The Crust and Skeleton Applications in GIS

Transcription

1 The Crust and Skeleton Applications in GIS Christopher Gold and Maciek Dakowicz School of Computing, University of Glamorgan Pontypridd CF37 1DL Wales UK Abstract This paper shows that the simple point Voronoi diagram, together with the extraction of crust and skeleton information, can be of considerable value in a GIS environment, and that these form fundamental tools for the evaluation of spatial relationships. The labelled skeleton and the Voronoi skeleton are described, and are shown to be useful in the extraction of features from digitized or scanned data. Terrain and runoff modelling, especially from contours, is described using Sibson interpolation and skeleton extraction. Finally, the crust and skeleton can be shown to be of use for regional analysis of river networks. Introduction GIS is fundamentally concerned with the spatial relations between objects, whether they are houses, data points or roads. Some of these objects are clearly isolated, while others require some specification of connectedness such as road or polygon boundary segments, or observations of a continuous field, such as elevation. The Voronoi diagram, which expresses the proximal regions of each object, often serves as an excellent mechanism to convert a set of objects into a field, and hence to provide adjacency relationships. This is well known for point data, and provides the most robust mechanism for determining the set of neighbours to be used to interpolate intermediate elevations. The method: insert the query point into the Voronoi diagram, find the Voronoi neighbours, calculate the areas stolen from their Voronoi cells, and use these as the weightings of the neighbour elevation values, is well known under various names: natural neighbour, Sibson or area-stealing interpolation (Sibson, 1980). However, for more complex objects perhaps composed of multiple points or line segments additional work is needed when considering their spatial relationships. The basic issue is how a set of points are to be associated with each other, for example along a line, and not associated with other points that are part of other objects. These associations may be made on the basis of attributes (labels) or geometry (e.g. the adjacency of their Voronoi cells, which means they are connected in the dual Delaunay triangulation). Sometimes (especially when reading data from a file) the input order of the points implies adjacency and connectivity, and this information may be used as well. We will first look at the labelled point skeleton and its applications. This was developed before the geometric skeleton was developed, and it may be useful in cases where geometric position is insufficient to specify the correct adjacency. We will then look at Blum s (1967) medial axis definition, and show how the Voronoi diagram provides a good model for obtaining both the crust (the connected points forming the outline of the shape of an object outline) and the skeleton (the discrete approximation of its medial axis). An application of the labelled point skeleton is given for the rapid digitizing of forest polygons. The Voronoi crust and skeleton are used for various aspects of scanned map processing, as well as for a variety of terrain modelling applications. Map Digitizing: Labelled Point Skeletons Gold et al. (1996) developed a method for the rapid digitizing of forest stands for the Quebec forest industry. Scanning would have been impossible, as the original paper maps had several layers of information. Traditional GIS polygon construction and editing was estimated to take three weeks for each of several hundred maps. Because the objective was approximate area calculation, a technique was devised where a dot at the centre of the digitizer cursor was rolled around the interior of each polygon, and the

2 generated fringe points were given the polygon label. Fig. 1 shows the resulting points, and Fig. 2 gives the resulting Voronoi cells. The dual Delaunay triangulation was then scanned to extract the labelled point skeleton, shown in Fig. 3 as the boundaries with solid lines the original boundaries are given as dashed lines. The scanning algorithm is indicated in Fig. 4. Starting with the topmost triangle of the map, each subsequent triangle is processed as one of the three cases shown in the leftmost column, generating two, zero or one child triangles to be processed later. The remaining columns show the process depending on how many vertices have identical labels, designed to extract only the boundaries between points with dissimilar labels, and to suppress those between points with the same label. A similar process was developed for scanned polygonal maps, where image processing techniques were used to generate fringe points and label them with the colour of the polygon interior. Labels thus identify sets of points that are part of the same object in this case a polygon interior and the boundaries form the skeleton between objects. Similar techniques have been used for geological maps (Okabe et al., 1 st ed., 1992) where the data points were observations of the rock type at outcrops. (The cover image for that edition was probably obtained in a similar fashion.) Fig. 1: Digitized polygon interiors Fig. 2: Voronoi cells for Fig. 1

3 Fig. 3: Extracted boundaries Fig. 4: triangulation scanning algorithm Blum s Medial Axis Blum (1967) first described the Medial Axis, as a descriptor of biological shapes (Fig. 5). Any point on this medial axis was equidistant from at least two points on the boundary, and thus could have a circle radius associated with it. He also pointed out that this radius could on occasion be interpreted as height (Fig. 6). Fig. 5: Medial axis (after Blum) Fig. 6: Medial axis height (after Blum) Crust and Skeleton Extraction Amenta et al. s seminal 1998 paper attempted to reconstruct the boundary ( crust ) from a set of unordered input points forming the boundary of a polygon. They showed that, subject to a sufficient sampling density, the crust could be formed from a subset of the edges of a Delaunay triangulation of the points. They noted that the Delaunay edges that needed to be suppressed were those that crossed the polygon (Fig. 7). In addition, the skeleton (discrete medial axis) was approximated by some of the Voronoi edges. They therefore inserted the Voronoi vertices into the original triangulation, destroying the unwanted Delaunay edges, and constructed the crust from the Delaunay edges connecting the original boundary points. Gold (1999) noted that every Voronoi/Delaunay edge pair was either part of the crust (Delaunay) or part of the skeleton (Voronoi) and showed that a simple incircle test sufficed to determine which (Fig. 8). (A crust edge has an empty circle without any Voronoi vertices in it Fig. 8, right while a skeleton edge has no

4 such empty circle Fig. 8, left.) Fig. 9 shows the polygon of Fig. 7 with this test applied. Both the crust and the skeleton have been extracted. (Note both the endo- and exo-skeletons.) Gold and Snoeyink (2001) refined the required sampling density of Amenta et al. (1998). Fig. 7: Voronoi and Delaunay edges Fig. 8: Skeleton/crust test Fig. 9: Skeleton and crust of Fig. 7 Object Separation by the Medial Axis In many cases the Voronoi-based skeleton acts as an excellent mechanism for separating objects formed from multiple (unlabelled) points. Fig. 10 contains several point clusters, and some of the skeleton boundaries separating them can be seen. Fig. 11 shows a sketch of houses and roads the skeleton serves to separate the individual objects: house outlines and road edges, and to indicate which houses are adjacent to which roads. Thus, if necessary, these objects can be defined by giving labels to the points within each skeleton boundary.

5 Fig. 10: Point clusters Fig. 11: Crusts, skeletons and adjacency Text Recognition and Topology from Scanned Maps As with Blum s original concept, the skeleton may be used for the identification of particular object types as well as for object separation. Ogniewicz and Ilg (1990), for example, processed images of objects on a conveyor belt, and used a form of skeleton both to identify the objects and to identify their relative positions. Fig. 12 shows the skeletons of the outlines of scanned text, and Fig. 13 shows a portion of a scanned cadastral map, where an edge detection filter has been used to separate the background area from the ink. Note that the skeleton may be used: (1) to help identify text characters; (2) to group them together to form numbers; (3) to form a connected topological structure representing the connectivity of the property boundaries; (4) to identify the buildings and (5) to position the text and the buildings within the correct land parcel. Burge and Monagan (1995) attempted to process scanned cadastral maps with a form of labelled point skeleton. Fig. 14 shows that the skeleton may still be useful where the original boundary is incomplete in some way. This map generalization problem requires the reduction of a river polygon to a single centreline. However, some of the desired boundary is missing where a bridge broke the original outline. If the separation is not too large, the skeleton may be reconstructed automatically after removing the bridge information (Fig. 15). Fig. 12: Text skeletons Fig. 13: Cadastral map skeletons

6 Fig. 14: River centreline with bridge Fig. 15: Fig. 14 with bridge removed Skeleton Retraction Particularly in the case of scanned data (see Fig. 9) the generated skeleton has many hairs due to slight perturbations in the boundary. Thibault and Gold (2000) developed a first-order skeleton retraction method to remove these hairs and smooth the crust. Fig. 16 shows how a crust vertex is moved onto the circle associated with the parent of a leaf node of the skeleton. This moves the leaf node to the parent node location, removing minor perturbations of the crust. This is an iterative process, due to the opposing influence of the skeleton on the opposite side of the crust. Fig. 17 shows the result of performing skeleton retraction on Fig. 9. Fig. 16: Skeleton retraction Fig. 17: Smoothed version of Fig. 9 Terrain Modelling The Voronoi diagram provides a model of spatial adjacency for individual data points, as well as for points forming part of an object. Sibson interpolation (Sibson 1980) takes advantage of this to produce a local interpolation where the weighting function matches the selection of neighbours. Fig. 18 illustrates how, by the insertion of query point Z into the diagram, a weighted average interpolation may be based on the area of each neighbouring cell stolen by Z. This function is smooth except at data points. It is ideal for poorly distributed data, where a simple search by distance is not satisfactory, as in Fig. 19, where the data is obtained from digitized contours. Despite the development of new surveying methods, in many projects data is still only available from contour maps. These are particularly difficult to process effectively partly because of the awkward data distribution, and partly because, in generating a triangulation model ( TIN ) any ridges or valleys are likely to produce flat triangles where all three triangle vertices are on the same contour (Fig. 20). However, in Amenta s crust extraction method mentioned earlier she showed that inserting Voronoi vertices into the

7 triangulation broke up the unwanted triangles in our case only those Voronoi vertices associated with the flat triangles. These vertices have an associated radius (Fig. 21) that may be used to provide estimated heights for the newly inserted skeleton points. Thibault and Gold (2000) showed that, on the assumption of constant ridge or valley wall slope, the elevation of any skeleton point may be estimated from the ratio of the radii of its associated circle radius and that of the base triangle s Voronoi vertex, where the base triangle touches contours of two different elevations. Indeed, even in the case of a single contour (and some assumption about slope) the radius associated with each skeleton point may be used to produce a surface model (Figs. 22 and 23), following the idea of Blum (Fig. 6 see Dakowicz and Gold, 2002). Fig. 24 shows the crust and skeleton of a (very implausible) contour map, and Fig. 25 gives a 3D view. Fig. 18: Sibson interpolation Fig. 19: Neighbouring point selection Fig. 20: Flat triangles Fig. 21: Skeleton height estimation Fig. 22: Skeleton of a closed contour Fig. 23: Height interpretation of Fig. 22

8 Fig. 24: Crust, skeleton and TIN Fig. 25: 3D view of Fig. 24 Flow Modelling and Hydrography Finite difference flow modelling of runoff on a terrain surface has usually been done using a regular grid. This has various disadvantages, as the regular pattern does not conform well to observed features such as watersheds, the runoff pattern is biased to the grid axes, and original data points are lost. Adding a random Voronoi pattern to the original data avoids these issues, as there is no axis bias, points may be added anywhere and original data points may be retained. The flow model simply requires a set of buckets to hold the water (the Voronoi cells) and slope information to provide the local runoff rate (the Delaunay edges). Fig. 26 shows a partially completed simulation based on the model of Fig. 25, with the addition of random Voronoi cells, whose heights were estimated using the Sibson method. Fig. 26: Finite difference runoff modelling In many cases of regional analysis, the primary source of information is often the river network itself. Perhaps surprisingly, even without additional elevation information, generating the skeleton and assigning heights based on Blum s idea (Fig. 6) can give very reasonable watershed estimates. (This approach has been used to produce preliminary maps in the Rocky Mountains of British Columbia.) The method works because, although the absolute slopes are not available, the slopes on each side of a watershed are often similar, and so the watershed is approximately equidistant to each river. Fig. 27 shows a sketched river network and the resulting skeleton. (For simplicity the map frame has been treated as another, exterior, river system.)

9 However, there is more information that can be extracted. When we drew a crust segment as part of the river network, we suppressed the associated dual Voronoi edge crossing it. But these, together with the skeleton edges, form the Voronoi cells around each data point on the river network. (The sum of these cells gives the cell for each map object in this case one branch of the river.) As each cell gives the map region closest to its particular data point, and as each suppressed Voronoi edge is perpendicular to a crust segment, this Voronoi cell gives the region whose runoff will flow towards the data point (Fig. 28). Thus the upstream sum of these cells gives the catchment area above any point on the river network. Fig. 29 shows a 3D view of the triangulated landscape estimated by this method, and Fig. 30 shows the cumulative upstream catchment area for each point on the network. Fig. 27: River network and watersheds Fig. 28: Voronoi cells of Fig. 27 Fig. 29: 3D view of Fig. 27 Fig. 30: Cumulative catchment areas This catchment area, however, is not a complete runoff model. In order to perform runoff simulation we need to generate a finite difference model as before, adding random Voronoi cells to the basic watershed and river data. Fig. 31 shows part of the resulting triangulated model, and Fig. 32 shows a nearly-completed simulation. Thus even with very limited data availability preliminary regional runoff analysis may be performed.

10 Fig. 31: Finite difference triangulation Fig. 32: Finite difference cells Conclusions This paper has demonstrated that the simple point Voronoi diagram, together with the extraction of crust and skeleton information, can be of considerable value in a GIS environment, and that these form fundamental tools for the evaluation of spatial relationships. The labelled skeleton and, more particularly, the Voronoi skeleton are of great help in the extraction of features from digitized, scanned or point dat a. Terrain and runoff modelling, especially from contours, is greatly assisted by Sibson interpolation and skeleton extraction. Finally, the crust and skeleton allows significant regional analysis of river networks, even in the absence of detailed elevation data. References Amenta, N., Bern, M. and Eppstein, D. (1998). The crust and the beta-skeleton: combinatorial curve reconstruction. Graphical Models and Image Processing 60, Blum, H. (1967). A transformation for extracting new descriptors of shape. In: W. Whaten-Dunn (ed.), Models for the Perception of Speech and Visual Form. MIT Press, Cambridge, Mass., Burge, M. and Monagan, G. (1995). Using the Voronoi tessellation for grouping words and multi-part symbols in documents. Proceedings, Vision Geometry IV, SPIE 2573, Dakowicz, M. and Gold, C.M. (2002). Extracting Meaningful Slopes from Terrain Contours, In: Proceedings, Computational Science - ICCS 2002, Lecture Notes in Computer Science 2331, (Ed.: Sloot, P.M.A. et al.) Springer-Verlag, Berlin, Gold C.M. (1999). Crust and anti-crust: a one-step boundary and skeleton extraction algorithm. Proceedings, ACM Conference on Computational Geometry, Miami Gold, C.M., Nantel, J. and Yang, W. (1996). Outside-in: an alternative approach to forest map digitizing. International Journal of Geographical Information Systems 10, Gold, C.M. and Snoeyink, J. (2001). A one-step crust and skeleton extraction algorithm. Algorithmica 30, Ogniewicz, R. and Ilg. M. (1990). Skeletons with Euclidean metric and correct topology and their application in object recognition and document analysis. Proceedings, 4 th International Symposium on Spatial Data Handling 1, Okabe, A., Boots, B. and Sugihara, K. (1992). Spatial Tessellations - Concepts and Applications of Voronoi Diagrams (1 st edition). John Wiley and Sons, Chichester. Sibson, R. (1980). A Vector Identity for the Dirichlet Tessellation. Math. Proc. Cambridge Philos. Soc. 87, Thibault D. and Gold, C.M. (2000). Terrain Reconstruction from Contours by Skeleton Construction. GeoInformatica 4,