ARTICLE IN PRESS. Journal of Visual Languages and Computing

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1 RTILE IN PRESS Journal of Visual Languages and omputing 2 (29) ontents lists available at ScienceDirect Journal of Visual Languages and omputing journal homepage: Deriving topological relations between regions from direction relations Luo Guo a, Shihong Du b,c, a ollege of Life and Environmental Sciences, entral University for Nationalities, eijing 8, hina b Institute of Remote Sensing and GIS, Peking University, eijing 87, hina c State Key Laboratory of Remote Sensing Science, Institute of Remote Sensing pplications, hinese cademy of Sciences, eijing, hina article info rticle history: Received 25 pril 27 Received in revised form 7 November 28 ccepted 27 January 29 Keywords: Qualitative spatial reasoning Topological relations Direction relations Direction relation matrix 9-intersection model abstract The integration of topological and direction relations plays an important role in many applications, like spatial databases and pictorial retrieval systems. The method for deriving composition of binary topological relations cannot always yield unique or interesting results. Therefore, to integrate efficiently topological and direction relations, some new mechanisms are required to derive topological relations from direction cases when the above situations occur. This paper presents the computation methods for deriving topological relations from direction relations. The methods fall into two categories: the derivation of topological relations from one direction relation and two direction relations. Our methods can provide topological information when topological relations are unavailable, or more precise results are expected. Thus they are helpful in the integration of the calculi for topological and direction relations. & 29 Elsevier Ltd. ll rights reserved.. Introduction Qualitative spatial relations have been widely used in many research areas, such as spatial data query and retrieval [,7], spatial data mining [6], equivalence and similarity of spatial relations [6], and consistency maintenance of spatial database [4,9], etc. In these areas, the knowledge about spatial objects can be modeled by spatial relations which are represented as a small set of symbols, such as inside, overlap, contain, north, and south, etc. Many applications, such as spatial databases, use a set of symbolic relations to define query sentences, or describe the knowledge about spatial objects. Qualitative spatial reasoning can automatically derive new spatial relations from symbolic representations of known spatial relations by orresponding author at: Institute of Remote Sensing and GIS, Peking University, eijing 87, hina. Tel.: ; fax: addresses: guoluo2@63.com (L. Guo), dshgis@hotmail.com (S. Du). exploiting some common rules among spatial relations. For example, the connectivity matrix was used to represent and derive spatial knowledge about a spatial scene []. The matrix allows each element to be empty, a set of relations, or a unique relation, which imply that the spatial relation represented by the element is unknown, incomplete, or known, respectively. Reasoning about topological relations conducted on the matrix can make an unknown spatial relation become an incomplete or known one, or make an incomplete relation become more precise. This is helpful in evaluating consistency of symbolic description of a spatial scene. Pictorial retrieval systems used the reasoning about spatial relations to check whether a set of spatial relations between image objects is consistent with the real relations in the real world [5,2,25]. urrent methods for computing reasoning about spatial relations can fall into two groups: one is the reasoning of single-type spatial relation; the other is the integration of multi-type spatial relations. The reasoning of single-type spatial relation means that both the known and unknown spatial relations are of X/$ - see front matter & 29 Elsevier Ltd. ll rights reserved. doi:.6/j.jvlc.29..2

2 RTILE IN PRESS L. Guo, S. Du / Journal of Visual Languages and omputing 2 (29) the same type and formalized by the same model. Therefore, this kind of reasoning mainly involves the reasoning about direction relations, reasoning about topological relations, etc. For the reasoning about direction relations, Frank [2] proposed a composition operator based on algebra to infer direction relations between points. Goyal [3] studied the composition operations of direction relation matrix. Unfortunately, the method proposed by them cannot obtain the correct composition for several examples [23]. To overcome the shortcoming, Skiadopoulos and Koubarakis [23] first defined a composition table for two single-item direction relations, which only contain one direction, and then discussed the computational methods for composing multi-item direction relations, which contain multiple directions. Papadias and Egenhofer [8] developed two algorithms to infer direction relations between two points organized hierarchically: the first one inferred direction relations between points from the relations between their parent regions; the second one inferred through common points. oth the two algorithms are based on the direction relations between the MRs of two objects. To evaluate the consistency of multi-resolution direction relations, the effects of spatial scale on direction relations are modeled explicitly [7]. Then, the methods are presented for deriving direction relations between coarse spatial objects from the relations between detailed objects. Finally, the directional consistency is evaluated by checking whether the derived relations are compatible with the relations computed from the coarse objects. For the reasoning about topological relations, Egenhofer [] presented a method to compute the composition of two binary topological relations between regions. bdelmoty and El-Geresy [2] pointed out that the existing methods for spatial reasoning only focus on spatial relations between objects with similar types, same dimensions, and simple shapes. Therefore, they extended the 9-intersection to a general intersection model for describing topological relations between regions with complex shapes [3], and then proposed a general method to compute composition tables between objects with arbitrary types and complexity shapes [2]. In the case of the composition of direction relations with other spatial relations, Hong et al. [4] presented a method for computing the composition of direction relations with approximate qualitative distance. Papadias et al. [9] investigated the composition of directions with quantitative distances, by which the distance range and direction relations can be obtained. Sharma [22] first decomposed qualitative topology, direction into two interval relations, and then discussed the reasoning about topological relations from directions between simple regions. Due to the approximation of regions by two intervals, the reasoning cannot represent correctly the real situation between simple regions. Especially, it only focused on the reasoning between two single-item direction relations, and overlooked the reasoning about multi-item relations. If a more exact model of direction relations is used, more precise results could be computed. s a result, it is necessary to investigate the method to compute topological relations from a more exact model. In this study, we pay our attention to the derivation of topological relations (like overlap, disjoint, inside), which describe the spatial relations that are invariant under the topological transformations, from direction relations (like is north of, is southeast of D), which describe the cardinal directions of a target object with respect to a given reference object. The derivation is needed because of the following facts: () many applications, such as spatial databases and pictorial retrieval systems, involve the topological and direction relations together, thus it is necessary to integrate calculi for topological and direction relations; (2) the composition of binary topological relations cannot always yield unique or interesting results. For example, if disjoint and disjoint, the topological relations to be derived is the disjunction of all possible topological relations between regions, so does the composition of overlap and overlap; (3) the interrelationships between direction and topology make deriving topological relations from direction relations possible. For example, for points,, and, if is southeast of, and is southeast of, then, and must be disjoint in terms of common knowledge. Moreover, it is possible to obtain more precise results by combining the knowledge derived from direction relations and the composition of topological relations than only from topological relations. For example, if meets, and is disjoint with, the set of possible relations between and is {disjoint, meet, contain, cover, overlap}; if is north of, and north of, is disjoint with or meet. y combining the information of topology and direction between and, and and, the more precise results, disjoint or meet, can be identified. The contributions in this paper consist of: (i) investigating the method for deriving topological relations from one direction relation; (ii) developing methods for computing topological relations in terms of two direction relations. We limit our investigation to the integration of two specific models: the 9-intersection and the direction relation matrix for simple regions, which are homeomorphic to a closed plane disk []. The regions to be handled are neither composed of multiple regions, nor complex regions with holes. The contributions of this paper can be used in the following: Detecting the inconsistencies of topological and direction relations between regions in multi-resolution or multi-source spatial databases. Determining whether certain objects in the physical world satisfy the topological and direction relations contained in a query sentence before it is submitted for processing. ssessing the consistency of a complete or incomplete symbolic description of a spatial scene. Section 2 discusses related work about formalization and reasoning of spatial relations. The methods for deriving topological relations from direction relation fall into two categories deriving topological relations from one direction relation and from two relations in Section 3. Section 4

3 RTILE IN PRESS 37 L. Guo, S. Du / Journal of Visual Languages and omputing 2 (29) investigates the method for deriving topological relations from one direction relations. Section 5 classifies the derivation of topological relations from two direction relations into four categories, and then presents the computational method for each category. 2. Spatial relations 2.. Topological relations To formalize topological relations, the 9-intersection model [8] first divides space embedded region into three subsets: the interior (), boundary (q) and exterior ( ), and then uses the matrix derived from the intersections among the three subsets of two regions and to determine topological relations (Eq. ()): 2 \ \ 3 6 T ¼ @ \ 7 5. () \ \ Since each intersection has two values: non-empty () and empty (), the 9-intersection model can determine 52 topological relations in theory. Most of them, however, are not significant for two regions in the twodimensional geometric plane. Only eight relations are valid (Fig. ). For convenience, universal set U is used to denote all of the eight topological relations between simple regions, i.e., disjoint, meet, overlap, inside, cover, coveredy, contain and equal (Fig. ). shapes of spatial objects, not by real shapes. For example, the cone-based method uses points to approximate regions, thus it could not consider influences of the sizes and shapes of regions on direction relations 2D-String model does not represent direction relations directly. Derivation is needed to obtain direction information from symbol stings. Direction relation matrix uses the minimum bounding rectangle (MR) of a region (so-called reference region) and the real shape of another region (so-called target region) to compute the direction relations [3]. TheMRismoreadaptedtothesizeandtheshapeof the reference region than points, and the real shape of the target region is considered, not approximate shapes. Therefore, direction relation matrix is more suitable to formalize direction relations between regions than other models. In terms of the MR of a reference region, direction relation matrix divides the space into nine regions, such as N, NE, E, SE, S, SW, W, NW, and O (Fig. 2). The eight regions around reference region represent eight cardinal directions, respectively, i.e., north, northeast, east, southeast, south, southwest, west, and northwest, while the region at the center is called same. The direction relations from target region to reference region can be represented by a set of directions, which falls inside. If a direction relation contains only one direction, it is a single item; if it contains multiple directions, it is a multi-item. Definition. Let and be two simple regions, and R be the relation between them. If R is a single item, the 2.2. Direction relations NW N NE There are many models, such as direction relation matrix [3], cone-based [2], projection-based [2], four semi-infinite area model [], and 2D-String [5], have been proposed to formalize direction relations. ll of these models compute the direction relations by approximate W O E SW S SE Fig. 2. Direction relation matrix. disjoint meet overlap contain equal coveredy inside cover Fig.. Topological relations between simple regions [9].

4 RTILE IN PRESS L. Guo, S. Du / Journal of Visual Languages and omputing 2 (29) inf x() NW sup x() NE inf x () NW N sup y() N sup x () NE b b 2 supy() b 3 b 4 W O E W O E SW S SE inf y() SW S SE inf y() Fig. 3. Direction relations and the MR of a reference region. nine directions can be defined by the MR of the reference region as follows [23]: R ¼ {O } iff inf x ðþpinf x ðþ, sup x ðþxsup x ðþ, inf y ðþp inf y ðþ, and sup y ðþxsup y ðþ R ¼ {N } iff inf x ðþpinf x ðþ, sup x ðþxsup x ðþ, and sup y ðþpinf y ðþ R ¼ {S } iff inf x ðþpinf x ðþ, sup x ðþxsup x ðþ, and sup y ðþpinf y ðþ R ¼ {E } iff inf y ðþpinf y ðþ, sup y ðþxsup y ðþ, and sup x ðþpinf x ðþ R ¼ {W } iff inf y ðþpinf y ðþ, sup y ðþxsup y ðþ, and sup x ðþpinf x ðþ R ¼ {NE } iff sup y ðþpinf y ðþ and sup x ðþpinf x ðþ R ¼ {NW } iff sup y ðþpinf y ðþ and sup x ðþpinf x ðþ R ¼ {SE } iff sup y ðþpinf y ðþ and sup x ðþpinf x ðþ R ¼ {SW } iff sup y ðþpinf y ðþ and sup x ðþpinf x ðþ In above equations, inf x (), sup x (), inf y (), and sup y () denote the minimum and maximum projections of the reference region on x- and y-axis, respectively, so do inf x (), sup x (), inf y (), and sup y () (Fig. 3). If R is multi-item with directions r, r 2, y, r n (2pnp9), the target region can be split into sub-regions b, b 2, y, b n (2pnp9), such that ¼[ n i¼ b i, R b ¼ {r }, R b2 ¼ {r 2 }, y, R bn ¼ {r n } hold. For example, in Fig. 3b, ¼ b [b 2 [b 3 [b 4 and R b ¼ {N }, R b2 ¼ {NE }, R b3 ¼ {O }, R b4 ¼ {E }, thus R ¼ {N, NE, O, E }. For two simple regions, there are realizable 28 direction relations [3,23], including 29 multi-item, and 9 single-item cases. 3. Deriving topological relations from direction relations Spatial reasoning can be used as a mechanism to derive new spatial relations from incomplete or complete symbolic description of a spatial scene, and to evaluate whether the incomplete or complete symbolic description is consistent. For example, given two symbolic descriptions: () direction relation R ¼ {O, W, SW, S } and topological relation T ¼ contain; (2) direction relation R ¼ {O, W, NW }, R ¼ {S }, and topological relation T ¼ overlap. If other spatial relations and geometrical information about the spatial scenes are unavailable, are these topological and direction relations consistent? consistent description can guarantee that the spatial relations in the description represent correctly knowledge between spatial objects. Thus the spatial query based on the symbolic description can be answered rightly. The consistency of the description can be evaluated by checking [9,5]: () node consistency, (2) arc consistency, and (3) path consistency. The node consistency means the description should contain the spatial relations between an object and itself, i.e., T ¼ equal, R ¼ {O }. The arc consistency means that the spatial relations from region to region must be compatible with that from to. The path consistency implies the relations, derived from relations between regions and, and and, should be compatible with the real relations between regions and. The node consistency can be easily evaluated. Therefore, to evaluate the consistency integrating topological and direction relations, two new reasoning mechanisms should be developed: Given the direction relation between regions and, R, how about the possible topological relations between and? i.e. R! T. Given the direction relation between regions and, R, and the relation between regions and, R, how about the possible topological relations between and? i.e. R R! T. In the first mechanism, the topological relations to be derived and the given direction relations are of the same pair of regions, thus the correspondences between topological and direction relations should be investigated. The mechanism of this kind is called deriving topological relations from one direction relation, and can be used to evaluate arc consistency. In the second mechanism, there are two known direction relations R ( is the reference region) and R ( is the reference region), the possible topological relations between and can be derived from R and R. The mechanism of this kind is called deriving topological relations from two direction relations, and can be used to evaluate path consistency. y utilizing the two reasoning mechanisms, topological relations can be derived first from direction relations in an incomplete or complete symbolic description; secondly, if the intersection of known and derived

5 RTILE IN PRESS 372 L. Guo, S. Du / Journal of Visual Languages and omputing 2 (29) topological relations between the same pair of regions is empty, the symbolic description is inconsistent; otherwise, consistent. For example, in terms of the first mechanism, the set of possible topological relations, T ¼ {disjoint, meet, overlap, cover}, can be derived from the direction relation R ¼ {O, W, SW, S }. Therefore, the relation T ¼ contain in the initial set is inconsistent. 4. Deriving topological relations from one direction relation To derive topological relations from direction relations, the extents of nine directions should be defined clearly (Definition 2). These definitions are consistent with the Goyal s [3] and Skiadopoulos and Koubarakis s [23] formalizations of direction relations. Definition 2. Let the MR of the reference region be [inf x (), sup x ()] [inf y (), sup y ()]. Then the spatial region of direction N (SRD(N )) is denoted by [inf x (N ), sup x (N )] [inf y (N ), sup y (N )] (Fig. 3a). ccording to direction partition around region (Definition ), the following are the SRDs of directions N, S, E, W, NE, SE, NW, SW, and O : SRDðN Þ¼½inf x ðþ; sup x ðþš ½sup y ðþ; þþ; SRDðS Þ¼½inf x ðþ; sup x ðþš ð ; inf y ðþš, SRDðE Þ¼½sup x ðþ; þþ ½inf y ðþ; sup y ðþš, SRDðW Þ ¼ ð ; inf x ðþš ½inf y ðþ; sup y ðþš, SRDðNE Þ¼½sup x ðþ; þþ ½sup y ðþ; þþ, SRDðSE Þ¼½sup x ðþ; þþ ð ; inf y ðþš, SRDðNW Þ ¼ ð ; inf x ðþš ½sup y ðþ; þþ; SRDðSW Þ ¼ ð ; inf x ðþš ð ; inf y ðþš; and SRDðO Þ¼½inf x ðþ; sup x ðþš ½inf y ðþ; sup y ðþš. The inf and sup of a SRD denote the minimum and maximum value of the spatial extent of a direction, respectively. For example, SRD(N ) ¼ [inf x (), sup x ()] [sup y (),+N) means that inf x (N ) ¼ inf x (), sup x (N ) ¼ sup x (), inf y (N ) ¼ sup y (), and sup y (N ) ¼ +N (Fig. 3). With the exception of direction O, the spatial regions of other directions are unbounded. Therefore, the inf of a SRD can be N, and the sup of a SRD can be+n. In the symbolic description, direction relations are described as symbols. Therefore, the shape information of regions should be estimated from these symbols. The direction relation matrix uses the MR of reference region and the real shape of the target region to determine direction relations between regions. Therefore, the set union of SRDs can be used to approximate the shape of the target region (Definition 3). Definition 3. Let R be a direction relation, then the spatial region of R is defined as the set union of SRDs of all directions in R, denoted by SRDðRÞ ¼ r2r SRDðrÞ. The operator is to merge the common border of SRDs of two connected directions, thus SRD(R) is a curve region composed of rectangular ones. In Fig. 4, the gray part is the SRD(R ), i.e., SRD(R ) ¼ SRD(N )SRDðNW ÞSRDðW ÞSRDðSW ÞSRDðS Þ. ecause the SRD of a direction relation R contains the possible shape where the target region can be located, and MR() the approximation of region, the rules for deriving topological relations from one direction relation can be obtained by analyzing the topological configurations between SRD(R ) and MR(). Rule. If R represents the direction relation between and, and O er, the topological relation between and is either disjoint or meet (Fig. 5a). s O er, the topological configuration between SRD(R ) and MR() is meet. When region does not touch the borders of MR(), the topological relation between and, T,isdisjoint (Fig. 5a); when touches the boundary, T is meet (Fig. 5b) or disjoint. ccordingly, the topological configuration between and is either disjoint or meet. Rule 2. If R represents the direction relation between and, and R ¼ {O }, the possible topological relations between and are {disjoint, meet, overlap, contain, cover, equal, coveredy}. In terms of Definition 2, SRD(R ) ¼ SRD(O ) ¼ MR() holds. In this situation, the topological configuration between SRD(R ) and MR() isequal. mong the eight basic topological relations, the seven relations, disjoint (Fig. 6a), meet (Fig. 6b), overlap (Fig. 6c), contain (Fig. 6d), cover (Fig. 6e), equal (Fig. 6f), and coveredy (Fig. 6g), can satisfy this condition. Rule 3. If R is the direction relation between and, O 2 R, and R includes at most three directions out of N, S, E, and W, the possible topological relations between and are {disjoint, meet, overlap, coveredy}. The topological configuration between SRD(R ) and MR() is cover. ecause R contains at most three directions out of N, S, E, and W, at least one border Fig. 4. The spatial region of a direction relation. Fig. 5. Possible topological relations of Rule.

6 RTILE IN PRESS L. Guo, S. Du / Journal of Visual Languages and omputing 2 (29) Fig. 6. Possible topological relations of Rule 2. Fig. 7. Possible topological relations for Rule 3. Fig. 8. The additional topological relations for Rule 4. of MR() cannot be inside. Therefore, the relations cover, inside, equal, and contain, are not possible. ccordingly, the possible relations are disjoint (Fig. 7a), meet (Fig. 7b), overlap (Fig. 7c), or coveredy (Fig. 7d). Rule 4. If R represents the direction relation between and, and fo ; N ; S ; E ; W gr (Fig. 8), the possible topological relations between and are {disjoint, meet, overlap, coveredy, inside}. The topological configuration between SRD(R ) and MR() iscontain. ecause fo ; N ; S ; E ; W gr, and the four borders of MR() can be inside, there is one additional relation inside (Fig. 8), besides the four relations of Rule 3. lthough the topological configuration between SRD(R ) and MR() can be one of the eight basic relations between regions, in terms of definitions of direction relations (Definition ), only the following four configurations are possible for SRD(R ) and MR(): Direction O shares boundaries with other directions, thus SRD(R ) and MR() cannot be disjoint. Direction O is bounded, other directions are unbounded; thus when R a{o }, SRD(R ) is also unbounded. s a result, SRD(R ) cannot be inside or covered by MR(). SRD(R ) is composed of spatial regions of directions. Therefore MR() ¼ SRD(O ) is either a part of SRD(R ) or not. s a result, SRD(R ) and MR() cannot overlap. When R ¼ {O }, SRD(R ) ¼ MR() ¼ SRD(O ) hold, SRD(R ) and MR() isequal. ccordingly, there are four possible topological configurations, between SRD(R ) and MR(), meet, equal, cover, and contain. The four configurations correspond to the Rules 4, respectively. The four rules are jointly exhaustive and pairwise disjoint (JEPD). Therefore, they cover all possible cases for deriving topological relations from one direction relation. 5. Deriving topological relations from two direction relations To evaluate path consistency of symbolic description, it is necessary to check whether the topological relation between and recorded in the symbolic description is compatible with the relations derived from that between and and and. The composition of topological relations can be used to perform this task []. However, the compositions cannot always yield interesting results. Some compositions correspond to the universal set of topological relations. For example, if T ¼ disjoint and T ¼ disjoint, according to the composition of disjoint and

7 RTILE IN PRESS 374 L. Guo, S. Du / Journal of Visual Languages and omputing 2 (29) disjoint, T ¼ {disjoint, meet, overlap, inside, contain, cover, coveredy, equal}. The compositions, overlap and overlap, inside and contain, are similar to disjoint and disjoint. ecause the universal set is the largest set of possible relations, the recorded relation is always compatible with derived relations. ccordingly, the path consistency cannot be evaluated efficiently for these compositions. Some other compositions correspond to large sets of possible topological relations. For example, if T ¼ meet and T ¼ meet, according to the composition of meet and meet, T ¼ {disjoint, meet, overlap, cover, coveredy, equal}. There are 9 compositions out of all 8 cases corresponding to derived sets consisting of more than five relations. The larger the derived set is, the smaller the efficiency for evaluating path consistency is. Moreover, if the topological relations are unavailable, the consistency will not be evaluated. If the direction relations between and, and between and are available, the reasoning mechanism, R 3R -T, can obtain possible topological relations. Therefore, this mechanism can be used to evaluate path consistency of a symbolic description when topological information is unavailable, or improve the efficiency when the compositions of topological relations have large reasoning results. Direction relation matrix uses the MRs to approximate reference regions, while it considers the real shapes of target regions. The shapes information of target regions implied in direction relations (multi-item relations contain more shape information than single-item ones) can be used to derive the set of possible topological relations. The smaller the set is, the more precise the reasoning result is. Therefore, to improve the precision, the computation can fall into four categories: the reasoning between two single-item direction relations (Fig. 9a), between a multi- and a single-item direction relation (Fig. 9b), between a single- and a multi-item direction relation (Fig. 9c), and between two multi-item direction relations (Fig. 9d). 5.. Reasoning between two single-item direction relations Let MR() be the MR of region, and let SRD(R )be the spatial region of R. Then there are eight possible topological configurations in theory. That is, the topological configuration between SRD(R ) and MR(), denoted by T(SRD(R ),MR()), is one of the eight relations, disjoint, meet, overlap, coveredy, inside, cover, contain, and equal. When R ¼ {N }, there are four cases: When R ¼ {NE }, {N }, and {NW }, SRD(R )isdisjoint with MR() (Fig. a); thus the topological relation between and is also disjoint. When R ¼ {E }, {O }, and {W }, SRD(R )isdisjoint with (Fig. b), or meets MR() (Fig. c); thus the topological relation between and is disjoint or meet; the topological configuration of this kind is denoted by T(SRD(R ), MR()) ¼ meet. When R ¼ {SE } and {SW }, SRD(R ) meets (Fig. d), or overlaps (Fig. e) MR(). When T(SRD(R ), MR()) ¼ meet, the topological relation between and is disjoint or meet; when T(SRD(R ), MR()) ¼ overlap, the topological relation is in the set U ¼ {disjoint, meet, overlap, contain, cover} (Fig. ). Therefore, the final set is U. ecause the derived relations from T(SRD(R ), MR()) ¼ meet is a proper subset of that from T(SRD(R ), MR()) ¼ overlap, the topological configuration of this kind is denoted by T(SRD(R ), MR()) ¼ overlap. When R ¼ {S }, SRD(R ) covers (Fig. f) or overlaps (Fig. g) MR(). When T(SRD(R ), MR()) ¼ overlap, the topological relation is in the set {disjoint, meet, overlap, contain, cover}; when T(SRD(R ), MR()) ¼ cover, the topological relation is in the set U 2 ¼ {disjoint, meet, overlap, contain, cover, coveredy, equal}. Therefore, the final set is U 2. The topological configuration of this kind is denoted by T(SRD(R ), MR()) ¼ cover. Similarly, when R ¼ {E }, {W }, and {S }, SRD(R ) and MR() have the similar topological configurations. Therefore, the derived sets are also the same. When R ¼ {NE }, there are three cases: When R ¼ {NE }, {N }, {NW }, {SE }, and {E }, SRD(R )isdisjoint with MR(); thus the topological relation is also disjoint. When R ¼ {S }, {O }, and R ¼ {W }, SRD(R ) is disjoint or meets MR(); thus the topological relation between and is also disjoint or meet; the topological configuration of this kind is denoted by T(SRD(R ), MR()) ¼ meet. When R ¼ {SW }, SRD(R ) covers or contains MR(). When T(SRD(R ), MR()) ¼ cover, the topological relation is in the set {disjoint, meet, overlap, contain, Fig. 9. Deriving topological relations from two direction relations: (a) reasoning between two single-item direction relations; (b) reasoning between a multi- and a single-item relation; (c) reasoning between a single- and a multi-item relation; (d) reasoning between two multi-item relations.

8 RTILE IN PRESS L. Guo, S. Du / Journal of Visual Languages and omputing 2 (29) Fig.. Topological configurations between SRD(R ) and MR() when R ¼ {N }. disjoint meet overlap contain cover Fig.. Possible topological relations between and when TðSRDðR Þ; MRðÞÞ ¼ overlap. cover, coveredy, equal}. When T(SRD(R ), MR()) ¼ contain, the topological relation is in the universal set U. Therefore, the final set is U. In the same way, when R ¼ {SE }, {NW }, and {SW }, SRD(R ) and MR() have the same topological configurations, and can derive the same sets of topological relations. When R ¼ {O }, there are two cases: When R ¼ {N }, {S }, {E }, {W }, {NE }, {NW }, {SE }, and {SW }, SRD(R ) meets or overlaps MR(). When T(SRD(R ), MR()) ¼ meet, the topological relation is either disjoint or meet; when T(SRD(R ), MR()) ¼ overlap, the topological relation is in the set U. Therefore, the final set is U. The topological configuration is denoted by T(SRD(R ), MR()) ¼ overlap. When R ¼ {O }, SRD(R ) is inside or equal to MR(). When T(SRD(R ), MR()) ¼ inside, the topological relation is in the set {disjoint, overlap, meet, contain, cover}; T(SRD(R ), MR()) ¼ equal, the topological relation is in the set {overlap, equal, cover, coveredy}. Therefore, the final set is {disjoint, overlap, meet, contain, cover, equal, coveredy}, denoted by U 2. The reasoning results between two single-item direction relations can be organized in a 9 9 composition table (Table ). The first column and row in Table represent single-item direction relation R and R, respectively; other items represent the reasoning results Table omposition table for inferring topological relations from two singleitem direction relations N NE E SE S SW W NW O N {d} {d} {d, m} U U 2 U {d, m} {d} {d, m} NE {d} {d} {d} {d} {d, m} U {d, m} {d} {d, m} E {d, m} {d} {d} {d} {d, m} U U 2 U {d, m} SE {d, m} {d} {d} {d} {d} {d} {d, m} U {d, m} S U 2 U {d, m} {d} {d} {d} {d, m} U {d, m} SW {d, m} U {d, m} {d} {d} {d} {d} {d} {d, m} W {d, m} U U 2 U {d, m} {d} {d} {d} {d, m} NW {d} {d} {d, m} U {d, m} {d} {d} {d} {d, m} O U U U U U U U U U 2 according to the directions in the corresponding row and column. In all tables of this study, the topological relations disjoint, overlap, meet, contain, cover, equal, inside, and coveredy, are abbreviated as d, o, m, ct, cv, e, i, and cb, respectively Reasoning between a multi- and a single-item direction relation Since deriving topological relations from directions are performed based on symbolic relations R and R, the possible extents, where regions and can occupy should be derived. Then, the relation between MR() and the extent of can be used to derive topological relations between and. The spatial region of the relation R represents the possible extent where can be located, and the relation R is defined around MR(). Therefore,

9 RTILE IN PRESS 376 L. Guo, S. Du / Journal of Visual Languages and omputing 2 (29) a rectangular direction relation of R, named minimum closure of R (Definition 5), should be defined to approximate MR() [7]. Then, the rectangular relation and R can be related to derive topological relations between and (Definition 6). Definition 4. Let R be a direction relation, and the minimum bounding region of R, denoted by MR(R), be [inf x (R), sup x (R)] [inf y (R), sup y (R)]. Then inf x ðrþ ¼minðinf x ðrþjr 2 RÞ, sup x ðrþ ¼maxðsup x ðrþjr 2 RÞ, inf y ðrþ ¼minðinf y ðrþjr 2 RÞ, sup y ðrþ ¼maxðsup y ðrþjr 2 RÞ. For example, in Fig. 2a, R ¼ {N, NW, W, SW, S }, inf x (R ) ¼ N, sup x (R ) ¼ sup x (O ) ¼ sup x (), inf y (R ) ¼ N, and sup y (R ) ¼ +N. That is, MR(R ) ¼ [ N, sup x ()] [ N,+N]. Definition 5. Let R be a direction relation then the relation composed of directions falling inside the MR(R) is called minimum closure of R, denoted by Mc(R). Therefore, McðRÞ ¼frj½inf x ðrþ; sup x ðrþš ½inf x ðrþ; sup x ðrþš^- ½inf y ðrþ; sup y ðrþš ½inf y ðrþ, sup y (R)]}. In Fig. 2a, direction relation R ¼ {N, NW, W, SW, S } and its minimum closure is {N, NW, W, SW, S, O }. ccording to Definition 4, the 28 realizable direction relations between two regions can be projected on 36 minimum closures. mong these closures, nine cases are associated with 9 single-item direction relations, and 27 cases (Fig. 3) are relevant to 29 multi-item directions. Definition 6. Let R be a direction relation, and Mc(R) be the minimum closure of R. Then, the relation composed of the westernmost directions in Mc(R) is called bounding relation, denoted by Most(W, Mc(R)), which is defined by MostðW; McðRÞÞ ¼ frjinf x ðrþ ¼inf x ðrþ^r 2 McðRÞg (revised from [23]). Similarly, other eight bounding relations can be defined: MostðE; McðRÞÞ ¼ frjsup x ðrþ ¼sup x ðrþ^r 2 McðRÞg, MostðS; McðRÞÞ ¼ frjinf y ðrþ ¼inf y ðrþ^r 2 McðRÞg, MostðN; McðRÞÞ ¼ frjsup y ðrþ ¼sup y ðrþ^r 2 McðRÞg, MostðNE; McðRÞÞ ¼ frjsup x ðrþ ¼sup x ðrþ^sup y ðrþ ¼sup y ðrþ^r 2 McðRÞg, MostðNW; McðRÞÞ ¼ frjinf x ðrþ ¼inf x ðrþ^sup y ðrþ ¼sup y ðrþ^r 2 McðRÞg, MostðSE; McðRÞÞ ¼ frjsup x ðrþ ¼sup x ðrþ^inf y ðrþ ¼inf y ðrþ^r 2 McðRÞg, Fig. 2. Minimum closures of direction relations. MostðSW; McðRÞÞ ¼ frjinf x ðrþ ¼inf x ðrþ^inf y ðrþ ¼inf y ðrþ^r 2 McðRÞg, MostðO; McðRÞÞ ¼ McðRÞ. For example, Most(E, Mc{N, NW, W, SW, S }) ¼ {N, O, S }(Fig. 2a); Most(W, Mc{N, NW, W, SW, S }) ¼ {NW, W, SW } (Fig. 2a); Most(SE, Mc{N,NW,W,O }) ¼ {O }(Fig. 2b). Theorem. Let R be a multi-item direction relation, R be a single-item one, and r be the exclusive direction in R, then R 3R ¼ Most(r, Mc(R ))3R. Symbol R 3R means deriving topological relations from direction relations R and R. Proof. Let R ¼ {E } and R ¼ Most(E, Mc(R )), MR(R ) ¼ MR(Mc(R )) holds according to Definition 4. Since R is composed of easternmost directions in R, sup x (R) ¼ sup x (R ), sup y (R) ¼ sup y (R ), and inf y (R) ¼ inf y (R ) hold in terms of Definitions 5 and 6. ccording to definitions of direction relations (Definition ), the direction E bounded by R and R are equal. Therefore, R 3R ¼ Most(E, Mc(R ))3R is true when R ¼ {E }. Similarly, when R is composed of any one of other eight directions, Theorem also holds. & Theorem shows that the reasoning between a multiitem relation R and a single-item relation R can be computed by Most(r, Mc(R )) (rr ) and R. Furthermore, Mc(R ) represents a group of direction relations with the same minimum closure. Therefore, the combinations of such multi-item relations with a single-item relation correspond to the same derived set of possible topological relations. The causes are the following: The direction relation matrix model uses the MR of a reference region to compute direction relations, not the real shape. That is, if two regions have the same MRs, they have the same direction partitions. Only symbolic information about direction relation R is available, while geometrical information about regions and is unavailable. Therefore, the minimum closure of R is the only way to estimate MR(). ccordingly, those direction relations with same minimum closure correspond to the same derived set as symbolic information and the approximate shape are adopted. In terms of the definitions of Most, when direction r (rr ) is NW, NE, SE, or SW, the relation Most(r, Mc(R )) is single item. Therefore, the reasoning can be computed directly by querying Table according to Most(r, Mc(R )) and R. When r is N, S, E,orW, Most(r, Mc(R )) is multi-item with 8 possible relations (Table 2). These relations are generated by Most from the 27 minimum closures (Fig. 3). ccording to Theorem, the reasoning between R and R can be derived from the combinations of the 8 relations with N, S, E, and W, as well as combinations of nine directions N, S, W, E, SW, SE, NW, NE, and O, with NW, NE, SE, and SW.

10 RTILE IN PRESS L. Guo, S. Du / Journal of Visual Languages and omputing 2 (29) {NW, N, NE } {NW, N, NE, {NW, N, NE, W, O, E, W, O, E } SW, S, SE } {NW, N } {NW, N, W, O } {NW, N, W, O, SW, S } I II III IV V VI {N, NE } {N, NE, O, E } {N, NE, O, E, S, SE } {W, O, E } {W, O, E, SW, S, SE } {W, O } VII VIII IX X XI XII {W, O, SW, S } {O, E } {O, E, S, SE } {SW, S, SE } {SW, S } {S, SE } XIII XIV XV XVI XVII XVIII {NW, W } {NW, W, SW } {N, O } {N, O S } {NE, E } {NE, E, SE } XIX XX XXI XXII XXIII XXIV {W, SW } {O, S } {E, SE } XXV XXVI XXVII Fig. 3. The 27 minimum closures of multi-direction direction relations. ased on the analysis above, Theorem 2 can be used to derive topological relations from a multi-item and a single-item direction relation. Theorem 2. If R is a multi-item direction relation, and R is a single-item direction relation, possible topological relations between and, derived from R and R, can be computed by the following three situations. Let rr, R ¼ Most(r, Mc(R )), then: When r ¼ NE, r ¼ SE, r ¼ NW,orr ¼ SW (Fig. 4a), for r 2 R, R 3R ¼ {r }3{r} ({r }3{r} can be obtained by querying Table according to r and r).

11 RTILE IN PRESS 378 L. Guo, S. Du / Journal of Visual Languages and omputing 2 (29) () When r ¼ N, r ¼ S, r ¼ E,orr¼W (Fig. 4b), if R includes only one element r, R 3R ¼ {r }3{r}; otherwise, R 3R ¼ R3{r} (R3{r} can be obtained by querying Table 2 according to R and r). (2) When r ¼ O holds, if O emcðr Þ (Fig. 4c), R 3R ¼ {disjoint, meet}; if Mc(R ) includes all nine directions (Fig. 4d), R 3R ¼ U; otherwise (Fig. 4e), R 3R ¼ U 2, where symbol U and U 2 have same meanings as in Table. Proof. () When r ¼ NE, r ¼ SE, r ¼ NW, or r ¼ SW, there is only one element, r,inr. ccording to Theorem, R 3R ¼ R3R holds, i.e., R 3R ¼ {r }3{r} holds. (2) In this situation, R may contain one, two or three elements. When R contains one element, the reasoning and proof are same with (). When R contains more than one element, there are 8 possible relations for R (Table 2). The left part of Table 2 is the combinations of Most(N, Mc(R )) and Most(S, Mc(R )) with r ¼ N and r ¼ S, respectively. The right part is the combinations of Most(E, Mc(R )) and Most(W, Mc(R )) with r ¼ E and r ¼ W, respectively. If Most(N, Mc(R )) ¼ {NW, N, NE }, and region is north of region, then and is disjoint (Fig. 3b); if Most(S, Mc(R )) ¼ {NW, N, NE } and region is south of, the set of possible topological relations between and is U. That is, the reasoning results of the two combinations, Most(N, Mc(R )) ¼ {NW, N, NE } with N, and Most(S, Mc(R )) ¼ {NW, N, NE } with S, are true. Other combinations in Table 2 can also be proofed by similar ways. (3) When r ¼ O,ifO emcðr Þ, i.e., SRD(O ) either is disjoint or meets MR(); i.e., T(SRD(R ), MR()) ¼ meet. Table 2 The topological relations derived from a multi- and a single-item direction relation. Most (r, Mc(R )) N S Most (r, Mc(R )) E W {NW, N, NE } {d} U {NW, W, SW } U {d} {NW, N } {d} U 2 {NW, W } U 2 {d} {N, NE } {d} U 2 {W, SW } U 2 {d} {W, O, E } U U {N, O, S } U U {W, O } U U {N, O } U U {O, E } U U {O, S } U U {SW, S, SE } U {d} {NE, E, SE } {d} U {SW, S } U 2 {d} {NE, E } {d} U 2 {S, SE } U 2 {d} {E, SE } {d} U 2 Thus region must be disjoint with or meets region (Fig. 3c). When Mc(R ) contains all nine directions of (Fig. 3d), T(SRD(R ), MR()) ¼ contain. Thus the topological relation is any one of the eight basic topological relations. When O 2 McðR Þ and Mc(R ) contains less than nine directions (Fig. 3e), T(SRD(R ), MR()) ¼ cover. Thus the topological relation between and is one of U 2 in Table 2. & In terms of Theorem 2, the reasoning between a multiand a single-item direction relation can be computed. ecause there are 27 distinct minimum closures for 29 multi-direction relations, the reasoning results of this kind can be organized into a 27 9 composition table (Table 3). In Table 3, the first column represents the 27 distinct minimum closures of multi-item direction relations, and the first row denotes the nine single-item direction relations. Other entries refer to the reasoning results of the corresponding multi- and single-item direction relations. Table 3 omposition table for a multi- and a single-item direction relation Mc(R ) N NE E SE S SW W NW O Fig. 3-I {d} {d} {d} {d} U {d} {d} {d} {d, m} Fig. 3-II {d} {d} {d} {d} U {d} {d} {d} U 2 Fig. 3-III {d} {d} {d} {d} {d} {d} {d} {d} U Fig. 3-IV {d} {d} {d, m} U U 2 {d} {d} {d} {d, m} Fig. 3-V {d} {d} U U U {d} {d} {d} U 2 Fig. 3-VI {d} {d} U {d} {d} {d} {d} {d} U 2 Fig. 3-VII {d} {d} {d} {d} U 2 U {d, m} {d} {d, m} Fig. 3-VIII {d} {d} {d} {d} U U U {d} U 2 Fig. 3-IX {d} {d} {d} {d} {d} {d} U {d} U 2 Fig. 3-X U {d} {d} {d} {d} {d} {d} {d} U 2 Fig. 3-XI U {d} {d} {d} {d} {d} {d} {d} U 2 Fig. 3-XII U U U U U {d} {d} {d} U 2 Fig. 3-XIII U U U {d} {d} {d} {d} {d} U 2 Fig. 3-XIV U {d} {d} {d} U U U U U 2 Fig. 3-XV U {d} {d} {d} {d} {d} U U U 2 Fig. 3-XVI U {d} {d} {d} {d} {d} {d} {d} {d, m} Fig. 3-XVII U 2 U {d, m} {d} {d} {d} {d} {d} {d, m} Fig. 3-XVIII U 2 {d} {d} {d} {d} {d} {d, m} U {d, m} Fig. 3-XIX {d} {d} U 2 U {d, m} {d} {d} {d} {d, m} Fig. 3-XX {d} {d} U 2 {d} {d} {d} {d} {d} {d, m} Fig. 3-XXI {d} {d} U U U U U {d} U 2 Fig. 3-XXII {d} {d} U {d} {d} {d} U {d} U 2 Fig. 3-XXIII {d} {d} {d} {d} {d, m} U U 2 {d} {d, m} Fig. 3-XXIV {d} {d} {d} {d} {d} {d} U 2 {d} {d, m} Fig. 3-XXV {d, m} U U 2 {d} {d} {d} {d} {d} {d, m} Fig. 3-XXVI U U U {d} {d} {d} U U U 2 Fig. 3-XXVII {d, m} {d} {d} {d} {d} {d} U 2 U {d, m} Fig. 4. Reasoning between a multi- and a single-item direction relation.

12 RTILE IN PRESS L. Guo, S. Du / Journal of Visual Languages and omputing 2 (29) Reasoning between a single- and a multi-item direction relation Reasoning between a single-item R and a multiitem R can be regarded as the combinations of reasoning between R and each direction in R. That is, the results can be computed by combining the results of reasoning between two single-item direction relations (Table ) ombinations of reasoning between two single-item direction relations There are 9 9 combinations of two single items, and the reasoning results are represented in Table. Therefore, the possible combinations of two single-item direction relations can be obtained from Table. For example, when R ¼ {N }, there are four cases: The combinations of R with directions N, NE,orNW correspond to one relation disjoint. The combinations of R with directions E, W,orO correspond to {disjoint, meet}. The combinations of R with directions SE,orSW correspond to the set U. The combination of R with direction S corresponds to the set U 2. ecause reasoning between a single-item R and a multi-item R can be considered as combinations of reasoning between R and each direction in R, the four cases, R ¼ {N }, {S }, {E }, and {W }, have 5 combinations ( ¼ ¼ 5). However, only nine combinations are realizable (Table 4), and others are unrealizable, because directions in R must be connected. For example, {disjoint} and U is an impossible combination when R ¼ {N }, as directions {NW, N, NE } corresponded by set {disjoint}, and directions {SE, SW } related by set U, are disconnected. That is, any direction relations composed of directions in {NW, N, NE }, and in {SE, SW } are unrealizable for two regions, such as {NW, SE }, {NW, SE }, and {N, SE }. In the same way, when R ¼ {S }, {E }, and {W }, there are still nine realizable combinations. When R ¼ {SE }, {SW }, {NE }, and {NW }, there are three cases: {disjoint}, {disjoint, meet}, and U. The three cases have five realizable combinations, among total 7 combinations ( ¼ 3+3+ ¼ 7). When R ¼ {O }, there are two realizable combinations. Table 4 only lists possible combinations for R ¼ {N }, {NE }, and {O }, others have the same ones. In Table 4, Function d(r, y, r n ) is a set of all realizable single- or multi-item direction relations generated by the disjunction of directions r, y, r n. That is, d(r, y, r n ) ¼ {R, y, R m }, and R i (pipm) is one of the 28 realizable direction relations. For example, d{nw, N, NE } ¼ {{NW }, {N }, {NE }, {NW, N }, {N, NE }, {NW, N, NE }}. Direction relation {NW, NE } is ruled out as it is not a case among the 28 realizable relations. Function d (r, y, r n ) represents a set of all realizable multi-item direction relations generated by the disjunction of directions r, y, r n. That is, d (r, y, r n ) ¼ {R, y, R m }, and R i (pipm) is one of the 29 realizable multi-item direction relations. For example, d {NW, N, NE } ¼ {{NW, N }, {N, NE }, {NW, N, NE }}. Direction relations, {NW }, {N }, and {NE }, are excluded as they are single-item relations; {NW, NE } is ruled out as it is not realizable. Function d 2 (r, y, r n ) refers to all single- or multi-item direction relations generated by the disjunction of directions r, y, r n. That is, d 2 (r, y, r n ) ¼ {R, y, R m }, and R i (pipm) is composed of any one or several directions among r, y, r n. d 2 {NW, N, NE } ¼ {{NW }, {N }, {NE }, {NW, N }, {N, NE }, {NW, NE }, {NW, N, NE }}. It is notable that d 2 {NW, N, NE } can generate an unrealizable direction relations between regions, while d and d cannot. For example, {NW, NE } is not a valid case for d and d, while it is for d 2. Operator denotes the Descartes production of two sets of direction relations. Let M ¼ {R, R 2, y, R m }, and M 2 ¼ {R 2, R 22, y, R 2n }, M M 2 is a set of all realizable direction relations by joining each R i 2 M and each R 2j 2 M 2. That is, M M 2 ¼ fr i [ R 2j jr i 2 M ^ R 2j 2 M 2 ^ R i [ R 2j 2 Dg, where D is the set of 28 realizable direction relations between simple regions. For example, M ¼ d{e, W, O } ¼ {{E }, {W }, {O }, {E, O }, {W, O }, {E, W, O }}, M 2 ¼ d 2 {SE, SW } ¼ {{SE }, {SW }, {SE, SW }}. M M 2 ¼ {{E, SE }, {W, SW }, {E, SE, O }, {W, SW, O }, {E, SE, O, W }, {E, O, W, SW }, {E, SE, O, W, SW }}. The following direction relations, {E, SW }, {E, SE, SW }, {W, SE }, {W, SW, SE }, {O, SE }, {O, SW }, {O, SE, SW }, and so on, are ruled out, because they are not realizable omputing topological relations between regions from that between their parts Let r be the exclusive direction of R, and let R be composed of r, r 2, y, r n (2pnp9). Then region can be split into n exclusive sub-regions c, c 2, y, c n, and [ n i¼ c i ¼ holds. Each reasoning rr i (pipn) can derive possible topological relations between and c i. Thus the possible topological relations between and can be inferred from that between and sub-regions c i. Let the set of derived topological relations between and c i be T i, and the set between and c j (piajpn) be T j. Then 8t l 2 T i and 8t m 2 T j, t l t m is to derive possible topological relations between and (c i [c j ) from that between and c i and and c j. Tryfona and Egenhofer [24] have presented a computational method for deriving topological relations between two regions and from that between and the parts of. The inference results can be organized into an 8 8 table. The original method is to infer topological relations between and from that between c i and, and c j and, which handled topological relations converse with the ones in this study. Therefore, the results are revised and presented in Table 5. t l t m ¼ + implies that the corresponding composition is unrealizable.

13 RTILE IN PRESS 38 L. Guo, S. Du / Journal of Visual Languages and omputing 2 (29) Table 4 Possible combinations of derived set of topological relations. R Possible combinations Possible R Reasoning results N {d} d {NW, N, NE } {d} {d, m} d {E, W, O } {d, m}} {d}, {d, m} d{nw, N, NE } d{e, W, O } {d, m} {d, m}, U d{e, W, O} d2{se, SW} {d, m, o} {d, m}, U 2 {S, O }} {d, m, o, cb} {d}, {d, m}, U d{nw, N, NE } d{e, W, O } d 2 {SE, SW } {d, m, o} U, U2 {S, SE}, {S, SW} U2 {SE, S, SW } U {d, m}, U, U2 d{e, W, O} {SE} {S}, d{e, W, O} {SW} {S} {d, m, o, cb} d{e, W, O} {SE, SW} {S} {d, m, o, cb, i} {d}, {d, m}, U, U 2 d{nw, N, NE } d{e, W, O } {{SE }, {SW }} {S } {d, m, o, cb} d{nw, N, NE } d{e, W, O } {SE, SW } {S } {d, m, o, cb, i} NE {d} d{nw, N, NE, E, SE} {d} {d, m} d {S, W, O }} {d, m} {d}, {d, m} d 2 {NW, N, NE, E, SE } d{s, W, O } {d, m} {d, m}, U d{s, W, O} {SW} {d, m, o, cb, i} {d}, {d, m}, U d{nw, N, NE, E, SE} d{s, W, O} {SW} {d, m, o, cb, i }} O U d{nw, N, NE, E, SE, S, SW, W } and jr jp4 U {NW, N, NE, E, W}, {E, SE, S, SW, W}, {NW, N, S, SW, W}, {N, NE, E, SE, S} U {NW, N, NE, E, SE }, {NE, E, SE, S, SW }, {NW, SE, S, SW, W }, {NW, N, NE, SW, W } U d{nw, N, NE, E, SE, S, SW, W } and jr jx6 U U, U2 d{nw, N, NE, E, SE, S, SW, W} {O} and jrjp5 U {NW, N, NE, E, W, O }, {E, SE, S, SW, W, O }, {NW, N, S, SW, W, O }, {N, NE, E, SE, S, O } U others U