Uncertainty of Spatial Metric Relations in GIs

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1 This file was created by scanning the printed publication. Errors identified by the software have been corrected; however, some errors may remain. Uncertainty of Spatial Metric Relations in GIs Xiaoyong hen', Takeshi ~oihara' and Mitsuru ~asu' Abstract.---In this paper, after an introduction to the notations of metric topology, a novel theory of spatial metric relations between sets is developed in which the relations are defined in terms of the intersections of the boundaries, interiors and exteriors of two dynamically generated sets. Then, the presented theory is extended to quantitatively deriving the spatial metric relations between sets in consideration of conceptual and positional uncertainties based on the fuzzy set theory. Finally, some potential applications of presented theories and the ideas for spatial and temporal reasoning in GIs are also suggested. INTRODUCTION One of the most fbndamental requirements for GIs is to modeling and communicating error in spatial databases. With increased research into error modeling in GIs over the past few years, there has a considerable body of models and techniques available for measurement spatial database error from researching to real applications [Goodchild and Gopal, 1989; Vergin, Spatial relationships (such as distance, direction, ordering, and topology) between objects, as very useful tools for spatial queries, constraints checking and sorting in GIs, are strongly influenced by the uncertainties of original spatial data. These practical needs in GIs have led to the investigation of formal and sound methods for driving spatial relations and their variations with uncertainties [Chen and et. al., 1 995, 1 996; Egenhofer and Franzosa, ; Frank, 1992; Kainz, et.al., 1993; Mark, et. al., 1995; Peuquet and Zhang, However, how to derive spatial relations between sets (non-point-like) in consideration of conceptual and positional uncertainties based on an mathematically well-defined algebra framework is still an open problem up to now. The lack of this comprehensive theory has been a major impediment for solving many sophisticated problems in GIs, such as formally deriving complex spatial relations among spatial objects with multiple representations, and generation of the related standards for transferring spatial relations. This paper focuses on the development of the theory and associated models of spatial metric relations between sets in consideration of data uncertainties. At first, based on the metric topology, a novel theory of spatial metric relations between sets is developed in which the relations are defined in terms of the intersections of the boundaries, interiors and exteriors of two dynamically generated sets. Then, the AAS Research Institute, Asia Air Survey Co., LTD.,&IO, Tarnura-Cho, Atsugi-Shi, Kanagatva 243, JAPAN 545

2 presented theory is extended to quantitatively deriving the spatial metric relations between sets in consideration of conceptual and positional uncertainties based on the fbzzy set theory. Finally, some potential applications of presented theories and the ideas for spatial and temporal reasoning in GIs area are also suggested. FUNDAMENTAL DEFINITIONS Metric Spaces A metric space is a pair consisting of a set E and a mapping (p,,p2) +d (p,,p2) of ExE into R+, having the properties: (1). p1=p20 d(p,,p2)=0;(2). d(p,,p2)=d(p2,p,) (symmetry); (3). d(p,,p2)<d(p,,p,)+d (p,,p2)(triangle inequality). The function d is called a metric and d(p,,p2) is called the distance between the points p, and p2. Distance between points pi (xi,.xi,,,x) in R~ is described in terms of the Minkowski d, -metric: Conventional Euclidean distance is defined by the d2 -metric. Similarly, the Manhattan distance defined by the dl -metric, and the maximum distance defined by the dm -metric. Metric Topological Spaces Topological spaces A topological space is a pair consisting of a set E and a collection Aof subsets of E called the open sets, satisfjmg the three following properties: (1). every union (finite or otherwise) of open sets is open; (2). every finite intersection of open sets is open; (3). the set E and the empty set 0 are open. Metric topology A metric d on a set E includes a topology on E, called metric topology defined by d. This topology is such that UCE is an open set if, for each p,&, there is an E>O such that the d-ball of radius E around p, is containedu. A d-ball is the set of points whose distance from p, in the metric d is less than E, i.e. {p2 dp(pl,p2)<&). Notice that the metric topological spaces are Hausdorffand separable. The Hausdorff Metric The Hausdorff metric is defined on the space K where each point is a non empty compact set of^'. If K, and K, denote two non empty compact set in R' (or equivalently two points in K ), and B ( E) is the closed ball with a radius E, then the quantity:

3 :,..., il disjoint inside contains I equal.:..: Q [8! il meet covers I coveredby Figure The eight topologkal relations between regions in IR2. [i I il overlay p(kl,k,)=inf(&:k, ck2@b(&),k2ck,ob(~)) [21 defines a metric p on K, known as the Hausdorff metric. From equation [2], p is the radius of the smallest closed ball B such that both K, is contained in the set K20B(&) generated by the morphological dilation [Serra, and K, is contained in the dilated set Serra (1 982) has proven that the Hausdorff distance p(kl,k2) satisfies all properties of distance functions. In particular case, when K, and K2 are reduced to two points, the Hausdorff distance p(kl,k2) coincides with the Euclidean distance. SPATIAL METRIC RELATIONS BETWEEN SETS Set Intersection Models 9-Intersection For driving binary topological relations between sets, Egenhofer et al., (1994) developed the 9-intersection model based on the usual concepts of point-set topology with open and closed sets, in which the binary topological relations between two objects, K, and K,, in IR' is based upon the intersection of K, 's interior ( K; ), boundary ( ik1 ),and exterior ( K; ) with K, 's interior ( K; ), boundary (2K2), and exterior (K;). A 3x3 matrix 3,, called the 9-intersection as follows: By considering the values empty (0) and non-empty (1) in equation [3], one can distinguish between 2'=5 12 binary topological relations in which only a small subset can be realized when the objects of concern are embedded inir2. Egenhofer and Franzosa (1 99 1) showed that, for two regions with connected boundaries embedded in IR2, the %intersection distinguishes just 8 different relations, i. e. disjoint, contains, inside, equal meet, covers, coveredby, and overlap (figure 1). However, when we apply the 9-intersection model to describing topological relations between other types of spatial objects, the situation will be more complicated. According to the results of Mark et. al. (1995), for two simple lines 33 different spatial relations are

4 possible, and for a line and a region, 19 are possible. Dynamic 9-intersection For integrally deriving different kinds of spatial relations between sets, Chen et al., (1995, 1996) developed the dynamic 9-intersection model based on the concepts of the metric topology with open and closed sets and the morphological dilation, in which the general 9-intersection of equation [3] is extended as follows: where the Kl and K2 are given two closed sets, the K,BB(&,) means relevant morphological dilation by the closed ball B with radius E,, and the 3:,,,) (E,) means dynamic %intersection with parameter E, from K to Kj. Based on the equation [4], we can derive dynamic topological relations by using the different parameter E,. In particular case, when E, =O, the structure element B(E,) is reduced to the original point {o}, according to the algebraic properties of morphological dilation [Serra, 19821, we have )=K@{o)=K,, then the dynamic 9-intersections i, j ( ~ t ) defined in equation [4], coincide with the general 9-intersection3, in equation [3]. Spatial Metric Relations between Sets Distance Relations between Sets According to the derived dynamic topological relations by the dynamic 9- intersections of equation [4] with different parameter E,, such as dynamic equal, dynamic covers (or dynamic corveredby) and dynamic contains (or dynamic inside), we can easily get the Hausdorff distance p(kl,k2) between two closed sets K, and 6 by calculating the minimum and maximum dilated distances: p(kl,k2)=max{min(s,), min(~,)i; when 1 where " * " means either empty (0) or non-empty (1). The binary distance relations derived by equation [5] are suitable for different types of spatial objects, such as point-objects, line-objects and region-objects, as well as combining different types of spatial objects such as a line and a region, a point and a line, or a point and a region. Some examples are shown in figure 2.

5 Figure The spatial metric relations between two planar sets A and B. Directional Relations between Sets Directional relations between sets can be defined by the Hausdorff metric of angular bearings, in which the direction fiom one spatial object to another is identical to that for metric fimction except that the angular bearing is computed for each ordered pair in the Cartesian product. The angular bearing is measured in the sense of navigation bearings (i.e. increasing clockwise fiom north). Similarly, for calculation of the directional relation q(ki,k, ) between two non empty compact sets Kl and K2 in R2, we can select the angular bearing set R(a,) as instead of the closed ballb(s,)in equation [3], then we can get the Hausdorff direction q(ki,kj) by calculating the minimum and maximum dilated angles as follows: [111][ , 001,[001] [ip[k][iy'] L contains

6 where " * " means either empty (0) or non-empty (1). The binary directional relations derived by [6] are also suitable for different types of spatial objects and their combining types. Some examples are shown in figure 2. UNCERTAINTY OF SPATIAL METRIC RELATIONS Conceptual Uncertainty of Spatial Metric Relations Unlike the spatial metric relations between points, the concepts of distances and directions between subsets are fuzzy [Zadeh, 19651, since the spatial objects may contain many subsets, the distances and directions between these subsets are difficult to be represented just by a single value. If we define the fuzzy memberships as the covering percentages of generated region areas ( or point numbers and line lengths ) by the dynamic intersection of sets in R', we can find that the Hausdorff metric is just the special case with the fuzzy membership value equal to one. Based on the changes of the covering areas of regions (or point numbers and line lengths) from an empty set to a complete set, we can estimate the fuzzy memberships from zero to one, then we can quantitatively derive the spatial metric relations between subsets. For reasons of simplicity the distances and directions between closed subregions discussed in this paper only, related models for estimation of conceptual fuzzy membership functions are defined as follows: where A(*) means covered area sizes by dynamic intersections with the parameters n and B for distances and directions separately, the functions Os@,(n)<l and o<a,(s)sl with the parameters O<<<~(K,,K,) and osesp(k,,~,)kx are called the size distribution functions [Serra, An example of conceptual uncertainty of spatial distance relations between two spatial regions is shown in figure 3. Positional Uncertainty of Spatial Metric Relations Generally, the locations of objects in spatial databases are not error- free, they may contain many kinds of errors, such as the errors of scanning, digitizing, selecting, projection, overlaying, and et.al. [Goodchild and et.al., For describing the uncertainty in the positions of spatial objects (such as lines and areas), we can use the error model of E -band developed by Chrisman (1982), in which the positional uncertainty of a spatial object K~ can be represented as K.=K@~(E), where E is the buffer distance of error distribution and p(~) is the fuzzy membership function derived by E. According to the equations of [2]-[7], if we use the uncertain spatial object K as instead of the general spatial object K in the equations of [2]-[7], we

7 Figure The conceptual fuzzy membership fhction between subsets. Figure The positional fuzzy membership fbnction between subsets. can drive the spatial metric relations between uncertain sets and the conceptual fuzzy membership functions between uncertain subsets. For quantitatively estimating the positional fuzzy membership functions of derived spatial metric relations between uncertain subsets, we can use the measurement of dynamic covering uncertain areas as follows: where ki =Ei -4 =[Ki BP (E)]-Ki means the uncertain area generated by the related E - band, A{*) means the covered size of area or volume when p(*) is a constant or a general fuzzy member function with the value o~j(*)~i separately. An example of positional uncertainty of spatial directional relations between two spatial regions is shown in figure 4. CONCLUSIONS AND OUTLOOKS As the natural extension of the general 9-intersection, the dynamic 9-intersection based on metric topology supplied a general fi-amework for studying metric spatial relations between sets. The presented integrated theory of spatial relations between sets makes a new way for formally deriving spatial relations among complex spatial objects with conceptual and positional uncertainties. Even though the presented approach is only focus on the applications in GIs field, the related results for deriving spatial relations between sets and uncertain sets can be also used for many other fields, such as CAD, computer vision, pattern recognition, robot space searching and so on. However, only the theoretical models have be presented in this

8 paper, a wide field of practical application for management and analysis of spatial data in 2-D and 3-D GIs environments has not been touched. Therefore, the reported results must be verified and extended in order to be used in different practical environments. Two main directions for further research shall be pointed here, one is the applications of the presented theoretical models and algorithms in 2-D and 3-D GIs environments for developing the new tools of spatial query and analysis; another one is the extensions of presented theories and models for formally deriving complex spatial relations among spatial objects with multiple representations. ACKNOWLEDGMENTS The authors wish especially to thank Prof. S.Murai and Mr. H.Yamamoto for their considerable helps and supports of this research project. REFERENCES Chen, X., et. al., Spatial Relations of Distances between Arbitrary Objects in 2-Dl3-D Geographic Spaces based on the Hausdorff Metric. LIESMARS'95, Wuhan, China p. Chen, X., Spatial Relations between Sets. ISPRS XVIII Congress. Vienna. Austria. (accepted) Chnsman, N. R., Methods of spatial analysis based on errors in categorical maps. Unpublished PhD thesis, University of Bristol. Egenhofer, M., and Franzosa, R., Point-Set Topological Spatial Relations. IJGIS. 5(2), p. Egenhofer, M., and et. al., A Critical Comparison of the 4-Intersection and 9- Intersection Models for Spatial Relations: Formal Analysis, AUTO CARTO p. Frank, A., Qualitative Spatial Reasoning about Distances and Directions in Geographic Space. J. of Visual Languages and Computing. 3(4), p. Goodchild, M., and Gopal, S., The Accuracy of Spatial Databases. Taylor & Francis. Kainz, W., and et. al., Modeling Spatial Relations and Operations with Partially Ordered Sets, IJGIS. 7(3), p. Mark, D. J., et. al., Toward a Standard for Spatial Relations in SDTS and Geographic Information Systems. GISLIS '95, Nashville. Tennessee p. Peuquet, D. J., and Zhan, C., An Algorithm to Determine the Directional Relationship between Arbitrarily-S haped Polygons in the Plane. Pattern Recognition. 20(1), p. Serra, J., Image Analysis and Mathematical Morphology. Academic Press. Vergin, H., Integration of Simulation Modeling and Error Propagation for the Buffer Operation in GIs. PE & RS. Vo1.60, No.4, p. Zadeh, L. A., Fuzzy Sets. Information and Control. vol. 8, p.