Pullar, D., Egenhofer, M. (1988) Toward Formal Definitions of Topological Relations among patial Objects. In the Proceedings of the 3rd International ymposium on patial Data Handling. Randell, D. A., Cui, Z., Cohn., A., (1992) A patial Logic Based on Regions and Connection. In the Proceedings of the 3rd International Conference on Principles of Knowledge Representation and Reasoning, KR 92. Morgan Kaufman. Tversky, B. (1993) Cognitive Maps, Cognitive Collages and patial Mental Models. In Frank, A.U., Campari, I. (eds.) Proceedings of the European Conference on patial Information Theory COIT, Elba, Italy. pringer-verlag.
Egenhofer, M. (1991) Reasoning about Binary Topological Relations. In Gunther, O. and chek, H.J. (eds.) Advances in patial Databases. econd ymposium, D 91. pringer- Verlag. Frank, A. U. (1992) Qualitative patial Reasoning about Distances and Directions in Geographic pace. Journal of Visual Languages and Computing, 3, pp 343-371. Frank,A.U.(1994) Qualitative patial Reasoning: Cardinal Directions as an Example. To appear in the International Journal of Geographic Information ystems. Taylor Francis. Freksa, C. (1992) Using Orientation Information for Qualitative patial Reasoning. In Frank, A.U., Campari, I., Formentini, U. (eds.) Proceedings of the International Conference GI - From pace to Territory: Theories and Methods of patio-temporal Reasoning in Geographic pace, Pisa, Italy. pringer-verlag. Hernandez, D. (1993) Maintaining Qualitative patial Knowledge. In Frank, A.U., Campari, I. (eds.) Proceedings of the European Conference on patial Information Theory COIT, Elba, Italy, eptember 1993. pringer-verlag. Holmes, P.D., Jungert, E. (1992) ymbolic and Geometric Connectivity Graph Methods for Route Planning In Digitized Maps. IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. PAMI-14, no 5, pp. 549-565. Kuipers, B.J. (1978) Modelling patial Knowledge. Cognitive cience, 2, pp. 129-153. Lee, -Y., Yang, M-C., Chen, J-W. (1992) ignature File as a patial Filter for Iconic Image Database. Journal of Visual Languages and Computing, 3, pp 373-397. Levine, M. (1978) A Knowledge-Based Computer Vision ystem. In Hanson, A. and Riseman, E. (eds.) Computer Vision ystems. New York: Academic. pp 335-352. Mark, D.M., Frank., A.U. (1990) Experiential and Formal Representations of Geographic pace and patial Relations. NCGIA Technical Report. McDermott, D.V., Davis, E. (1984) Planning Routes through Uncertain Territory. Artificial Intelligence, 22, pp 107-156. Papadias, D., Glasgow, J.I. (1991) A Knowledge Representation cheme for Computational Imagery. Proceedings of the 13th Annual Conference of the Cognitive cience ociety, Lawrence Erblaum Associates. Papadias, D., ellis, T. (1993) The emantics of Relations in 2D pace Using Representative Points: patial Indexes. In Frank, A.U., Campari, I. (eds.) Proceedings of the European Conference on patial Information Theory COIT, Elba, Italy, eptember 1993. pringer-verlag. Papadias, D., Kavouras, M. (1994) Acquiring, Representing and Processing patial Relations. To appear in the Proceedings of the 6th International ymposium on patial Data Handling. Taylor Francis. Papadias, D., ellis, T. (1994) A Pictorial Language for the Retrieval of patial Relations from Image Databases. To appear in the Proceedings of the 6th International ymposium on patial Data Handling. Taylor Francis. Puequet, D. (1986) The Use of patial Relationships to Aid patial Database Retrieval. In the Proceedings of the 2nd International ymposium on patial Data Handling.
and the destination points are in different maps. For instance, bus-stop T3 can be reached from C1 by highway line H2 (or H1) to and then by bus trip R1: 22 (C1H2) 34 (H2) T 22 (R1) T 23 (T3R1). Additional types of connections can be defined to express situations involving routes from a station to another. In procedural terms the previous definitions reduce to a search for intersections. Direction information can be used in choosing a line which is in the direction of the destination, when several choices are available. 6. Conclusion This paper describes how symbolic spatial indexes can be used in several computational tasks involving direction and topological relations in 2D space. In particular we have dealt with spatial information retrieval, composition of spatial relations, route planning and update operations. everal extensions can be developed to increase the applicability of the concepts presented in the paper. For instance, PQBE can be extended with the inclusion of topological expressions for queries of the form how the route from station to that involves the smallest number of intermediate connection. The composition operator can also be extended to handle composition of both direction and topological relations. Although these problems have been studied independently (e.g., a study about composition of topological relations can be found in Egenhofer, 1991), to our knowledge there does not exist previous work that combines both approaches. There are also additional application domains where symbolic spatial indexes can be used. Lee et al. (1992) developed algorithms for image similarity retrieval based on variations of 2D strings. imilar algorithms can be developed for spatial indexes. Information processing using spatial indexes, and relation-based representations in general, involves symbolic and not numerical computation and avoids the usual problems of geometric representations, like finite resolution and geometric consistency. Although relation-based systems cannot be used in all applications involving spatial knowledge (e.g., applications involving quantitative reasoning and visualisation), we believe that there is a wide scope of potential applications ranging from Qualitative patial Reasoning and patial Databases to Robot Navigation and Computational Vision. Bibliography Chang,.K, Erland, J., Li, Y. (1989) The Design of Pictorial Database upon the Theory of ymbolic Projections. In Buchmann, A., Gunther, O., mith, T., Wang, T., (eds.) Proceedings of the First ymposium on the Design and Implementation of Large patial Databases (D 89). pringer-verlag. Davis, E. (1986) Representing and Acquiring Geographic Knowledge. Morgan Kauffman. Dutta,. (1989) Qualitative patial Reasoning: A emi-qualitative Approach using Fuzzy Logic. In Buchmann, A., Gunther, O., mith, T., Wang, T., (eds.) Proceedings of the First ymposium on the Design and Implementation of Large patial Databases (D 89). pringer-verlag. Egenhofer, M., Herring, J. (1990) A Mathematical Framework for the Definitions of Topological Relationships. In the Proceedings of the 4th International ymposium on patial Data Handling.
5. Route Planning A system which can only represent direction relations in 2D space is inadequate for several practical applications that also involve neighbourhood, inclusion, overlap or other topological queries. In this section we will show how we can use symbolic spatial indexes that incorporate direction and topological information in route planning. Previous systems designed to deal with this problem, include TOUR (Kuipers, 1978) and PAM (McDermott and Davis, 1984). Holmes and Jungert (1992) have demonstrated how symbolic projections can be applied to knowledgebased route planning in digitised maps. The symbolic spatial indexes of Figure 3 can be used for the retrieval of direction relations but are not adequate for route planning, since they do not preserve connections among subway stations and bus-stops. In order to have the expressive power to answer queries regarding connections we need indexes that also preserve topological information. Figure 13 illustrates two spatial indexes which in addition to the direction information of the previous indexes, also preserve intersection information about the maps of Figure 1. H2 C2H1 C1H1H2 H1 H1H2 T R1 R1 T3R1R2 T2R2 T4R2 Fig. 13 ymbolic spatial indexes that preserve direction and topological relations We use subscripts to denote the individual cells of a symbolic spatial index; ij denotes the cell at row i and column j. ij and kl refer to the same cell iff i=k and j=l and 11 is the lower left cell of the array. The predicate ij () denotes that cell ij contains the string or substring ; 23 (H1), for instance, denotes that cell 23 contains the H1. A direct connection between stations C k and C l corresponds to a subway line that intersects with both stations and can be defined as: C k d_connects C l H m ij gf [ ij (C k H m ) gf (C l H m )]. H1 directly connects stations and since 34 (H1) 23 (H1). Notice that the symbolic spatial indexes of Figure 13 do not preserve the order of stations in a subway trip, i.e., and are directly connected through subway line H1 but the indexes do not show whether there are intermediate stations, or whether and are next to each other. If such information is needed, it should be added as subscripts in the names of lines denoting the order of traversal. We will use the notation H m reaches H n to denote that subway line H n can be reached from H m using subway line intersections (we assume bi-directional subway lines). The relation reaches can be defined as: H m reaches H n ij [ ij (H m H n )] H k [H m reaches H k H k reaches H n ]. H1 reaches H2 since 22 (H1H2). An indirect connection is achieved through a subway line that intersects with the first station, and a second subway line which reaches the first one (directly or indirectly) and passes from the second station. Indirect connection through subway lines can be defined as: C k i_connects C l H m H n ij gf [ ij (C k H m ) H m reaches H n gf (H n C l ) ]. and are indirectly connected through H1 and H2 since 23 (H1) H1 reaches H2 11 (H2). Indirect connections are necessary when the start
IC6 C2 C6 C1 a. Query b. Result Fig. 10 Object insertion using PQBE In some cases, the insertion of an object may result in several output arrays corresponding to the possible positions of the new object. For example, the insertion of C6 RestrictedNorth of (but not necessarily outhwest of ) expressed by the query in Figure 11a would result in the three spatial indexes of Figure 11b, since the relation of C6 with respect to C2 and is not precisely specified. An alternative way to insert an object in a symbolic spatial index is by explicitly specifying the cell where the object is to be added. Furthermore, object movements within the array can be handled by removing the object from its old position and inserting it in the new one. IC6 C2 C6 C2 C6 C1 C6 C2 C1 C1 a. Query b. Possible Results Fig. 11 Object insertion involving uncertainty Related to update operations are the problems of spatial knowledge assimilation and error correction (Davis, 1986). The goal of assimilation is, given an accurate representation of spatial knowledge and an accurate fact, to augment the representation in order to include the new fact. The goal of error correction is, given a spatial knowledge representation which is not quite accurate and a new fact which is more precise, to improve the representation. Figure 12 illustrates how new spatial knowledge regarding the position of two additional subway stations C6 and C7 is incorporated in the existing symbolic spatial index. C2 C1 C6 C7 C2 C6 C1 old state new facts new state C7 Fig. 12 Example of knowledge assimilation Both these problems can be treated by operators, similar to the composition operator, that take two symbolic spatial indexes as arguments (i.e., one representing the initial state and one representing the new fact) and generate one or more output indexes that describe the possible new states.
There are also cases where composition does not produce new information about the possible relation between two objects. These cases are denoted with the symbol T in the composition table. For instance, com(, W,,, ) will generate the arrays of Figure 8. The nine arrays correspond to the relations of M D1 between and when it is also known that is NorthEast of both and. In this case composition does not rule out any possible relations. Fig. 8 patial indexes corresponding to com(, T,,, ) The same composition operator can handle symbolic spatial indexes preserving direction relations using two, or more, representative points per object but the number of possible outputs grows exponentially with the number of points used to represent the objects. 4. Update Operations Any spatial knowledge representation system is expected to function in a dynamic environment in which a change in a single item of knowledge may have widespread effects. In monotonic spatial logics, for example, the assertion of a new fact may invalidate previous inferences. With minor extensions, PQBE of can be used to handle update operations in symbolic spatial indexes. Object deletion from an index can be treated in a way similar to object retrieval. Instead of the character P that causes the contents of a variable to be printed, we can use a special character R that causes all the objects that satisfy a spatial query to be removed from the index array. After all the symbols that belong to the objects that satisfy the spatial conditions are deleted, the empty rows or columns are removed. For instance, the skeleton array of Figure 9a will delete from spatial index all the subway stations outhwest of tation C1 (i.e., station ). The new symbolic spatial index, after the removal of, is illustrated in Figure 9b. R_C C1 C2 C1 a. Query b. Result Fig. 9 Object removal using PQBE Object insertion can also be handled by PQBE provided that the position of the object to be added is precisely specified. For example, the query in the skeleton array of Figure 10a will add a station C6 RestrictedNorth of and outhwest of in map (the special character I denotes object insertion). The new symbolic spatial index that results after the insertion operation is illustrated in Figure 10b.
indexes to be composed, the common object with respect to which the composition is made, and two other objects each belonging to one array, whose composition relation is to be found. The operator creates one or more output arrays which contain only the three objects preserving their relative positions. For instance, com(, T,,, T4) will generate the array of Figure 6. T4 Fig. 6 patial index com(, T,,, T4) In order to compute the composition relation, the operator uses the following composition table. Table 1 illustrates the composition relation between objects X and Y when their relation with a third object Z is known (instead of the full name of the relation we use only the capital letters e.g., NW instead of NorthWest). For instance, the array of Figure 6 is generated using the facts that is outhwest of in index, is outhwest of T4 in index T and the entry (7,7) of the composition table. 1 2 3 4 5 6 7 8 9 NW(Z,Y) RN(Z,Y) NE(Z,Y) RW(Z,Y) P(Y,Z) RE(Z,Y) W(Z,Y) R(Z,Y) E(Z,Y) 1 NW(X,Z) NW NW NW RN NE NW NW NW RN NE NW RW W NW RW W T 2 RN(X,Z) NW RN NE NW RN NE NW RW W RN P R NE RE E 3 NE(X,Z) NW RN NE NE NE NW RN NW NE NE T NE RE E NE RE E 4 RW(X,Z) NW NW NW RN NW RW RW RW P RE W W W R E 5 P(X,Z) NW RN NE RW P RE W R E 6 RE(X,Z) NW RN NE NE NE RW P RE RE RE W R E E E 7 W(X,Z) NW RW W NW RW W T W W W R E W W W R E 8 R(X,Z) NW RW W RN P R NE RE E W R E W R E 9 E(X,Z) T NE RE E NE RE E W R E E E W R E E E Table. 1 Composition Table In cases where there are multiple possible outputs, the composition operator, as in Freksa s and Egenhofer s systems, will generate a list of output arrays corresponding to the possible disjunctions. For instance, com(, T,,, T2) will generate the arrays of Figure 7. The three arrays preserve the same order on the x axis, that is, they all contain the same east-west direction information as well as the same information regarding s position with respect to and T2. The composition relations are generated using the facts that is outhwest of in index, is NorthWest of T2 in index T and the entry (7,1) of the composition table. The arrays correspond to the three possible relations of the disjunction. Notice that symbolic spatial indexes do not preserve metric information; otherwise only one relation (T2 NorthEast ) could be generated as the result of composition. T2 T2 T2 Fig. 7 patial indexes corresponding to com(, T,,, T2)
to verbal Query-By-Example, PQBE generalises from the example given by the user, but instead of having queries in the form of skeleton tables showing the relation scheme, PQBE uses skeleton arrays. In skeleton arrays variables are preceded by the character "_", while constants appear without qualification. The character P before a variable causes its value to be retrieved and printed. For instance, the query in the skeleton array of Figure 5a will retrieve the subway stations (_C is a station variable) outhwest of station in map (i.e., stations C1, and ). On the other hand, the query in Figure 5b will retrieve all subway maps (_M is a map variable) in the database that contain a station outhwest of station. P_C P_M _C a. Object retrieval b. Map retrieval Fig. 5 Example queries in PQBE PQBE has the ability to express negation, independent sub-conditions in the form of multiple skeleton arrays, union, intersection and join operations, image and relation retrieval, and inference capabilities. Furthermore, with the incorporation of some additional features, PQBE can be used as an intuitive and easy to use GI interface, independently of the underlying representation. A detailed description of PQBE can be found in (Papadias and ellis, 1994). M D1 is adequate when high resolution is not needed but consider that we want the ability to answer queries of the form "are there parts of object X which are south of some parts of Y?". One symbol per object is not sufficient for such queries and we need representations that preserve two or more representative points for each object. Examples of symbolic spatial indexes that use multiple points/symbols per object can be found in (Papadias and ellis, 1993). Related is the work on symbolic projections and 2D strings (one dimensional encodings of symbolic images) by Chang et al., (1989). 3. Composition of patial Relations - Map overlay The problem of composition can be defined as "if the spatial relation between X and Z, and between Z and Y is known what are the possible relations between X and Y?". Frank (1992) presented composition tables for direction relations based on the concepts of projections and angular directions. Freksa (1992) also studied composition of direction relations in 2D space and Egenhofer (1991) composition of topological relations. Freksa`s and Egenhofer's systems deal with the composition problem in similar ways. When various cases are possible the systems generate a disjunction of the possible relations between the two objects. In this subsection we will show how symbolic spatial indexes can deal with composition of spatial relations. Consider that we want to perform an map overlay using the symbolic spatial indexes of Figure 3, i.e., we would like to generate an index that preserves information existing in and in T. The two indexes have an object ( in this case) in common. Let be the set of representation structures (symbolic spatial indexes preserving M D1 ) and O the set of symbolic object representations. Then we can define a composition operator com: xxoxoxo, that takes as input the two
applications where high resolution is not needed one point (symbol) per object is enough for the representation of the spatial relations of interest. In applications where a large number of spatial relations is to be preserved more points/symbols per object are used. An extensive discussion about representational issues related to symbolic spatial indexes can be found in (Papadias and ellis, 1993). This paper concentrates on information processing using spatial indexes. ection 2 describes how they can be used in storage and retrieval of direction relations in 2D space. ection 3 demonstrates how symbolic spatial indexes can deal with composition of spatial relations and map overlay. ection 4 discusses update operations and ection 5 route planning using spatial indexes. ection 5 concludes with a discussion and comments about future research directions. 2. Retrieval of Direction Relations For the representation system of this section we assume that the spatial relations of interest are direction relations such as north, east, northeast etc. Previous studies on direction relations can be found in (Peuquet, 1986), (Dutta, 1989) and (Frank, 1992). In the rest of the section will show how we can map direction relations onto relations among representative points stored in symbolic spatial indexes and how these relations can be retrieved using a pictorial query language. In our first example, we are interested in retrieving the relative positions of subway stations and bus-stops existing in one or more maps. If we use one representative point per object, each station and bus-stop is represented as one symbol. Figure 3 illustrates spatial indexes and T that represent the subway stations and the bus-stops of maps s and t using one symbol per object. C2 C1 T T4 T3 T2 Fig. 3 ymbolic spatial indexes that use one symbol per object M D1 denotes the set of mutually exhaustive and pair-wise disjoint direction relations which are represented in the previous indexes. M D1 contains nine direction relations that correspond to the highest resolution we can achieve using one point per object: NorthWest, RestrictedNorth, NorthEast, RestrictedWest, ameposition, RestrictedEast, outhwest, Restrictedouth, outheast. Figure 4 illustrates how direction relations among object symbols are represented in a symbolic spatial index. The symbol Y denotes the reference object and the other symbols refer to the direction relation depending on the position of the primary object in the array. NorthWest RestrictedNorth NorthEast RestrictedWest Y RestrictedEast outhwest Restrictedouth outheast Fig. 4 Direction relations encoded in symbolic spatial indexes Papadias and ellis (1994) developed a Pictorial Query-by-Example (PQBE) for the retrieval of direction relations from databases of symbolic spatial indexes. imilar
Fig. 1 Example maps Numerous relation-based systems have been developed to represent and process spatial relations. Levine (1978), for instance, developed a semantic network where the arcs encode relations such as left, above, or behind. Randell et al., (1992) developed a theory for topological reasoning expressed in a many-sorted logic. Egenhofer and Herring (1990) presented a system which deals binary topological relations. In their formalism, the topological relation between any two objects (point sets) X and Y is described by the four intersections of X's boundary and interior with the boundary and the interior of Y. Graph representations have been used in the area of spatial constraint networks. A spatial constraint network is a graph-based description of a scene, where the nodes represent objects and the arcs represent sets of relations corresponding to disjunctions of possible spatial relations between objects (Hernandez, 1993). A discussion about relation-based representation systems and the processes involved can be found in (Papadias and Kavouras, 1994). A large amount of research in relation-based systems has concentrated on array structures, like symbolic images (Chang et al., 1987) and symbolic arrays (Papadias and Glasgow, 1991). A symbolic image is an array representation of a map (or an image in general) where each object is denoted by one or more symbols. ymbolic arrays are hierarchical symbolic images. Figure 2 illustrates two symbolic images and T representing the conceptual maps of Figure 1. The symbolic images preserve spatial relations such as the fact that is northeast of C1, and discard shape, size and metric information. H2 C2 H1 H1 H1 H2 H1 H2 C1 H1 T R1 Fig. 2 ymbolic images R1 T3 R2 R2 R2 R2 T2 everal questions arise about the symbolic images, as for instance, which are the spatial relations that are preserved, how symbolic images can be constructed and how spatial relations of higher resolution can be incorporated. ymbolic spatial indexes are a special kind of symbolic images that were developed as an attempt to provide answers to the previous questions. Each object in spatial index is represented by one or more symbols that correspond to special points of the object called representative points. The choice of representative points depends on the application needs; in R2 T4
Technical Report KDBLAB-TR-93-06 Relation-Based Information Processing With ymbolic patial Indexes Dimitris Papadias 1,2 and Theodoros Andronikos 2 1 Department of Geoinformation, Technical University of Vienna, Gusshausstrasse 27-29, Austria A-1040. e-mail: dpapadia@fbgeo1.tuwien.ac.at 2 Dept. of Electrical and Computer Engineering, National Technical University of Athens, Zographou, Athens, Greece 15780. e-mail: {dp, andron}@theseas.ntua.gr Abstract. The topic of the paper is spatial information processing using representations of direction and topological relations, called symbolic spatial indexes. The paper describes how symbolic spatial indexes can be used in information retrieval, composition of spatial relations, route planning and update operations. Information processing using spatial indexes involves symbolic, and not numerical, computation and avoids the usual problems of geometric representations such as finite resolution and geometric consistency. 1. Introduction The representation and processing of spatial relations has recently gained much attention in areas involving patial Reasoning (Freksa, 1992), Image Databases (Chang et al., 1989), Geographic Applications (Mark and Frank, 1990), and Cognitive cience (Tversky, 1993). Frank (1994), for instance, points out the importance of spatial relations in Geographic Interfaces and patial Query Languages. Pullar and Egenhofer (1988) classified spatial relations in: - direction relations that describe order in space (e.g. north, northeast), - topological relations that describe neighbourhood and incidence (e.g. disjoint), - comparative or ordinal relations that describe inclusion or preference (e.g. in, at), - distance relations such as far and near and - fuzzy relations such as next to and close. Consider, for example, the conceptual maps s and t of Figure 1. Map s is a subway map where Ci denotes a station and Hi a subway line. Map t illustrates the bus routes around station ; Ti denotes a bus-stop and Ri a bus route. The maps convey information about objects' characteristics (e.g., size of the subway stations, shape of bus routes) as well as spatial relations in 2D space among the objects (e.g., northeast of C1). Relation-based representations of the maps discard object characteristics and irrelevant spatial relations and preserve only a set of spatial relations of interest. In this paper we will deal with the representation of binary topological and direction relations among rigid objects. This research has been supported by the Esprit Basic Research Program 6881 (AMUING)