9 -intersection calculi for topological reasoning on ...mschneid/files/arif/yohei...

JOURNAL OF SPATIAL INFORMATION SCIENCE

Submitted to the Journal of Spatial Science

© Yohei Kurata Licensed under Creative Commons Attribution 3.0 License

9+-Intersection Calculi for Topological Reasoning on Heterogeneous Objects

Yohei Kurata

SFB/TR8 Spatial Cognition, Universität Bremen, Postfach 330 440, 28334 Bremen, Germany

Manuscript for the first review

Abstract: This paper develops a series of qualitative spatial calculi that feature topological relations. While most qualitative spatial calculi have targeted spatial relations between single-type objects, our calculi support spatial relations between heterogeneous combinations of objects. Thus, with our calculi, we can disambiguate topological arrangements of objects without regard to the types of the objects. Topological relations between points, lines, regions, and their combinations are distinguished systematically by the 9+-intersection and assigned unique names that indicate their characteristics. The 9+-intersection is a refinement of the 9-intersection, which is has been frequently used in GIScience communities as a model of topological relations. Based on the 9+-intersection we can generalize the rules for deriving compositions and, accordingly, a variety of composition tables are generated systematically as a foundation of new calculi. By integrating all relevant sets of topological relations, composition tables, and lists of converse relations, the algebraic framework of ordinary spatial calculi is successfully reused in our new spatial calculi for conducting constraint-based spatial reasoning. We demonstrate that our 9+-intersection calculi realize finer spatial reasoning than the former 9-intersection calculi.

Keywords: 9+-intersection, 9+-intersection calculi, qualitative spatial reasoning, qualitative spatial calculi, topological relations, heterogeneous objects

2 KURATA

www.josis.org

1 Introduction

Topological relations, which essentially concern how two objects connect or overlap,

are one of the most studied sorts of spatial relations in GIScience communities. This

paper develops a series of qualitative spatial calculi (QSC) that target such topological

relations. QSC are the frameworks of qualitative spatial reasoning, with which we can

detect inconsistency in information about the spatial relations that hold between a set

of objects and, if inconsistency is not detected, we can further obtain candidates for the

unspecified spatial relations. Such reasoning is called constraint-based spatial reasoning,

because the spatial relations whose types are given or already determined serve as the

constraints on the types of other spatial relations in the reasoning process. The

remarkable feature of our QSC is the ability to conduct constraint-based spatial

reasoning even when the objects are heterogeneous (i.e., when the objects are a mixture

of points, lines, and regions). Usually QSC target spatial relations between single-type

objects. For instance, Allen’s interval algebra [1], Region Connection Calculus [2],

Cardinal Direction Calculus [3], and Double Cross Calculus [4, 5] target the relations

between two intervals, two regions, two points, and three points, respectively. Such

spatial calculi fit nicely into an algebraic framework (relation algebra or its family) and,

accordingly, constraint-based spatial reasoning is achieved by algebraic computation

(Section 3). In geographic information systems, however, we often deal with

heterogeneous geographic features, whose relative positions are semantically

important (e.g., rivers and wildlife habitats). Naturally, a question arises as to how we

can disambiguate spatial arrangements of such heterogeneous objects. We answer this

question with regard to topological relations.

Our work extends Kurata’s [6] pilot work on heterogeneous QSR. He featured

topological relations distinguished by the 9-intersection [7]. In GIScience communities,

both the 9-intersection and Region Connection Calculus [2] have been frequently used

as models of topological relations. While Region Connection Calculus has a fixed

granularity, the 9-intersection distinguishes different numbers of topological relations

depending on the types of two objects. For instance, in an 2D Euclidian space ℝ�, the 9-intersection distinguishes 8 region-region relations, 19 line-region relations, and 33

line-line relations [7]. We extend Kurata’s work by shifting the foundation from the 9-

intersection to the 9+-intersection [8]. The 9+-intersection is an extension of the 9-

intersection, which enables us to identify the complete set of topological relations, as

well as their conceptual neighborhood graph, for arbitrary pairs of object domains by

semi-automated methods [9, 10]. We show that the 9+-intersection also allows us to

generalize the rules for deriving the compositions of topological relations and,

9+-Intersection Calculi for Topological Reasoning on

Heterogeneous Objects

3

Submitted to JOSIS

accordingly, the composition tables, used as a foundation of new calculi, can be

derived in a systematic way (Section 4.2). Another merit of the 9+-intersection is that

we can represent and reason about spatial arrangements of objects more precisely than

the 9-intersection, as demonstrated in Section 5.

We develop two sorts of QSC based on the 9+-intersection: homogeneous 9+-

intersection calculi, which target topological relations between single-type objects, and

heterogeneous 9-intersection calculi, which target topological relations between

heterogeneous objects. We show that by simple extensions the algebraic framework of

ordinary spatial calculi can be reused in our new calculi for conducting constraint-

based spatial reasoning on topological relations between heterogeneous objects.

The remainder of this article is structured as follows: Section 2 introduces the 9+-

intersection and prepares several sets of topological relations for later discussion.

Section 3 summarizes basic concepts of QSC. Section 4 develops the 9+-intersection

calculi, starting from the development of conversion and composition operations on a

variety of topological relations. Section 5 demonstrates the use of new calculi for

qualitative spatial reasoning. Finally, Section 6 concludes the discussion.

In this paper, lines and regions refer to simple lines and simple regions [11],

respectively. Simple lines are single-component lines without branches or loops. They

are represented by a one-to-one continuous mapping from [0, 1] to the space. We

distinguish the start and end points of each line explicitly, even though this distinction

is lost in the 9-intersection-based topological relations. Simple regions are single-

component regions without spikes, cuts, or disconnected interiors. They are

homeomorphic to two-dimensional discs.

2 The 9+-Intersection

The 9+-intersection [9] is a model of topological relations that refines the 9-intersection

[7]. Both models presume the distinction of interior, exterior, and boundary of each object,

which are also called the topological parts of the object. Based on point-set topology [12],

the interior, exterior, and boundary of a spatial object X, denoted �°, ��, and ∂�, are defined as the union of all open sets contained in X, the union of all open sets that do

not contain X, and the difference between ��’s complement and �°, respectively. By this definition, the boundary of a region refers to its circular edge, the boundary of a

line refers to its two endpoints, and the boundary of a point is the point itself,

regardless of the dimension of the embedding space.

4 KURATA

www.josis.org

Given two objects A and B embedded in a space �, the 9-intersection characterizes the topological relation between A and B based on the topological

properties of the 3×3 set intersections between A’s topological parts and B’s topological

parts. These 3×3 set intersections are concisely represented in the 9-intersection matrix

(Equation 1). Usually, topological relations are distinguished simply by the emptiness

or non-emptiness of these 3×3 set intersections, which we call the 9-intersection pattern.

The 9-intersection pattern is represented by an icon with 3×3 cells, where each cell is

marked if the corresponding matrix element is non-empty (Figure 1a).

M, � = �° ∩ �° ° ∩ ∂� ° ∩ ��∂ ∩ �° ∂ ∩ ∂� ∂ ∩ �� ∩ �° � ∩ ∂� � ∩ �� (1)

The 9+-intersection refines the 9-intersection by considering the set intersections

between the topological primitives of two objects. The topological primitives are self-

connected and mutually-disjoint subparts of the topological parts of objects. For

instance, the boundary of a simple line L consists of two topological primitives (the

start point ∂�� and the end point ∂��), while L’s interior and exterior each consist of one topological primitive. The interior, exterior, and boundary of a simple region R

consist of one topological primitive, respectively. Accordingly, the topological relation

between L and R is characterized by a nested 9-intersection matrix in Equation 2. Just

like the 9-intersection, the topological relations are distinguished by the emptiness or

non-emptiness of the set intersections in the nested 9-intersection matrix, which we call

the 9+-intersection pattern, and this pattern is represented by an icon like Figure 1b. This

icon is similar to the icon of the 9-intersection pattern (Figure 1a), but the cells

corresponding to multiple set intersections are partitioned.

M��, � = � �° ∩ �° �° ∩ ∂� �° ∩ ��∂�� ∩ �°∂�� ∩ �°� �∂�� ∩ ∂�∂�� ∩ ∂�� ∂�� ∩ ��∂�� ∩ �� ∩ �° �� ∩ ∂� �� ∩ �� (2)

(a) (b)

Figure 1: Topological relations between a line and a region in ℝ2 characterized by (a) a

9-intersection pattern and (b) a 9+-intersection pattern

L∂oL

R∂ oR

L∂

oR R∂oL

−L

−R

oLLs∂Le∂

R∂ oR oR R∂oL

−L

−R

Ls∂Le∂



5

Submitted to

Submitted to JOSIS

Topological primitives are classified by their dimensionality (0D, 1D, 2D, and 3D)

and their boundedness (bounded, unbounded, looped) [9]. By showing all topological

primitives that form the interior, boundary, and exterior of an object and their

connections, we can represent the topological structure of each object (Figure 2).

Figure 2: Topological structures of a point, a line, and a region in ℝ2

Given two object domains DA and DB and a space �, the sets of topological relations between an object in DA and an object DB, distinguished by the 9- and 9+-

intersection patterns, are denoted ��A�B ̵� and ��!�B ̵� , respectively. For instance, ��"# ̵ℝ$ is the set of topological line-region relations in ℝ� distinguished by the 9+-intersection patterns (note that P, L, and R represent the domains of points, lines, and

regions). Table 1 shows the number of topological relations between every pair of

simple objects distinguished by the 9- and 9+-intersection patterns [9]. The 9+-

intersection has higher granularity when either or both objects are lines or when the

embedding space is one-dimensional. Otherwise, two models are equivalent.

Table 1: Numbers of topological relations distinguished by the 9-/9+-intersection

patterns [9] (ℝ%: n-dimensional Euclidian space, &%:n-sphere) Relation Types

9-intersection 9+-intersection ℝ' ℝ� ℝ( &' &� ℝ' ℝ� ℝ( &' &� Point-Point 2 2 2 2 2 6 2 2 2 2

Point-Line / Line-Point 3 3 3 3 3 10 4 4 4 4

Point-Region / Region-Point – 3 3 – 3 – 3 3 – 3

Point-Body / Body-Point – – 3 – – – – 3 – –

Line-Line 8 33 33 11 33 26 80 80 28 80

Line-Region / Region-Line – 19 31 – 19 – 26 45 – 26

Line-Body / Body-Line – – 19 – – – – 26 – –

Region-Region – 8 43 – 11 – 8 43 – 11

Region-Body / Body-Region – – 19 – – – – 19 – –

Body-Body – – 8 – – – – 8 – –

exterior

boundary

interior

0D

bounded 1D bounded 2D

unbounded 2D

0D

unbounded 2D

0D looped 1D

unbounded2 D

Point Line Region

6 KURATA

www.josis.org

In the following discussion, we use nine sets of topological relations that may

hold between points, lines, regions, and their combinations in ℝ�—��)) ̵ℝ$, ��)" ̵ℝ$, ��)# ̵ℝ$, ��") ̵ℝ$, ��"" ̵ℝ$, ��#) ̵ℝ$, ��#) ̵ℝ$, and ��## ̵ℝ$ . These nine sets are shown in Figs. 3-4. In these figures, pairs of relations which can be obtained by exchanging the

objects A and B (e.g., djPR and djRP) are illustrated together, since each pair shares a

sample configuration. Each relation is assigned a unique name, like �[+]�A�B. �A and �B represent the object domains. X represents the class of the relation. We consider ten

classes of topological relations, namely equal (eq), disjoint (dj), meet (mt), overlap (ov),

coveredBy (cb), covers (cv), inside (in), contains (ct), adhereTo (ad), and adheredBy (ab). The

first eight classes follow the names of eight topological region-region relations given by

Egenhofer and Al-Taha [13]. Two additional classes, adhereTo and adheredBy, are used

when one object is completely in the boundary of another object. These ten classes are

distinguished by the flowchart in Figure 6. The optional superscript Y is used when

there are multiple relations that belong to the class X. If either object is a line (Figures

4b-c), Y basically represents the position of the line’s start and end points. For instance, /012LR is a line-region relation where the line starts from the region’s exterior and ends at the region’s interior (note i: interior, x: exterior, b: boundary, s: start point, and e: end

point). Similarly, if both objects are lines (Figures 3c), Y represents the position of the

start and end points of the two lines (e.g., 5671/17LL). In this way, we systematically assigned unique names to the 9+-intersection-based topological relations. Such names

are highly useful for understanding the characteristics of the relations, as well as

discovering similarities between the topological relations in different relation sets.



7

Submitted to JOSIS

(a)

(b)

(c)

Figure 3: (a) Topological point-point relations, (b) topological region-region relations,

and (c) topological line-line relations, which are distinguished by the 9+-intersection

djPPeqPPdjRReqRR mtRR ovRR cbRR inRRcvRR ctRR

mtix/xiLL mtxi/ix

LL mtxe/ieLL mtxs/eiLLmtie/xeLL mtei/xsLL

mtii/xxLL mtxx/ii

LL mtsx/siLL mtex/isLLmtsi/sxLL mtis/exLLmtex/xsLL mtxs/ex

LL mtix/xxLL mtxi/xxLLmtxx/ixLL mtxx/ixLL

mtei/isLL mtis/eiLL mtix/iiLL mtxi/iiLLmtii/ixLL mtii/xiLL

mtix/ixLL mtsi/siLLmtxi/xiLL mtie/ieLL mtii/iiLLmtse/seLL mtes/esLLdjLL mtxe/xeLLmtsx/sxLL

ovix/ixLL ovsi/siLLovxi/xiLL ovie/ieLL ovii/iiLLovse/seLL oves/esLLovxx/xxLL ovxe/xeLLovsx/sxLL

ovex/xsLL ovxs/ex

LL ovix/xxLL ovxi/xxLLovxx/ixLL ovxx/ixLL ovii/xxLL ovxx/ii

LL ovsx/siLL ovex/isLLovsi/sxLL ovis/exLL

ovix/xiLL ovxi/ix

LL ovxe/ieLL ovxs/eiLLovie/xeLL ovei/xsLL ovei/isLL ovis/eiLL ovix/iiLL ovxi/iiLLovii/ixLL ovii/xiLL

cbis/exLL cbei/xsLLcvex/isLL cvxs/eiLLinLL ctLL cbie/xeLL cbbsi/sxLLcvxe/ieLL cvsx/siLLeqLL eq'LL

8 KURATA

www.josis.org

(a)

(b)

(c) Figure 4: (a) Topological point-region/region-point relations, (b) topological point-

line/line-point relations, and (c) topological line-region/region-line relations, which

are distinguished by the 9+-intersection

Figure 5: Distinction of ten classes of topological relations (�9: �’s closure)

adPRdjPR inPLabRPdjRP ctLP

stxbPL sbxbLPdjPL djLP inPL ctLPadbxPL abbxLP

ovixLR ovixRLadLR abRL

ovibLR ovibRLovbxLR ovbxRL

cbbbLR cvbbRLcbbb+LR cvbb+

RL cbiiLR cviiRLcbbiLR cvbiRL

inLR ctRL

cbbi+LR cvbi+RLcbibLR cvibRL cbib+LR cv

ib+RL

ovxiLR ovxiRL

ovbiLR ovbiRLovxbLR ovxbRL ovxxLR ovxxRL ovbbLR ovbbRL oviiLR oviiRL

mtxbLR mtxbRL

djLR djRL

mtxb+LR mtxb+

RLmtbxLR mtbxRL mtbx+LR mt

bx+RL mtxxLR mtxxRL mtbb+LR mt

bb+RLmtbbLR mt

bbRL

=

∩∩

∩∩−−−

−

BABA

BABA

¬

¬

φφ

φφ

¬¬

¬

φφ

φφ

¬¬

¬¬

φφ

φφ

¬¬

¬

φφ

φφ

¬

¬¬

φφ

φφ

=∩ oo BA

φ φ¬

All relations

equal(eq)

disjoint (dj)

meet(mt)

overlap(ov)

adhereTo(ad)

φ φ¬=∩ oo BA

φ φ¬

adheredBy(ab)

=∂∩∂ BA

φ φ¬=∂∩∂ BA

φ φ¬

inside(in)

coveredBy(cB)

=∩ oo BA

contains(ct)

covers(cv)



9

Submitted to JOSIS

3 Qualitative Spatial Calculi

Qualitative spatial calculi (and their lower dimensional counterparts, qualitative

temporal calculi) have been studied extensively in AI communities [14, 15]. Each QSC

targets a specific set of qualitative spatial relations between two or three objects. A

binary QSC normally supports conversion and (weak or strong) composition operations, in

addition to ordinary set-theoretic operations. Let r and s be the spatial relation between

A and B and that between B and C. The conversion of :, denoted :˘, gives the spatial relation between B and A (or possibly a set of all spatial relations that may hold

between B and A, if it is not uniquely determined). On the other hand, the weak

composition of : and <, denoted : ⋄ <, gives a set of all spatial relations that may hold between A and C. The weak composition : ⋄ < is an upper approximation of the strong composition : ∘ < as defined in Equation 3. Ternary QSC also have counterparts of these operations [5], called inverse, homing, shortcut, and composition. : ∘ < = ?, @ ∈ B | ∃� ∈ � , � ∈ : ∧ �, @ ∈ <F (3)

Each QSC normally targets a jointly exclusive and pairwise disjoint set of spatial

relations that may hold between two objects in an object domain D. These spatial

relations are called base relations and denoted ℬℬℬℬ as a set. Many calculi also consider the

general relations (or simply called relations), each referring to a subset of ℬℬℬℬ. By introducing such general relations, every state of knowledge about the possible spatial

relations between two objects is represented as a single general relation. For instance, if

we know that A is north or northeast of B, the general relation between A and B is the

set {north, northeast}. If nothing is known about the possible spatial relations between A

and B, the general relation between A and B is equivalent to ℬℬℬℬ. The set of all general relations for ℬℬℬℬ (i.e., ℬℬℬℬ’s power set) is denoted ℛℛℛℛℬℬℬℬ.

The conversion and composition operations on ℛℛℛℛℬℬℬℬ are defined based on those on ℬℬℬℬ as Equations 4-5. As a consequence, the conversion and compositions on ℛℛℛℛ is closed under ℛℛℛℛℬℬℬℬ. ∀�U ∈ VB �U˘ = W :˘X∈YZ (4)

∀�U , �[ ∈ VB �U; �[ = W : ⋄ <X∈YZ �∈Y] (5)

The set ℛℛℛℛℬℬℬℬ, together with its conversion and composition operations, gives rise to an algebra. Normally, a binary QSC forms a non-associative algebra (or even its stronger

form, a relation algebra or a semi-associative algebra). Actually, Ligozat and Renz [15]

defined a QSC as a tuple of a non-associative algebra and its weak representation. Such

10 KURATA

www.josis.org

an algebraic framework allows us to conduct spatial reasoning computationally in the

finite domain ℛℛℛℛℬℬℬℬ. Spatial/temporal relations that may not hold between objects are removed in a step-by-step manner by enforcing the constraint of algebraic closeness [15]

(i.e., for any trio of objects (X, Y, Z), the relation between X and Z must be a subset of

the composition of the relation between X and Y and that between Y and Z). By this

stepwise process we can also detect inconsistency in the given information, if it exists.

We should be wary that even if inconsistency is not detected, the derived solution is

not always consistent (i.e., the derived candidates for possible spatial relations may

have no geometric realizations). Nevertheless, QSC are still useful for disambiguating

spatial arrangements of objects with regard to a certain relation set. There are already

some effective tools that enable us to conduct computationally such spatial reasoning

on user-defined QSC (e.g., SparQ [16] and GQR [17]).

4 9+-Intersection Calculi

As a basis of our new calculi, we first develop conversion and composition operations

on the 9+-intersection-based topological relations. We then formulate a series of calculi

that target the 9+-intersection-based topological relations, following the framework of

QSC sketched in the previous section.

4.1 Conversion

By conversion, a topological relation in ��^�$ ̵� is mapped to a topological relation in ��$�^ ̵�. For instance, ctRP in ��#) ̵ℝ$ is mapped to inPR in ��)# ̵ℝ$, which implies that if a region A contains a point B, then B is at the inside of A. The conversion list CL ̵��^�$� shows the mappings from �+

D1D2-� to �+D2D1-� by conversion. For instance,

Table 2 shows the conversion list CL ̵��#) ̵ℝ$, which shows the mappings from region-point relations ��#) ̵ℝ$ to point-region relations ��)# ̵ℝ$.

Table 2. Conversion list CL ̵��#) ̵ℝ$ : djRP abRP ctRP :˘ djPR adPR inPR

The conversion lists of the 9+-intersection-based topological relations are derived

easily, because for any relation : in ��^�$ ̵� , : and :˘ are represented by the transposed 9+-intersection patterns (Figures 3-4).



11

Submitted to JOSIS

4.2 Composition

By composition, a pair of a topological relation in ��^�$ ̵� and that in ��$�b ̵� is mapped to a subset of ��^�b ̵�. The composition table CT ̵��^�$�b ̵� shows the mapping from ��^�$ ̵� × ��$�b ̵� to the power-set of ��^�b� by composition. For instance, Table 3 shows the composition table CT ̵��)## ̵� , where point-region relations in ��)# ̵ℝ$ and region-region relations in ��## ̵ℝ$ are composed and mapped to the subset of ��)# ̵ℝ$.

Table 3. Composition table CT ̵��)## ̵� (UPR={djPR, adPR, inPR}) eqRR djRR mtRR ovRR cvRR cBRR ctRR inRR

djPR djPR UPR UPR UPR djPR UPR djPR UPR

adPR adPR djPR {djPR, adPR} UPR {djPR, adPR} {adPR, inPR} djPR inPR

inPR inPR djPR djPR UPR UPR inPR UPR inPR

The composition table is derived as follows: first, for each pair of :ef ∈ ��^�$ ̵� and :fg ∈ ��$�b�, we prepare all tuples of :ef, :fg , :eg where :eg ∈ ��^�b ̵�. Then, we remove invalid tuples using certain geometric constraints. After this filtering

process, :eg in each remaining tuple becomes a candidate for an element of the composition of :ef and :fg. Each candidate is approved if people can draw an instance of the composition (although this process may be omitted at the risk of commission

errors in spatial reasoning). By repeating this process for every pair of :ef and :fg, we obtain the composition table CT ̵��^�$�b ̵� .

For the filtering process, we use four geometric constraints. Suppose that three

objects A, B, and C (A∈D1, B∈D2,C∈D3) are located in the space �. Let X, Y, and Z be arbitrary sets of topological primitives of A, B, and C (in arbitrary order). Then, we

have the following two constraints:

Constraint 1: If X contains Y and Y intersects with Z, then X and Z intersect.

(Proof) Let h ∈ i ∩ j≠ ∅ . Naturally, h ∈ i and h ∈ j . Since i ⊆ j , h ∈ � . Thus, h ∈ � ∩ j, which indicates that � ∩ j ≠ ∅. Constraint 2: If X contains Y and X does not intersect with Z, then Y and Z do not

intersect.

(Proof) Let h ∈ i. Since i ⊆ �, h ∈ �. Meanwhile, since � ∩ j ≠ ∅, h ∈ � implies h ∉ j. Therefore, i ∩ j = ∅.

12 KURATA

www.josis.org

Similarly, let x, y, and z be arbitrary topological primitives of A, B, and C (in arbitrary

order). Then, we have the following two constraints:

Constraint 3: If x contains both y and z, x and y are of the same dimension, and z

partially overlaps with y, then z also intersects with at least one of y’s adjacent lower-

dimensional primitives (Figure 6a).

(Proof) Since x contains both y and z, x can be regarded as a sub-space of � that embeds y and z. If o ⊂ q , y is bounded on the space x by y’s adjacent lower-dimensional primitive(s), since x and y are of the same dimension. Thus, if z overlaps y, z must

intersect with y’s bound formed by y’s adjacent lower-dimensional primitive(s). On the

other hand, y = q is not possible, since z cannot overlap with y. Constraint 4: If x contains y, y contains z, and y’s adjacent lower-dimensional

primitives and z’s adjacent lower-dimensional primitives do not intersect, then x’s

adjacent lower-dimensional primitives and z’s adjacent lower-dimensional primitives

also do not intersect (Figure 6b)

(Proof) Let �h be the set of adjacent lower-dimensional primitives of a primitive p. In general, s ⊆ o implies �s ⊂ to ∪ �o v. In this case, �o ∩ �s = ∅. Thus, �s ⊂ y. Since o ⊆ q, �s ⊂ x. Since q ∩ �q = ∅, �q ∩ �s = ∅.

(a) (b)

Figure 6: Example configurations for (a) Constraint 3 and (b) Constraint 4

Constraints 1-2 generalize Egenhofer’s [18] constraints used for deriving the

composition table of the 9-intersection-based topological region-region relations in ̵ℝ� (i.e., CT ̵�### ̵ℝ$ ), while Constraints 3-4 generalize Kurata’s [6] additional constraints used for deriving the composition table of the 9-intersection-based topological line-line

relations in ̵ℝ� (i.e., CT ̵�""" ̵ℝ$ ). By our generalization Constraints 1-4 apply to any composition of the 9+-intersection-based topological relations, regardless of the types

of the three objects. Such generalizations become possible thanks to the concept of

topological primitives introduced by the 9+-intersection.

4.3 Homogeneous 9+-Intersection Calculi

Homogeneous 9+-intersection calculus for an object domain D and a space �, denoted Homo9�� ̵�, is a qualitative spatial calculus that targets the set of topological relations �+DD-� and supports the conversion and composition operations on �� ̵� as defined

x

y zx

y

z



13

Submitted to JOSIS

in CL ̵�� ̵� and CT ̵�� ̵� , respectively These elements satisfy the framework of ordinary QSC, as �� is a jointly exclusive and pairwise disjoint set of spatial relations with an identity relation eqDD, while the conversion and composition

operations on �� ̵�-based general relations V�{�� ̵� are closed under these general relations V�{�� ̵� (i.e., ∀�U , �[ ∈ V�{�� ̵� �U˘ ∈ V�{�� ̵� , �U ; �[ ∈ V�{�� ̵� ). Consequently, constraint-based topological reasoning on single-type objects can be conducted by

algebraic computation (See Section 5.1 for demonstration).

4.4 Heterogeneous 9+-Intersection Calculi

Heterogeneous 9+-intersection calculus for a space �, denoted Het9�� , is a qualitative spatial calculus that targets the 9+-intersection-based topological relations between

arbitrary simple objects embedded in �. Thus, Het9�ℝ~ (d ≤3) is built on: • d+1 object domains ?��, … , ��F where �� = ), �' = ", �� = #, and �( = �; • (d+1)2 sets of topological relations ��Z�]ℝ~�; • (d+1)2 conversion lists �CL ̵��Z�]ℝ~�; and • (d+1)3 composition tables �CT ̵��Z�]��ℝ~�.

where �, �, � ∈ ?0, … , �F. These elements are adapted to the framework of ordinary QSC, following the adaptation technique proposed in [6]. First, we introduce a generalized

object domain D* and a set of generalized base relations ℬℬℬℬ* as follows:

• �∗ = � �UU∈?�,…,�F

• B∗ = �� Z�]ℝ~U∈?�,…,�F � ∪ ?��∗F ℬℬℬℬ* refers to all topological relations between two arbitrary simple objects in D*, while it

also contains a global identity relation eq*. The presence of an identity relation is a

requirement of the algebraic framework (non-associative algebra) of QSC. At the same

time, we decided to keep domain-level identity elements (i.e., eqPP, eqLL, …) without

integrating them into eq*, in order not to reduce the reasoning power. For instance, the

composition of a point-point relation and a line-line relation is impossible ( ∴��)); ��"" = ∅), but this fact is no longer used in spatial reasoning if ��)) is replaced by a global identity relation (since ��∗; ��"" ≠ ∅). Thus, we introduce ��∗ as an abstract identity relation, which has no geometric interpretation (i.e., ∀�U ∈ �∗ �U, �U ∉ ��∗) and is used only for computation.

14 KURATA

www.josis.org

Next, the conversion list and composition table for ℬℬℬℬ* are prepared. The

conversion list CL ̵��∗�∗ ̵ℝ~ is derived simply by concatenating all relevant conversion lists �CL ̵��Z�] ̵ℝ~� and adding an item telling ��∗˘ = ��∗ (Table 4). Similarly, the composition table CT ̵��∗�∗�∗ ̵ℝ~ is derived simply by adjoining all relevant

composition tables �CT ̵��Z�]�� ̵ℝ~� and adding a row that indicates ��∗ ⋄ : = : and a column that indicates : ⋄ ��∗ = : for any relation r in ��∗�∗ ̵ℝ~ (Table 5).

Table 4: Outline of the integrated conversion list CL ̵��∗�∗ ̵ℝ$ : eq* eqPP djPP djPL … inPL djPR … inPR … eqRR … inRR :˘ eq* eqPP djPP djLP … ctLP djRP … ctRP … eqRR … ctRR CL ̵��)) ̵ℝ$ CL ̵��)" ̵ℝ$ CL ̵��)# ̵ℝ$ CL ̵��## ̵ℝ$

Table 5: Outline of the integrated composition table CT ̵��∗�∗�∗ℝ$

eq* eqPP djPP djPL ⋯ inPL djPR … inPR ⋯ eqRR ⋯ inRR eq* eq* eqPP djPP djPL ⋯ inPL djPR … inPR ⋯ eqRR ⋯ inRR eqPP eqPP CT ̵��))) ̵ℝ$ CT ̵��))" ̵ℝ$ CT ̵��))# ̵ℝ$ ⋯ ∅ ⋯ ∅ djPP djPP ⋯ ∅ ⋯ ∅ djPL djPL ∅ ∅ ∅ ⋯ ∅ ∅ ⋯ ∅ ⋯ ∅ ⋯ ∅ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋱ ⋮ ⋯ ⋮ ⋱ ⋮ inPL inPL ∅ ∅ ∅ … ∅ ∅ … ∅ ⋯ ∅ … ∅ djPR djPR ∅ ∅ ∅ ⋯ ∅ ∅ ⋯ ∅ ⋯ CT ̵��)##ℝ$ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋱ ⋮ ⋯ inPR inPR ∅ ∅ ∅ … ∅ ∅ … ∅ ⋯ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮

eqRR eqRR ∅ ∅ ∅ ⋯ ∅ ∅ ⋯ ∅ ⋯ CT ̵��### ̵ℝ$ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋱ ⋮ ⋯ inRR inRR ∅ ∅ ∅ … ∅ ∅ … ∅ ⋯

Now we have a set of base relations ℬℬℬℬ* and conversion and composition

operations on ℬℬℬℬ*. These elements satisfy the framework of ordinary qualitative spatial

calculi, because ℬℬℬℬ* is a jointly exclusive and pairwise disjoint set of spatial relations

with an identity relation eq*, and the conversion and composition operations on ℬℬℬℬ*-

based general relations VB∗ are closed under VB∗ . Consequently, constraint-based spatial reasoning can be conducted by algebraic computation, regardless of objects’

types (See Section 5.2 for demonstration).



15

Submitted to JOSIS

5 Demonstration

This section demonstrates the applications of the developed 9+-intersection calculi

(Homo9�" ̵ℝ$ and Het9�ℝ$) for spatial reasoning problems. Our examples follow those used in [6] for demonstrating the 9-intersection calculi [6], such that we can compare

the reasoning power of these calculi.

5.1 Reasoning with a Homogeneous 9+-Intersection Calculus

In the first example, we consider the highway network in the Boston metropolitan area,

which is simplified as illustrated in Figure 7a. Note that the direction of each highway

is explicitly given, since the 9+-intersection presumes the direction of lines. Imagine

that we drive two of the four highways and observe how each highway is connected to

other highways. Base on this observation, what can we tell about the connections

between unvisited highways? For instance, in Figure 7b, which illustrates the

information obtained by a drive on I-93S and I-495S, how are I-90E and I-95S

connected?

(a) (b)

Figure 7: (a) A highway network in Boston metropolitan area and (b) an illustration of

the information obtained by a drive on I-93S and I-495S.

To solve this problem, we used Homo9�" ̵ℝ$ , since the connections between highways are captured as topological line-line relations in ℝ�. Table 6 lists the actual topological relations between the four highways. For computation, we first registered

the data of Homo9�" ̵ℝ$ (the names of relations in ��"" ̵ℝ$ (Figure 3c), the conversion list CL ̵��"" ̵ℝ$ , and the composition table CT ̵��""" ̵ℝ$) to SparQ [16]. Then, for each drive scenario, we inputted all data in Table 6, except the relation between unvisited

highways, and computed all consistent scenarios in SparQ.

I-90EI-90E

I-95S

I-90E

I-95S

I-95S

I-95S

16 KURATA

www.josis.org

Table 6: Topological relations between pairs of highways in the network of Figure 7a.

I-90E I-93S I-95S I-495S

I-90E – mtxi/xxLL ovxx/xxLL mtix/xxLL

I-93S mtxx/xiLL – ovxi/xxLL mtix/xxLL

I-95S ovxx/xxLL ovxx/xiLL – mtse/seLL

I-495S mtxx/ixLL mtxx/ixLL mtse/seLL –

For each drive scenario, the derived consistent scenarios show the candidates for

the relations between unvisited highways (Table 7). These candidates successfully

contained the actual relation. For instance, when we assumed that I-90E and I-95S were

unvisited, the derived candidates for their relations were djLL, ovxx/xxLL, mtxi/xxLL, and

ovxi/xxLL, which successfully contained ovxx/xxLL (i.e., I-90E and I-95S are crossing). From

Figure 7b people may think that the other three candidates are impossible. Actually, it

is possible to remove mtxi/xxLL and ovxi/xxLL from the candidates if we further use the

information that I-93S is crossing I-95S and merged by I-90 at different locations—

practically, this is achieved by segmenting I-93S at every junction and conduct the

reasoning at a finer level. On the other hand, to remove djLL, we need such knowledge

as “I-90E diverges from the left side of I-495E” and“I-90E merges into the right side of I-

93S”, which is not captured by the topological relations.

Table 7: Candidates for topological relations between pairs of unvisited highways and

their number

Unvisited highways

Candidates for topological relations between unvisited highways derived by Homo9�" ̵ℝ$

# of candidates derived by Homo 9�" ̵ℝ$ Homo 9" ̵ℝ$

I-90E & I-93S djLL cvsx/siLL cbsi/sxLL ovxx/xxLL ovxi/xiLL ovsx/sxLL

mtxi/xiLL mtsi/siLL ovsi/siLL mtxi/xxLL ovxi/xxLL mtxx/xiLL ovxx/xiLL mtsi/sxLL ovsi/sxLL mtsx/siLL ovsx/siLL

18 18

I-90E & I-95S djLL ovxx/xxLL mtxi/xxLL ovxi/xxLL 4 4

I-90E & I-495S djLL ovxx/xxLL mtix/xxLL mtix/xxLL 4 4

I-93S & I-95S djLL ovxx/xxLL mtxi/xxLL ovxi/xxLL 4 4

I-93S & I-495S djLL ovxx/xxLL mtix/xxLL ovix/xxLL 4 4

I-95S & I-495S all but eqLL eq−LL inLL ctLL cbie/xeLL cvxe/ieLL cbsi/sxLL

cvsx/siLL cbis/exLL cvex/isLL cbei/xsLL cbxs/eiLL 68 28



17

Submitted to JOSIS

Similarly, in [6], the candidates for the relation between unvisited highways were

derived based on a 9-intersection calculus Homo9" ̵ℝ$. The right two columns in Table 7 show the numbers of candidates derived under Homo9�" ̵ℝ$ and Homo9" ̵ℝ$ ,

respectively. In five of six cases, both calculi derived the same numbers of candidates.

However, the results of Homo9�" ̵ℝ$ capture the directional characteristics of highway connections (e.g., merging or diverging) and, therefore, they are more informative than

those of Homo9" ̵ℝ$ . When we assumed that I-95S and I-495S were unvisited, Homo9�" ̵ℝ$ derived many more candidates than Homo9" ̵ℝ$ , but the results are essentially equivalent, since the result of Homo9�" ̵ℝ$ implies that we cannot tell anything about the directional characteristics of the relation between I-95S and I-495S

from the given information.

5.2 Reasoning with a Heterogeneous 9+-Intersection Calculus

The second example uses the map in Figure 8a, in which highway junctions (points)

and two districts (regions) are added to the previous map. At this time we imagine the

drive on three of the four highways and infer the connection between the unvisited

highway and the two districts. For instance, when we assume that we drive I-90E, I-95S,

and I-495S and obtain the knowledge illustrated in Figure 8b, can we guarantee that I-

93S goes through the two districts?

(a) (b)

Figure 8: (a) Spatial arrangement of highways, highway junctions, and two districts in

Boston metropolitan area and (b) an illustration of the information obtained by the

drive on I-90S, I-95S, and I-495S.

To solve this problem, we used Het9�ℝ$, since we have to deal with highway-highway, highway-district, and highway-junction connections, which are captured as

topological line-line, line-region, and line-point relations in ℝ�, respectively. We also

used the information about district-district relations (Boston urban district covers its

municipal district), district-junction relations (observable from highways), and

I-90

P1P2

P4

P5

P6

P7

P8P3

Boston

Municipal

District

Boston

Urban

DistrictI-90E

P1P2

P4

P5

P6

P7

P8P3

I-93

18 KURATA

www.josis.org

junction-junction relation (fixed to be djPP). For computation, we first registered the

data of Het9�ℝ$ to SparQ (the names of relations in ��∗�∗ ̵ℝ$ (Figures 3-4), the conversion list CL ̵��∗�∗ ̵ℝ$ (Table 4), and the composition table CT ̵��∗�∗�∗ ̵ℝ$ (Table 5)). Then, for each drive scenario, we inputted all topological relations in Figure 8a,

except the relations between the unvisited highway and the two districts, and

computed all consistent scenarios in SparQ.

For each drive scenario, the derived consistent scenarios show the candidates for

the pair of topological relations between the unvisited highway and the two districts

(Table 8). These candidates successfully contained the actual relation pair. For instance,

when we assumed that I-93S was unvisited (Figure 8b), we obtained only the correct

answer in which I-93S goes through the two districts (i.e., the relation between I-93S

and each district is ovxxLR).

Table 8: Candidates for the topological relations between an unvisited highway and

two districts and their numbers

Unvisited highway x

Candidates for [(x, municipal), (x, urban)] derived by Het9�ℝ$

Number of candi-dates derived by Het9�ℝ$ Het9ℝ$

I-90E [ovxiLR, ovxiLR] 1 1

I-93S [ovxxLR, ovxxLR] 1 6

I-95S [djLR, ovxxLR] [mtxxLR, ovxxLR] [ovxxLR, ovxxLR] 3 3

I-495S [djLR, djLR] [djLR, mtxxLR] [djLR, ovxxLR] [mtxxLR, mtxxLR] [mtxxLR, ovxxLR] [ovxxLR,

ovxxLR] 6 6

Table 8 also shows the numbers of the derived candidates, in comparison with

those in [6] based on a 9-intersection calculus Het9ℝ$. When we assumed that I-93S was

unvisited, Het9�ℝ$ gave a unique result, while Het9ℝ$ did not. As Kurata [6] pointed out, the 9-intersection omits the information that each line has two endpoints and,

accordingly, Het9ℝ$ derives invalid candidates which presume that the line has more than two endpoints (e.g., I-93S goes through the urban district and goes into the

municipal district at the same time). This problem is successfully avoided in Het9�ℝ$, thanks to the clear distinction of the line’s two endpoints as a start point and an end

point. In the other three drive scenarios, the numbers of the candidates derived by Het9�ℝ$ and Het9ℝ$ are the same, even though the results of Het9�ℝ$ are more



19

Submitted to JOSIS

informative, since they capture directional characteristics of highway-district relations

(e.g., entering or leaving).

6 Conclusions

In geographic information systems, we often deal with heterogeneous objects, whose

arrangement is of concern. Naturally, GIScience communities have developed a

number of spatial relation models that target heterogeneous objects, such as

topological line-region relations [7, 8], path relations [21], and generalized cardinal

direction relations [22]. How to equip such cross-domain relation models with the

capability of spatial reasoning has been left as a research challenge. In this paper, one

of such cross-domain models, the 9+-intersection, was built into the framework of

qualitative spatial calculi. The resulting 9+-intersection calculi enable us to

disambiguate spatial arrangements of heterogeneous objects from a topological

viewpoint. The 9+-intersection calculi are especially useful when the reasoning problem

concerns the connections between paths and landmarks, as paths should be treated as

lines with qualitatively different two endpoints. The 9+-intersection calculi realize finer

spatial reasoning than the 9-intersection calculi [6] when the target of the reasoning

problem includes lines, or potentially any object whose interior, boundary, or exterior

consist of multiple disjoint subparts.

We can consider that a heterogeneous calculus (say, Het9�ℝ$) integrates several homogeneous calculi (Homo9�) ̵ℝ$ , Homo9�" ̵ℝ$, Homo9�# ̵ℝ$), since the operations in each homogeneous calculus are succeeded to the heterogeneous calculus. However, we

should not confuse this integration with the calculi integration discussed in AI

communities (e.g., [19, 20]), where multiple calculi targeting different sorts of relations

(e.g., topological relations and size relations [19]) for the same pairs of objects are

combined. In our case, heterogeneous calculi combine the calculi that target the same

sort of spatial relations (e.g., 9+-intersection-based topological relations) for various

pairs of objects.

The recent spread of spatially-distributed sensor networks and collaborative

mobile robots increases the need for integrating incomplete and inconsistent

information gathered by multiple sensors/agents. Qualitative spatial calculi are the

key technology for such information integration. Previous spatial calculi, however,

limit the target to single-type objects. Development of spatial calculi that can deal with

heterogeneous objects, including our 9+-intersection calculi, will expand the

applicability of spatial reasoning to practical problems.

20 KURATA

www.josis.org

Acknowledgment

This work is supported by DFG (Deutsche Forschungsgemeinschaft) through SFB/TR

8 Spatial Cognition. The authors also appreciate the attendees of IJCAI-09 Workshop

on Spatial and Temporal Reasoning for useful comments and discussions on our

earlier work.

References

[1] ALLEN, J. Maintaining Knowledge about Temporal Intervals. Communications

of the ACM 26, 11 (1983), 832-843.

[2] RANDELL, D., CUI, Z., AND COHN, A. A Spatial Logic Based on Regions and

Connection. In 3rd International Conference on Knowledge Representation and

Reasoning (San Mateo, CA, USA, 1992), Morgan Kaufmann, pp. 165-176.

[3] FRANK, A. Qualitative Spatial Reasoning about Cardinal Directions. In

AutoCarto 10 (Baltimore, MD, USA, 1991), ACSM/ASPRS, pp. 148-167.

[4] FREKSA, C. Using Orientation Information for Qualitative Spatial Reasoning.

In International Conference GIS - From Space to Territory: Theories and

Methods of Spatio-Temporal Reasoning in Geographic Space: Lecture Notes in

Computer Science, vol. 639 (New York, NY, USA, 1992), Springer, pp. 162-178.

[5] ZIMMERMANN, K., AND FREKSA, C. Qualitative Spatial Reasoning Using

Orientation, Distance, and Path Knowledge. Applied Intelligence 6, 1 (1996), 49-

58.

[6] KURATA, Y. 9-Intersection Calculi for Spatial Reasoning on the Topological

Relations between Multi-Domain Objects. In IJCAI Workshop on Spatial and

Temporal Reasoning (Pasadena, CA, USA, 2009), AAAI Press.

[7] EGENHOFER, M., AND HERRING, J.: Categorizing Binary Topological

Relationships between Regions, Lines and Points in Geographic Databases. In

NCGIA Technical Reports 91-7, Egenhofer, M., Herring, J., Smith, T., andPark,

K., Eds. National Center for Geographic Information and Analysis, Santa

Barbara, CA, USA, 1991.

[8] KURATA, Y., AND EGENHOFER, M. The 9+-Intersection for Topological

Relations between a Directed Line Segment and a Region. In 1st Workshop on

Behaviour Monitoring and Interpretation: TZI-Bericht, vol. 42 (Osnabrück,

Germany, 2007), Technogie-Zentrum Informatik, Universität Bremen, pp. 62-76.



21

Submitted to JOSIS

[9] KURATA, Y. The 9+-Intersection: A Universal Framework for Modeling

Topological Relations. In GIScience 2008: Lecture Notes in Computer Science,

vol. 5266 (Park City, Utah, USA, 2008), Springer, , pp. 181-198.

[10] KURATA, Y. Semi-Automated Derivation of Conceptual Neighborhood Graphs

of Topological Relations. In COSIT 2009: Lecture Notes in Computer Science

(Brest, France, to appear), Springer.

[11] SCHNEIDER, M., AND BEHR, T. Topological Relationships between Complex

Spatial Objects. ACM Transactions on Database Systems 31, 1 (2006), 39-81.

[12] ALEXANDROFF, P. Elementary Concepts of Topology. Dover Publications,

Mineola, NY, USA, 1961.

[13] EGENHOFER, M., AND AL-TAHA, K. Reasoning about Gradual Changes of

Topological Relationships. In Theories and Methods of Spatio-Temporal

Reasoning in Geographic Space: Lecture Notes in Computer Science, vol. 639

(Pisa, Italy, 1992), Springer, pp. 196-219.

[14] COHN, A., AND HAZARIKA, S. Qualitative Spatial Representation and

Reasoning: An Overview. Fundamenta Informaticae 46, 1-2 (2001), 1-29.

[15] LIGOZAT, G., AND RENZ, J. What Is a Qualitative Calculus? A General

Framework. In PRICAI 2004: Lecture Notes in Artificial Intelligence, vol. 3157

(Auckland, New Zealand, 2004), Springer, pp. 53-64.

[16] WALLGRÜN, J.O., FROMMBERGER, L., WOLTER, D., DYLLA, F., AND

FREKSA, C. Qualitative Spatial Representation and Reasoning in the SparQ-

Toolbox. In Spatial Cognition V: Lecture Notes in Artificial Intelligence, vol.

4387 (Bremen, Germany, 2006), Springer, pp. 39-58.

[17] GANTNER, Z., WESTPHAL, M., AND WÖLFL, S. GQR - a Fast Reasoner for

Binary Qualitative Constraint Calculi. In AAAI Workshop on Spatial and

Temporal Reasoning (Chicago, USA, 2008), AAAI Press.

[18] EGENHOFER, M. Deriving the Composition of Binary Topological Relations.

Journal of Visual Languages and Computing 5, 2 (1994), 133-149.

[19] GEREVINI, A., AND RENZ, J. Combining Topological and Size Information for

Spatial Reasoning. Artificial Intelligence 137, 1-2 (2002), 1-42.

[20] WÖLFL, S., AND WESTPHAL, M. On Combinations of Binary Qualitative

Constraint Calculi. In International Joint Conference on Artificial Intelligence

(Pasadena, CA, USA, 2009), AAAI Press, pp. CD-ROM.

22 KURATA

www.josis.org

[21] KRAY, C., AND BLOCHER, A. Modeling the Basic Meanings of Path Relations.

In 16th International Joint Conference on Artificial Intelligence (San Francisco,

CA, USA, 1999), Morgan Kaufmann, pp. 384-389.

[22] GOYAL, R., AND EGENHOFER, M. Consistent Queries over Cardinal

Directions across Different Levels of Detail. In 11th International Workshop on

Database and Expert Systems Applications (Greenwich, UK, 2000), pp. 876-880.

9 -intersection calculi for topological reasoning on ...mschneid/files/arif/yohei...

Documents