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1 $<, ) {^^r^l^^^aj^u \ ) a-ta-<l^<^a/e^-'< &Xl^f. Geometric Matching of Areas Comparison Measures and Association Links Francis Harvey Swiss Federal Institute of Technology Institute of Geomatics GR-Ecublens CH-1015 Lausanne Switzerland Francis.Harvey@epfl.ch Francois Vauglin IGN COGIT Laboratory 2 avenue Pasteur F Saint-Mande France Francois.Vauglin@ign.fr i iuiiiji&ilmn AtefBelHadjAli IGN COGIT Laboratory 2 avenue Pasteur F Saint-Mande" France atef@cogit.ign.fr Abstract Increasingly, GIS user communities are faced with problems in combining data from multiple sources. Current implementations of GIS overlay simply do not provide the necessary control for combining data required in these complex, heterogeneous environments. Areas are merged solely on geometric criteria. In this paper we look at a procedure for analyzing and describing semantic relationships between different areal data sets. This work links semantic considerations through geometric operators on feature inclusion and proximity. It connects to previous work dealing with the geometric matching of lines, but focuses on semantic issues. Issues in areal geometric matching are significantly different and therefore require different GIS operations. This paper describes a methodology that uses geometric operators to elaborate semantic similarities in different coverages through geometric operations. Implemented through association links and filter operations, these operators relate individual polygons from different sources. A table contains these association links for the filtering operations. We look in this paper at fundamental issues and basic operations for matching areas using geometric measures that address semantic issues. Keywords: geometric matching, semantics, overlay integration 1. Areas are more than polygon boundaries Geographic information technology communities are faced with large problems in combining and integrating data from diverse sources. Areal data poses special problems. Overlay is not sufficiently developed to fulfill current and future user requirements. This is in part a result of emphasizing line geometry and topology at the cost of semantics. In heterogeneous GIS environments, that increasingly rely on the re-use and modification of data sets, additional requirements are placed on GIS operations in order to facilitate the 557

2 most effective and efficient use of limited data, processing, and personal resources. These issues lead directly to semantics and quality. Without local knowledge of the actual phenomena ("Ground Truth"), GIS users require operations and control possibilities that include semantics. Metadata is a means of adding this information, but because of weak links to data structures and operations is not well implemented for most users. The first basic consideration guiding our work is that until the objectives behind metadata become reality, viable solutions are required. This work proposes such a solution for improving capabilities to control and manipulate areal data and combination (integration) procedures. A second basic thought accompanying this work is that semantics are fundamental to any kind of operation on geographic data and information. In an ideal world statistical and positional data would be complemented by local knowledge. Issues of representation and data quality addressed by a question like "Is that area really prone to flooding like the GIS shows us?" would perhaps be answered by the following response in this ideal world, "Yes, well I've seen that spot under water at least twice in the last few years after they built up there." This knowledge would be the ideal complement to questions we have again and again about the reliability and accuracy of data. These are all issues addressed by semantics. At present this is largely attempted through metadata. However, metadata remain an awkward ersatz. For this reason we find it is necessary to improve the GIS 'toolbox' in a manner that facilitates the inclusion of relevant semantic information and enables more control and interaction with GIS operations. The third, and final, guide accompanying this work relates semantics specifically with generalization research insights. Our consideration of semantics builds on recent cartographic generalization research (Goodchild, 1996; Lagrange & Ruas, 1994; Mackaness, 1994; Miiller, Lagrange, & Weibel, 1995a; Ruas & Plazanet, 1996). Our conceptual framework for bringing these research results into work on combining and matching geographic information basically "distinguishes between scale and purpose in examining the semantics of geographic data (Harvey & Vauglin, 1996). The method we propose here builds and extends our previous work on geometric matching by linking semantic operations with geometric analysis techniques. In particular, this work specifically sets out to operate at the feature level. In this sense it is also distinct from earlier work on semantical issues connected to areas. Generally authors focused on comparing the accuracy of lines and preformed quantitative analysis of total areas associated with features or categories (Chrisman, 1982; Chrisman, 1989; Lanter & Veregin, 1992; MacDougall, 1975; Veregin, 1989; Veregin, 1995). This work provided important insights into causes of error, accuracy, and error propagation, but overlooked the specific semantics of areal phenomena by overemphasizing the geometry of boundaries and their accuracy. The focus on the accuracy of linear elements often left out other elements of quality, notably semantics, out of the picture. This is a topic requiring fundamental consideration, our pragmatic viewpoint is that just as different kinds of phenomena (hydrographic flow, soils, land use, wells) have different cartographic representations, different kinds of features need different kinds of GIS operations. 558 The next section of this paper (2) explores the fundamental conceptual issues connected to a more balanced consideration of semantics and spatial similarity. The following section (3) describes the procedures and algorithms used in a preliminary study. Then (4) we present some examples of applications, before coming to the conclusion and describing future work (5). 2. Conceptual Issues 2.1 Geometric matching Briefly, geometric matching is an algorithm for controlling the movement of nodes by employing multiple tolerances. It was developed for use with lines or polylines, that is without any consideration of issues specific to polygonal representations. In geometric matching, we distinguish between a match tolerance for the more accurate data set, and a shift tolerance for the less accurate data set. These tolerances can be set based on statistical evaluation or user judgement. Generally, features are aligned (without any loss of positional accuracy) from the less accurate data set with features from the more accurate data set. Cluster analysis is key to geometric matching of lines (Harvey & Vauglin, 1996; Harvey & Vauglin, 1997). As described earlier, geometric matching is part of a larger GIS 'toolbox' that is extendable and modifiable. The link between the work described in this paper and this previous work lies, on one hand, in the toolbox interface, and, on the other hand, in connecting different tools in flexible fashions. Because of fundamental semantic differences, areal geometric matching is not 'integrated' with linear geometric matching, but can be used together. 2.2 Semantics and Geometric Operators The concept of semantics that geometric matching rests on differentiates between scale and purpose (Harvey & Vauglin, 1996). Important insights come from cartographic generalization literature and point to fundamental role of semantics for modeling and representing geographic methods (Buttenfield, 1991; Lagrange, 1997; McMaster & Shea, 1992; McMaster, 1991; Muller, Lagrange, Weibel, & Sdge, 1995b; Ruas & Plazanet, 1996). Basically, generalization, following McMaster and Shea (1992) aims for a balance between spatial accuracy and attribute accuracy. For instance, where the real world distance between a lake and village is too small at a given scale to graphically display the railroad and the road that run between the edge of the lake and the village there is a representational conflict. Displacement is a solution, but leads to a loss in positional accuracy. Generalization modifies the geometry of a cartographic data set in order to keep important semantic relationships. For a particular purpose, other semantics may be expunged, modified, or enhanced. The choice of categories, the specification of interpretation guidelines, the definition of minimal feature sizes all affect the semantics of geographic information. The literature on generalization not only examines these issues? but also has developed tools and methods for describing generalization operations and automating them as well (Goodchild & Proctor, 1997; McMaster & Comenetz, 1996; Plazanet, 1995; Plazanet, 1997; Ruas & Lagrange, 1995; Ruas & Plazanet, 1996). These methods often use geometric operations to examine semantics. 559

3 The relationship between geometry and semantics is anchored in the fundamental role of distance in geography. Spatial arrangements can be readily analyzed geometrically, for example alignments, proximities, and distribution structures can all be described in geometric terms (Ruas & Lagrange, 1995). An important issue to consider is that while generalization relies heavily on geometric operations, it does so only to analyze spatial arrangements of the same types of features with another type of feature possibly forming a constraining element, i.e. streets surrounding a block (Timpf, 1997). This works fine for generalization, usually with well-defined purposes and precisely determined scales, but leaves open issues for cross-purpose and cross-scale geometric matching. Areal geometric matching calls for analytical techniques that facilitate the description and comparison of differences between polygon surfaces to assess semantic relationships. Our previous work on the geometric matching of lines and this work on areas draws on this literature in a broad way, but in particular through our adoption of shape and position indicators originating in this research. 2.3 Shape and position parameters Following the work of Anne Ruas a geometrically meaningful difference can be made between position and shape (Ruas & Plazanet, 1996). These components are not only fundamental for describing the quality of geographic data, but crucial in assessing the initial situation and semantic aspects essential to a particular constellation of cartographic elements. In generalization and quality assessment several parameters describe each of these components (see Table 1). Angular Measures Variograms Boundary Measures Compactness Measures Component Measures Table 1: Selected Shape and Position Parameters Position Surface Distance Hausdorff Distance Bias Standard Deviation Frechet Distance There are many shape measures and position measures, far too many to go into here (for a good overview of shape measures see (Wentz, 1997), for position parameters see (Vauglin, 1997)). Shape similarity measures require additional work. In this preliminary study, we developed hybrids of the two types of parameters. The functions for the geometric matching of areas draw on existing position parameters; they also provide indicators of inclusion and shape. In this preliminary study emphasis is on determining fundamental issues in matching areas using these techniques. The functions and operations we use are described in section 3. Issues (such as degenerate cases) we encountered during the development and testing of these measures (see section 4) are the basis for further research. 2.4 Association links explained To operationalize the geometric matching of areas, we utilize a concept called association links. These are links between intersecting polygons from the two data sets (see Figure 1). In this figure, polygons A and C from the first data set intersect with polygon B from the second data set. The intersection of A and B is identified by association link 1 and the intersection of polygons C and B by association link 2. Figure 1 Association links / / 1 / - /"-. The association links are used in the definition and construction of a matching table, where the filtering operations are carried out (see the following section). Association links are useful in discussing processing, but also as a metaphorical device for interface design, an important aspect of 'toolbox' construction. 3. Algorithms and Procedures After presenting the components of the approach in the previous section, we now move on to describe the algorithms and the overall procedures. In the preliminary study, a function of inclusion and a surface distance operation were used to filter the association links. Coming from our opening remarks about semantics, the procedures and algorithms we describe here extend the basic geometric intersection operator used in overlay to include semantic operators. To fulfill this objective, we precede the basic intersection operator extended by our clustering algorithm with an inclusion function and surface distance calculation for matching objects from each data set. The inclusion function preformed on each association link takes this form Equation 1 Inclusion function dft i( P B\ = _AfS24i_ 'V1"V mln(a(fl)^(ji))

4 This function computes the ratio of the smallest polygon's inclusion in the larger polygon. In equation 1, A refers to area, Pi and P2 are polygons, and I is the name of the inclusion function. The inclusion function only measures inclusion, and is not sufficient for determining the similarity between polygons or the relationships between groups of polygons. To complement the inclusion function, a surface distance is also calculated Equation 2 Surface distance D(P P 2 ) = 1 A(^l^) A(/>,u/>2) This distance is not a Euclidean distance, but instead measures the inverse proportion of the ratio of the union of polygon areas to their intersection areas (see Figure 2) k, ofobjecjt^ I 10 5 Number of OCSobjects= 1046 Number of CLC objects = 922. Number of association links = 4143 ilmmnri nocs - CLC EA 0 ' [0.05,0.((J.15,0.gI.25,0.*P-35.0.«}.45.0.H ,0.[a.75,O.B}-85,0.9( [ ry IF Figure 3 Comparison of Inclusion function results for land use data (see section 4.3) At this point in processing, it is possible to evaluate the matches, ending processing, or merge features in a further stage of processing following a heuristic that intersects and resolves multiple matches through coalescence or separation of polygon features. We especially think of using previous work on the geometric matching of lines. This issue throws up numerous issues beyond the context of this paper. 4. Sample Applications Figure 2 Visual representation of the surface distance calculation These two operations are used in filtering the association links. These links are stored in a structure called the matching table. The inclusion function operates as the first filter on the table. All association links are removed from further consideration if the inclusion function result lies below a particular threshold. The results of the surface distance calculation are filtered in a similar manner. The links that remain indicate the polygons from the two data sets that are semantically similar. The examples in section 4 illustrate the use of these operations and processing in greater detail. The selection of thresholds considers various aspects. For the inclusion function, the histograms for each data set follows a similar pattern to Figure BDtopo/Cadastre The first comparison examines the similarities between two data sets with information about buildings for the same area. The Cadastre data set is an extract from a cadastral data set. In this case, the data was originally collected at a scale of 1:4128, but generally this data would be collected at a scale of 1:10,000. The BDTopo building data is an extract of the much larger BDTopo data set (160 categories)(institut Geographique National (IGN), 1994a). Figure 3 shows the superimposition of these two data sets. 56?

5 E ^ V^jRl M interpretation specifications were used. The discrepancies here are larger, only 68% of the objects could be matched automatically. The remainder must be examined on a case by case basis. The reasons for this lie obviously in different operator decisions. A study to qualify and quantify these interpretation differences and their effects on semantics is obviously necessary to resolve the problems encountered through areal geometric matching. 0 Figure 4 BDTopo data superimposed with cadastral data 9 Generalization effects are plainly visible in the simpler blue lines that surround the filled areas identified as buildings in the cadastre. Several problems also are apparent. Although mismatches between lines are the results of generalization (where thicker lines approximate filled polygons) especially different guidelines for delimiting buildings, other problems lack apparent reasons. For example, the darker polygons have no BDTopo data set match. These are due to the different semantics specified for each database. The cadastre includes vegetable gardens with buildings (as an integral part of the home), but BDTopo does not. In a few cases, BDTopo includes buildings that the cadastre does not. These cases cannot be explained through differences in the specifications, they could be due to very recent building. They could also be completeness problems in the data sets. Ultimately, they will probably need to be examined in situ. All-in-all, this study (over a much larger area than illustrated in Figure 3) was very successful. It is important to point out that problem matches (degenerate cases) made up only 1.2% of the objects evaluated in this comparison (Bel Hadj Ali, 1997). BDTopo Cadastre No. Objects I>0.45, D< D>0.6 Table 1 Results from matching two specifications for buildings (BDTopo/Cadastre) a W/o intersect 4.2 BDTopo - Two interpretations Land use is a different matter than cadastre, and two studies were carried out to evaluate issues in geometric matching here. In the first study, two different interpretations of land use in the same area were examined. The land use data was originally collected for the BDTopo data set and were interpreted by different people, using different equipment, and at different equipment. The same remote sensing imagery was used and, most importantly, the same Rollin FIT No. Objects I>0.45, D< D>0.65 Table 2 Results from matching two interpretations of land use (BDTopo/BDTopo) W/o intersect ' 4.3 BDCarto/Corine This study also looked at land use, but emphasizes different specification semantics. One data set comes from BDCarto (Scale 1:50,000) with eight layers, one dedicated to land use and 11 categories(institut Geographique National (IGN), 1994b). It was compared with data for the same area from the EU Corine land use database. The data from BDCarto was resampled to fit the classification scheme of Corine, introducing another source of error. The matching of these data sets also encountered various problems. Only 68% of BDCarto objects could be matched automatically to Corine objects (Bel Hadj Ali, 1997). BDCarto Corine No. Objects I>0.45, D< D>0.7 Table 3 Results from matching two different land use specifications (BDCarto/Corine) W/o intersect 5. Conclusion and Future Work. These methods set out, in the larger framework of geometric matching to contribute to a better understanding of error and raise user awareness of representational and quality issues related to semantics. Specifically, this paper documents preliminary studies of the special issues connected to matching areal features in geographic data bases. We have shown that geometric operations can provide vital information about geographic relationships and semantics that current overlay operations do not consider. - The methods discussed here closely parallel ongoing work on generalization. Still, as generalization focuses on semantics of a product and not on the differences and similarities of semantics in different data sets, additional work is necessary. There still remain many important questions, some of which we will briefly describe here. How should semantically similar features be merged? Coalescing all small features to the largest corresponding feature may be a simple rule of thumb applicable in some instances, but certainly leading to other problems. Semantics

6 could certainly be used for improving general processing tasks by differentiating between feature types and attributes. Heuristic strategies for merging is an important research question in this sense. An interesting approach to be considered in linear geometric matching called nominal contours limits matches to only elements of features that fit certain criteria (Abbas, 1994). For example generalization operators that simplify complex facades to straight lines might only lead to partial matching of those features which coincide. This could be implemented, but calls for a more detailed examination first. A more fundamental question deals with methods for determining and measuring semantic coherence, especially in relationship to geometric quality. Deviation from specifications, and therefore semantics modeling, is one issue. Alternative methods for describing the form of polygons such as the Turning function (Arkin, Chew, Huttenlocher, Kedem, & Mitchell, 1991) need to be examined as approaches to evaluating shape resemblance and semantic similarity and extending this preliminary approach (Vauglin & Bel Hadj Ali, 1998). Degenerate cases amply indicate the limits of procedures discussed in this paper and in the future call for more complete documentation. Research on these issues would also address some of the problems encountered in this work such as positional offsets of features and situations where information is simply missing (degenerate cases). Combining and integrating geographic information in heterogeneous environment exacerbates old problems and leads to new issues that current implementations of overlay are ill-adapted to deal with. Clearly, there are strong needs for improved methods and approaches that make more robust tools accessible for GIS operations. References Abbas, I. 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7 M. J. Krakk & M. Molenaar (Ed.), The Seventh International Symposium on Spatial Data Handling (SDH'96), 1 (pp ). Delft, Holland: International Geographical Union (IGU). Timpf, S. (1997). Exploring the life screen objects. In N. R. Chrisman (Ed.), AutoCarto (pp ). Seattle: ASPRS/ASCM. Vauglin, F. (1997) Modeles statistiques des imprecisions geometriques des objets geographiques lineaires. PhD, Universite de Marne-la-Vall6e. Vauglin, F., & Bel Hadj Ali, A. (1998). Geometric matching of polygonal surfaces in GISs. In ASPRS (Ed.), ASPRS Annual Meeting, (to appear). Tampa, FL: ASPRS. Veregin, H. (1989). Error modeling for the map overlay operation. In M. Goodchild & S. Gopal (Eds.), The Accuracy of Spatial Databases (pp. 3-18). London: Taylor & Francis. Veregin, H. (1995). Developing and testing of an error propagation model for GIS overlay operations. International lournal of Geographical Information Systems. 9.(6), Wentz, E. A. (1997). Shape analysis in GIS. In N. R. Chrisman (Ed.), AutoCarto (pp ). Seattle: ASPRS/ASCM. Implementing Exact Line Segment Intersection in Map Overlay* Andreas Brinkmann and Klaus Hinrichs FB 15, Informatik, Westfalische Wilhelms-Universitat Einsteinstr. 62, D Mlinster, Germany {brinkan, khh)@math.uni-muenster.de tel.: (++49) , fax: (++49) Abstract: We present a method for exact line segment intersection in map overlay. This method is based on techniques to perform exact geometric computation and to generate correct parentage information for line segments. The parentage information provides the numerical input data for the exact geometric operations whereas vice versa the geometric operations guarantee the correct construction of the parentage information. The realization of the exact geometric computation is similar to the exact integer arithmetic developed by Fortune and van Wyk [FvW96], The performance of our method is competitive to implementations based on ordinary floating-point arithmetic. Keywords: line segment intersection, map overlay, exact geometric computation 1 Introduction Map overlay is a crucial operation in vector-based geographic information systems (GIS). One of the main tasks of a map overlay operator consists of splitting line segments which can be performed by algorithms solving the/me segment intersection problem: Given a set of n line segments in the Euclidean plane, find all pairs of line segments that intersect. Theoretical papers on algorithms which solve this problem often assume that the input is in general position JSH76], [B079], [B81], [C86], [M88], [F89], [CE92] et. al.). Line segments parallel to the sweep line in a plane sweep algorithm, overlapping segments or a point being the intersection of more than two segments are often considered as degeneracies. In map overlay processing these degeneracies are normal cases and occur frequently. The design of geometric algorithms is usually based on the assumption that numerical computations with real numbers can be performed exactly. However, computers with their limited finite memory are restricted to represent only finitely many numbers of bounded size. Therefore the exact real arithmetic is replaced by finite precision floating-point arithmetic [IEEE87]. Caused by (accumulated) rounding errors many implementaions of geometric algorithms crash or compute wrong results if no further provisions are made. An implementation of an algorithm is considered to berobust if it produces the correct result for some perturbation of the input. If the perturbation is small an implementation is calleds/awe [S97]. ' This work is supported by the DFG (Deutsche Forschungsgemeinschaft) under grant Hi 441/7-1. <;AS 569