Schematizing Maps: Simplification of Geographic Shape by Discrete Curve Evolution

Shape simplification in map-like representations is used for two reasons: either to abstract from irrelevant detail to reduce a map user’s cognitive load, or to simplify information when a map of a smaller scale is derived from a detailed reference map. We present a method for abstracting simplified cartographic representations from more accurate spatial data. First, the employed method of discrete curve evolution developed for simplifying perceptual shape characteristics is explained. Specific problems of applying the method to cartographic data are elaborated. An algorithm is presented, which on the one hand simplifies spatial data up to a degree of abstraction intended by the user; and which on the other hand does not violate local spatial ordering between (elements of) cartographic entities, since local arrangement of entities is assumed to be an important spatial knowledge characteristic. The operation of the implemented method is demonstrated using two different examples of cartographic data. 1 Map Schematization Maps and map-like representations are a common means for conveying knowledge about spatial environments that usually cannot be surveyed as a whole, like, for example, local built areas, cities, or entire states or continents. Besides the large spatial extent of these geographic spaces (Montello, 1993) an important characteristic is their complexity what regards possible aspects that can be depicted in a map. Dependent on its scale, a general topographic map is intended to depict as much spatial information as possible, since the specific purpose the map will be used for is not known in advance. 1 This work is supported by the Deutsche Forschungsgemeinschaft (DFG) under grants Fr 806-8 (‘Spatial Structures in Aspect Maps’, Spatial Cognition Priority Program) and Kr 1186-1 (‘Shape in Discrete Structures’). Performing a given task, however, usually only requires a rather small subset of spatial knowledge aspects extractable from a general purpose geographic map. Therefore special purpose schematic maps are generated which are only suitable for restricted purposes, but which, on the other hand, ease their interpretation by concentrating on relevant aspects of information by abstracting from others. Schematic public transportation network maps are a common example of schematic maps (e.g. Morrison, 1996). In this type of map-like representations most entities not directly relevant for using busses, underground trains, etc. are omitted (see Fig. 1 as an example). So, these kinds of maps concentrate on stations, the lines connecting them, and some typical features helpful for the overall orientation within the city at hand. Especially, they usually abstract from detailed shape information concerning the course of the lines and other spatial features, like waters. As a consequence, schematic maps often convey qualitative spatial concepts thus adapting to common characteristics of mental knowledge representation (Freksa et al., 1999). Fig. 1. Example of a schematic public transportation network map (Paris) Intending to derive a schematic map with reduced shape information from detailed spatial data, we need techniques for reducing spatial accuracy of shape aspects to abstract from the exact course of linear entities or the detailed shape of areal geographic objects. Shape simplification processes are needed not only for the generation of schematic map-like representations but also in general cartographic contexts. When a map of a smaller scale (e.g. an overview map of a larger area) is intended to be constructed using reference map data of larger scale, it is usually necessary to perform a reduction in spatial detail. Therefore, apart from the use of symbolic replacement, displacement, and resizing of cartographic entities, an important step in cartographic generalization is the simplification of details of spatial features (Hake & Grünreich, 1994). Automated cartographic generalization is a major research issue in the area of geographic information systems (GISs) (for an overview, see Müller et al., 1995; Jones, 1997). We employ discrete curve evolution (Latecki & Lakämper, 1999a, 1999b, 1999c) as a technique for simplifying shape characteristics in cartographic information, which will be presented in the following section. 2 Discrete Curve Evolution The main accomplishment of the discrete curve evolution process described in this section is automatic simplification of polygonal curves that allows to neglect minor distortions while preserving the perceptual appearance. The main idea of discrete curve evolution is a stepwise elimination of kinks that are least relevant to the shape of the polygonal curve. The relevance of kinks is intended to reflect their contribution to the overall shape of the polygonal curve. This can be intuitively motivated by the example objects in Fig. 2. While the bold kink in (a) can be interpreted as an irrelevant shape distortion, the bold kinks in (b) and (c) are more likely to represent relevant shape properties of the whole object. Clearly, the kink in (d) has the most significant contribution to the overall shape of the depicted object. Fig. 2. The relevance measure K of the bold arcs is in accord with our visual perception There exist simple geometric concepts that can explain these differences in the shape contribution. The bold kink in Fig. 2 (b) has the same turn angle as the bold kink in (a) but is longer. The bold kink in (c) has the same length as the one in (a) but its turn angle is greater. The contribution of the bold kink in Fig. 2 (d) to the shape of the displayed object is the most significant, since it has the largest turn angle and its line segments are the longest. It follows from this example that the shape relevance of every kink can be defined by the turn angle and the lengths of the neighboring line segments. We have seen that the larger both the relative lengths and the turn angle of a kink, the greater is its contribution to the shape of a curve. Thus, a cost function K that measures the shape relevance should be monotone increasing with respect to the turn angle and the lengths of the neighboring line segments. This assumption can also be justified by the rules on salience of a limb in (Siddiqi & Kimia, 1995). Based on this motivation we give a more formal description now. Let s s 1 2 , be two consecutive line segments of a given polygonal curve. More precisely, it seems that an adequate measure of the relevance of kink s s 1 2 ∪ for the shape of the polygonal curve can be based on turn angle β( , ) s s 1 2 at the common vertex of segments s s 1 2 , and on the lengths of the segments s s 1 2 , . Following (Latecki & Lakämper, 1999a), we use the relevance measure K given by ) ( ) ( ) ( ) ( ) , ( ) , ( 2 1 2 1 2 1 2 1 s l s l s l s l s s s s K + = β (1) where l is the length function. We use this relevance measure, since its performance has been verified by numerous experiments (e.g. Latecki & Lakämper, 2000a, 2000b). The main property of this relevance measure is the following: • The higher the value of K s s ( , ) 1 2 , the larger is the contribution of kink s s 1 2 ∪ to the shape of the polygonal curve. Now we describe the process of discrete curve evolution. The minimum of the cost function K determines the pair of line segments that is substituted by a single line segment joining their endpoints. The substitution determines a single step of the discrete curve evolution. We repeat this process for the new curve, i.e., we determine again the pair of line segments that minimizes the cost function, and so on. The key property of this evolution is the order of the substitution determined by K. Thus, the basic idea is the following: • In every step of the evolution, a pair of consecutive line segments s s 1 2 , with a smallest value of K s s ( , ) 1 2 is substituted by a single line segment joining the endpoints of s s 1 2 ∪ . Observe that although the relevance measure K is computed locally for every stage of the evolution, it is not a local property with respect to the original input polygonal curve, since some of the line segments have been deleted. As can be seen in Fig. 3, the discrete curve evolution allows to neglect minor distortions while preserving the perceptual appearance. A detailed algorithmic definition of this process is given in (Latecki & Lakämper, 1999a). A recursive set theoretic definition can be found in (Latecki & Lakämper, 1999c). Online examples are given on the web page www.math.uni-hamburg.de/ home/lakaemper/shape. This algorithm is guaranteed to terminate, since in every evolution step, the number of line segments in the curve decomposition decreases by one (one line segment replaces two adjacent segments). It is also obvious that this evolution converges to a convex polygon for closed polygonal curves, since the evolution will reach a state where there are exactly three line segments in the curve decomposition, which clearly form a triangle. Of course, for many closed polygonal curves, a convex polygon with more then three sides can be obtained in an earlier stage of the evolution. Fig. 3. Examples of discrete curve evolution. Each row shows only a few stages 3 Application to Map Schematization In the previous section we have seen how discrete curve evolution is used to simplify the shape of closed curves. Entities depicted in geographic maps are point-like, linear, or areal. Shape simplification can be applied both to linear and areal entities. However, when dealing with geographic information, the discrete curve evolution method has to be extended in several respects. We have seen that the relevance measure of a kink is computed from the two adjacent line segments. Therefore, for geographic objects represented by a single isolated point and for the endpoints of linear objects in spatial data sets no relevance measure can be computed as there is none or only one adjacent line segment (cf. Fig. 4a). As a consequence, these points cannot be included in the discrete curve evolution process (they should not be intended to be eliminated, either). On the other hand, if points belong to more than one object, we may not want to eliminate them, as this might seriously change spatial information. Consider as an example a linear object (e.g. a state border) being in part connected to the boundary of an areal object (e.g. a lake) as illustrated in Fig. 4b. When such points are eliminated or displaced in either of the two objects by the discrete curve evolution the spatial feature of being connected to each other may be (at least in part) modified. We will call such points fix points. These two cases make it plausible that not every point in an object may be eliminated or displaced by the discrete curve evolution. Therefore, fix points are introduced as points not to be considered by the simplification procedure, be it that it is not possible to assign a relevance measure to them, be it that they must not be eliminated.