, if the line that passes the point and is perpendicular to the segment line intersects the segment, or the smaller of the distances from the point to the segment vertices. The distance from a point to a polyline is the minimum of all the distances from the point to the polyline segments. Similarly, the distance from a point to a layer is the minimum of all the distances from the point to the layer polylines. X 3 u 3 n 3 X 1 Figure 1: Polyline sides A polyline divides the processed rectangle into two subsets. A normal vector can be constructed for each segment in two ways: provided the segment vector is, the vectors and are orthogonal to it. Since, WLOG, the vertex ordering within a polyline (the vector directions in Figure 1) and the normal vector construction can be set invariant; all the normal vectors point into the same subset. Therefore, it can be defined that the points X i and X j lie on the same side of a polyline if sgn(u i n i ) = sgn(u j n j ), where n i, and n j, is a normal to the segment closest to the point X i, and X j respectively, and u i, and u j, is the vector from the starting vertex of the closest segment to the point X i, and X j respectively. E.g. the points X 1 and X 2 in Figure 1 lie on the same side of the polyline because sgn(u 1 n 1 ) = sgn(u 2 n 2 ), and the points X 1 and X 3 do not lie on the same side of the polyline because sgn(u 1 n 1 ) sgn(u 3 n 3 ). Even though it is not properly defined, polylines will also be said to lie on the positive or negative side of a polyline in terms of the sign of the dot product. V 1 u 1 n 1 n 1 s 1 V 3 A s2 Figure 2: Correct segment identification In order to apply the definition correctly, the right closest segment must be identified in all cases. In Figure 2, the distances from the points A and B to the segments s 1 and s 2 are the same ( As 1 = AV 2 = As 2, Bs 1 = BV 2 = Bs 2 ), i.e. both u 1 V 2 B n 2 u 2 X 2 ISBN:

3 segments are, by definition, the closest segments to both points. In terms of lying on a polyline side, however, s 1 is the correct closest segment for the point A and s 2 for the point B. This situation can occur whenever the angle between two segments is smaller than. If the distances from the point A to the segments s i and s i+1 are equal, then s i is the correct candidate for the closest segment for the point A, if sgn(n i u i ) sgn(n i (V i+2 - V i+1 )), s i+1 otherwise. Figure 2 shows the case when i = 1. a c a c is reachable from the first relevant polyline over all other polylines of the first relevant layer. E.g. in Figure 4, the elevation is for the closest layer, for the second closest layer with the relevant polyline on the same side of closest polyline, but for the second closest layer with the relevant polyline on the same side of closest polyline and reachable from the closest polyline over all other polylines of the closes layer. Clearly, the elevation of the point must fall into. 40 b Figure 3: Unreachable and reachable polylines Finally, the polyline is reachable from the polyline over the polyline, if the polylines and lie on the same side of the polyline, i.e. if the vertex of the polyline closest to lies on the same side as the vertex of closest to. On the left of Figure 3 the polyline is not reachable from over, on the right it is. 3.2 The algorithm The algorithm estimates the elevation grid point by point. For each point the two layers relevant for its elevation are determined. The elevation of the point is then interpolated based on the elevations of the relevant layers and the positions of the relevant polylines of the layers. The first relevant polyline (and hence the first relevant layer) is the polyline closest to the point. The line connecting the point to the closest point on the polyline is not intersected by any other polyline (it would not be the closes one otherwise). Therefore, if is the elevation of the first relevant layer and is the difference between the elevations of neighbouring layers, the elevation of the point must fall into or depending on the next relevant layer. The second relevant polyline (and hence the second relevant layer) is the polyline that is closest to the point being estimated, lies on the same side of the first relevant polyline as the point, and b Figure 4: Relevant polylines The second relevant layer can be precomputed for each polyline before the individual grid point elevations are estimated. This significantly reduces computational time. So, for the positive and negative side of each polyline it is determined, which polyline is the closest one on the respective side and is reachable from the polyline over all other polylines of the layer of the polyline. Some polylines, such as the one with elevation in Figure 4, have no second relevant layer assigned to one or both sides. However, the elevation on one side can be estimated based on the elevation of the layer assigned to the other side. The interpolation is then based on the elevation symmetric with respect to the first relevant layer elevation. 4 Results The algorithm was implemented and tested on ESRI Shapefile input data. The implementation reconstructs the terrain per tiles that are joined to the final elevation grid. Prior to reconstruction, polylines in each tile are checked for discontinuities and other deviations from definitions, and fixed. The implementation was tested on several sets of input data. The following results show the reconstruction of data provided by Krkonose National Park administration. A rectangle of meters was used as input. It had vertices in polylines. An elevation 40 ISBN:

4 grid with resolution of split into tiles with resolution of was reconstructed from the input. To compare the results with existing methods, the open source GRASS GIS v. 6.4 and the commercial ESRI ArcGIS v. 9.2 were used. The same data was used as input. The same elevation grid was reconstructed in GRASS, an elevation grid with cell size of meters, i.e. with resolution of, was reconstructed in ESRI ArcGIS. The data had to be preprocessed in both systems, rasterized in GRASS GIS and converted to a point cloud in ESRI ArcGIS. The methods IDW from raster points and Raster contours were used in GRASS GIS. In ESRI ArcGIS, the 3D Analyst extension methods Inverse Distance Weighted, Spline, Kriging and Natural Neighbours were tested. The default parameters were left in all methods except for the rasterr cell size. Table 1 shows the time the individual algorithms and implementations took to reconstruct the elevation grid starting with the same input data and measured on the same hardware. The results are in minutes:seconds. The Kriging method did not finish in the recorded time. Method Proposed algorithm GRASS IDW from raster points GRASS Raster contours ESRI Inversee Distance Weighted ESRI Spline ESRI Krigingg ESRI Natural Neighbours Figure 5 shows the Lambertian-shaded results of the individuall methods. The fastest methods aree the raster oriented GRASSS GIS algorithms with all the obvious drawbacks. The proposed algorithmm is visually closest to the Natural Neighbours algorithm and reaches comparable computational performance. The algorithm, however, fits the input data moree closely, which becomes apparent when a high resolution grid is reconstructed, see Figure 6. 5 Conclusion Preprocessing Reconstruction Table 1: Computational times t 4:42 0:25 1:36 5:20 10: >110:00 >1 1:55 Total 4:51 0:34 1:45 14:33 19:48 110:00 11:08 An algorithm for elevation grid reconstruction from contours is proposed in the paper. The algorithm takes intoo account nott only the vertices of contours as most of the currently used methods do, butt the whole linear segments they define. Thus, the algorithm produces closer fitting reconstructions. The algorithm is also comparably fast. Figure 5: Results, left to right: proposed algorithm, GRASS IDW I from raster points, GRASS Raster contours, ESRI Inverse Distance Weighted, ESRI Spline, ESRI Kriging, ESRI Natural Neighbours Figure 6: High resolution grid, left to right: proposed algorithm, ESRI Natural Neighbours ISBN:

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