Fuzzy semantics for direction relations between composite regions

Information Sciences 160 (2004) 73–90

www.elsevier.com/locate/ins

Fuzzy semantics for directionrelations between composite regions

Christophe Claramunt a,*, Marius Th�eriault b

a Naval Academy Research Institute, Lanv�eoc-Poulmic, BP 600, 29240 Brest Naval, Franceb Laval University, Planning and Development Research Centre, Que., Canada G1K 7P4

Received 18 February 2003; received in revised form 10 June 2003; accepted 23 July 2003

Abstract

This paper proposes a model that combines qualitative spatial reasoning with fuzzy

semantic typing to derive direction relations between two composite regions in a spatial

configuration. It extends Goyal and Egenhofer’s direction relation model, initially

proposed for simple regions, towards composite regions. A fuzzy semantics typing

qualifies the overall direction relations of two composite regions. We introduce flexible

fuzzy measures that allow for both a qualitative and metric study of direction relation

similarities.

� 2003 Elsevier Inc. All rights reserved.

1. Introduction

Within the GIS community, qualitative spatial reasoning has long been

recognised as a valid complement to Euclidean geometry [22]. Qualitative

spatial reasoning supports inferences to reason with spatial entities in the ab-

sence of complete spatial knowledge by not differentiating between quantities

unless there is sufficient evidence to do so [5]. As nicely stated by Egenhofer

and Mark [7]: ‘‘topology matters, metric refines’’. In geographical space, rea-soning on spatial entities is supported by representations that involve direction,

topological, ordinal, distance, size and shape relationships [29].

* Corresponding author. Tel.: +33-2-98-23-42-06; fax: +33-2-98-23-38-57.

E-mail address: [email protected] (C. Claramunt).

0020-0255/$ - see front matter � 2003 Elsevier Inc. All rights reserved.

doi:10.1016/j.ins.2003.07.013

mail to: [email protected]

74 C. Claramunt, M. Th�eriault / Information Sciences 160 (2004) 73–90

Those representations support spatial inferences and the development of

spatial query languages, allowing for the analysis of similarities between dif-

ferent spatial configurations [1,12,14,25]. So far, these approaches have been

mainly oriented to the evaluation and retrieval of a small number of regions,while enforcing topological, directional and/or distance constraints. To the best

of our knowledge, an important issue that has not been addressed by quali-

tative spatial reasoning is the study of direction relationships between several

regions and analysis of similarities/differences between several spatial config-

urations.

Applying numerical analysis to understand the properties of a given spatial

configuration is also the scope of geographical science [4,6]. Although analysing

spatial patterns has been long investigated by geographers, qualitative reason-ing can provide new formal insights to the understanding of relationships in

geographical spaces. Despite some important work on the integration of

directional data in spatial analysis (e.g. [19,21,23,32]), few statistical analysis

integrate direction relations at the disaggregated level. Many applications need

qualitative approaches compatible with the essence of phenomena and pro-

cesses as they are observed in nature. The study and analysis of direction

relations is one of the areas to explore especially for applications in which part

of the phenomena represented is influenced by directionality. Amongst manyexamples of applications let us mention the analysis of direction relations be-

tween the distribution of ecological or biological species (e.g. studying the

spatial diffusion of Eastern spruce budworm, which is highly detrimental to

spruce and balsam fir forests in United States and Canada, from year to year

[18]) and the land-use in a given region of study, respective performance of ships

during a race or concurrent navigation monitoring for maritime authorities.

This paper introduces a fuzzy-based model that characterises direction

relations between composite regions in a two-dimensional space. It also anal-yses degrees of similarity between different direction relation configurations.

This approach is based on a direction relation model, introduced by Goyal and

Egenhofer [14,15], that characterises direction relations between two simple

regions, where a simple region is defined as a closed connected point set with no

holes in a two-dimensional space. We combine this model of direction relations

with a fuzzy semantics, and generalise them to composite regions. This gives a

form of numerical analysis and integrates fuzziness in the qualification of

direction relations. The modelling techniques employed are based on fuzzyvariables [39,40] and fuzzy semantic typing [35]. The approach involves:

• the choice of a frame of reference to reason with directional relations, i.e.,

Goyal and Egenhofer’s model;

• a scalar metric and a fuzzy-based semantics to model the overall direction

relations between two composite regions;

• fuzzy manipulation of resulting directional relationships and

C. Claramunt, M. Th�eriault / Information Sciences 160 (2004) 73–90 75

• an a-level set metric to estimate direction relation similarities/dissimilarities

between several spatial configurations.

The remainder of the paper is organised as follows. Section 2 gives somebasics in modelling direction relations and briefly introduces Goyal and Egen-

hofer’s direction model. Section 3 develops the fuzzy semantics approach used

to model directional relationships between two composite regions, and iden-

tifies basic properties and manipulation operators. Section 4 introduces a

metric for assessing direction similarities between different spatial configura-

tions. Section 5 illustrates the potential of our model by a case study. Finally

Section 6 draws some conclusions.

2. Direction relations

Direction––also called orientation––relationships are important and com-

mon-sense linguistic and qualitative properties used in everyday situations and

qualitative spatial reasoning [9]. Direction relations are defined according to aninternal or external frame of reference, i.e., whether the orientation system is

defined locally or globally [10]. External frames of reference are often based on

cardinal directions with respect to a local meridian in large-scale spaces.

Compass directions have been also used to partition space around a reference

simple region, and then to analyse the intersections between a target simple

region and the resulting tiles around a reference simple region using either a

cone [27] or projection-based approach [2,9,11,14,26,34,36]. Internal frames of

reference use relative orientation in which positioning of a simple region ismade with respect to an oriented line or an ordered set of points forming a

vector [11,13,16,30,33] or to some intrinsic properties of the reference simple

region, e.g. front vs. back [24]. The direction relations derived from those

frames of reference are binary per nature even if a third simple region is used in

some case to define those relations.

The objective of our research is a qualitative and computational exploration

of a form of generalised direction relation between two composite regions.

Without loss of generality we consider an external frame of reference, althougha local frame of reference can be also used for the development of our mod-

elling approach. The approach is based on the direction relation model

introduced by Goyal and Egenhofer [14,15]. Egenhofer and Goyal’s model is

based on a partition of space around a compass-oriented minimum bounding

rectangle (MBR) of a given simple region defined as a closed connected point

set with no holes in a two-dimensional space (a simple region is hereafter de-

noted a S-region). Direction tiles are denoted by cardinal directions (Na, NWa,

Wa, SWa, Sa, SEa, Ea, NEa) and the minimum bounding rectangle (Oa) withreference to a S-region a. Direction relations are derived from the intersection


of these tiles with a target S-region b. Those direction relations are represented

by a 3 · 3 matrix Dirða=bÞ as follows

Dirða=bÞ ¼NWa \ b Na \ b NEa \ bWa \ b Oa \ b Ea \ bSWa \ b Sa \ b SEa \ b

24

35 ð1Þ

where a is the reference S-region and b the target S-region.Specialisation or simplification of this model is obtained by either extending

or reducing the number of direction relations, respectively. Other variations

include the cases where the reference S-region is either a point or a line, and the

one where a cone-based partition is used to structure space [14]. For the design

of our model and further manipulation purposes we slightly modify the above

notation (1) using Boolean values. For any tile of a, denoted tilea, such as

tilea 2 fNWa;Na;NEa;Wa;Oa;Ea; SWa; Sa; SEag a tile relation tileða=bÞ is given asfollows

tileða=bÞ ¼ 1 if tilea \ b 6¼ ;0 otherwise

�ð2Þ

This gives a slightly modified 3 · 3 matrix Dirbða=bÞ where

Dirbða=bÞ ¼NWaða=bÞ Naða=bÞ NEaða=bÞWaða=bÞ Oaða=bÞ Eaða=bÞSWaða=bÞ Saða=bÞ SEaða=bÞ

24

35 ð3Þ

where for example NW ða=bÞ is equal to one if the target S-region b intersects

the tile North-West of the reference S-region a, zero otherwise.Let us introduce an example illustrated by Fig. 1. It relates a reference

S-region a and a target S-region b.

O a

b

N aNW a

W a

SW S a SE a

E a

NE a

a

Fig. 1. Direction relations between two S-regions.


The matrix representation of direction relations gives

Dirða=bÞ ¼:; :; ;:; ; ;; ; ;

24

35 and Dirbða=bÞ ¼

1 1 01 0 0

0 0 0

24

35

Dirðb=aÞ ¼; ; ;; :; :;; :; :;

24

35 and Dirbðb=aÞ ¼

0 0 0

0 1 1

0 1 1

24

35

Among interesting features of this model of direction relations, one should

note that:

• because the partition of space depends on the shape and extent of the refer-

ence S-region, a matrix of direction relations is partially antisymmetric (a

matrix of direction relations is symmetric, i.e., Dirða=bÞ ¼ Dirðb=aÞ, fortwo S-regions whose MBRs are equal);

• as several tiles are unbounded in one ðEa; Sa;Na;WaÞ, or two

ðNEa; SEa; SWa;NWaÞ directions, the likelihood to intersect them is a function

of the distance between the target S-region and the MBR of the reference

S-region. This is especially the case when the reference S-region is small in

one of its spatial dimensions; and,

• the central tile Oa is bounded in all directions so the case where a target

S-region intersects the central tile is the less likely for a random distributionof the target S-region in the two-dimensional space.

3. Direction relations among composite regions

3.1. Principles

Direction relations give a qualitative support to evaluate the relative posi-

tion of two S-regions. A composite region A, hereafter denoted C-region, isdefined as a closed subset of a two-dimensional space made of several S-regions(a1; a2; . . . ; an) such as [3]:

• each ai is a S-region• ai� \ aj� ¼ ;, 8i 6¼ j, where ai�, aj� denotes the interior of ai, aj, respectively• oai \ oaj ¼ ; or equal to a finite set of points, 8i 6¼ j, where oai, oaj denotes

the boundary of ai, aj, respectively

In order to study the direction relation between two C-regions, we go further

and introduce an approach whose principles are inspired by fuzzy semantic


typing. 1 Semantic typing techniques were first used in natural language pro-

cessing (NLP) for characterising word affects in text analysis (cf. [8] for a

survey). They have been recently integrated with fuzzy logic to analyse word

contents in textual documents [35]. The basic idea behind semantic typingapplied to NLP is to evaluate to which degree a word is used in a document,

while keeping ambiguity and imprecision. Fuzzy inferences support compu-

tational analysis of word categories, evaluation of intensities and similarities in

document. Fuzzy semantic typing is particularly adapted to cases where lin-

guistic terms are the object of study, and where approximation and ambiguity

should be kept while providing computational resources.

Fuzzy semantic properties are somehow related to direction relations as

whose are also linguistic properties per nature, marked by imprecision andambiguity when applied to composite regions. Direction relations can be also

grouped per category favouring thus the analysis of patterns at different levels

of abstraction (e.g. directions relations can be categorised in two groups

NW ;N ;NE and SW ; S; SE). The objective of this model is the exploration and

application of fuzzy-based typing for modelling and computing direction

relations between C-regions, that is, to evaluate to which degree a direction

relation is valid between two C-regions. In order to develop the modelling

approach we first define some basic notations.

Definition 1. Assume TILE denote the crisp set of cardinal directions

fN ;NW ;W ; SW ;O; S; SE;E;NEg and tile 2 TILE. Let us consider two C-regionsA ¼ fa1; a2; . . . ; amg and B ¼ fb1; b2; . . . ; bng. The fuzzy matrix of direction

relations of B relative to A, denoted DirbðA=BÞ, is given by

1 N

regions

[26].

DirbðA=BÞ ¼NW ðA=BÞ NðA=BÞ NEðA=BÞW ðA=BÞ OðA=BÞ EðA=BÞSW ðA=BÞ SðA=BÞ SEðA=BÞ

24

35 ð4Þ

where tileðA=BÞ ¼Pm

i¼1

Pnj¼1

tileðai=bjÞm�n .

For example, a membership value NW ðA=BÞ is derived from the normalised

sum of the number of direction relations NW ðai=bjÞ where ai 2 A and bj 2 B.This value is drawn from the unit interval [0,1], thus giving a fuzzy membership

degree that evaluates to which degree B is at the North-West of A. A mem-

bership value of 0 for a given tileðA=BÞ means that for any S-region ai of A,tileðai=bjÞ is null for any S-region bj of B. Conversely, a membership value of 1

means than all tileðai=bjÞ are equal to one for all ai 2 A, bj 2 B. The higher the

ote that, for this work, we consider crisp regions and not spatial relationships among fuzzy

as suggested by Prewitt [28] and Rosenfeld [31], or fuzzy relations between two crisp regions


membership value tileðA=BÞ the higher the relationship of B relative to A for

that tile. The sum of the tileðA=BÞ values for a given matrix is higher than 1

when at least one of the target S-regions intersects more than one tile of one of

the reference S-regions.The fuzzy matrix of direction relations is a fuzzy subset of the crisp set

TILE. The Universe of discourse is given by the nine reference tiles (note that

although we adopt a matrix presentation of these membership values, it is

equivalent to a non continuous membership grade). One can remark that

binary direction relations identified for two given S-regions represent a crisp set

specialisation of these fuzzy membership functions. For illustration purposes,

let us study the direction relations between a reference C-regionA ¼ fa1; a2; a3; a4g and a target C-region B ¼ fb1; b2g (Fig. 2).

Applying (4) gives the following fuzzy matrix of direction relations between

the reference C-region A and the target C-region B (one can note the emergence

of an East, North-East direction relation pattern of B relative to A):

DirbðA=BÞ ¼0 0:25 0:620 0:12 0:12

0:12 0:12 0:25

24

35

3.2. Measures of similarities

The fuzzy semantic approach supports comparison of direction relations

between two given C-regions. Additional operations include the analysis of

direction relations with respect to several target C-regions and conversely. In

a 1

a 2

a 3

b 1

b 2

a 4

Fig. 2. Direction relations between two C-regions.


order to facilitate these manipulations, two basic theorems are introduced.

Theorem 1 evaluates the fuzzy matrix of direction relations of the union of two

disjoint C-regions relative to a third C-region while Theorem 2 gives the

counterpart, that is, the fuzzy matrix of direction relations of a C-region rel-ative to the union of two disjoint C-regions. These theorems support flexible

operations in the analysis of direction configurations (e.g. adding/deleting some

simple regions to/from reference or target C-regions).

Theorem 1. Assume A, B and B0 are C-regions where A ¼ fa1; a2; . . . ; amg,B ¼ fb1; b2; . . . ; bng and B0 ¼ fbnþ1; bnþ2; . . . ; bnþpg, B and B0 being disjoint non-empty C-regions. The cardinalities of A, B and B0 are respectively m, n and p, andtile 2 TILE, hence

tileðA=B [ B0Þ ¼ nnþ p

� �tileðA=BÞ þ p

nþ p

� �tileðA=B0Þ ð5Þ

Proof

tileðA=B [ B0Þ ¼Xmi¼1

Xnþp

j¼1

tileðai=bjÞmðnþ pÞ

¼Xmi¼1

Xn

j¼1

tileðai=bjÞmðnþ pÞ þ

Xmi¼1

Xnþp

j¼n

tileðai=bjÞmðnþ pÞ

¼ nnþ p

� �1

m� n

� �Xmi¼1

Xn

j¼1

tileðai=bjÞ

þ pnþ p

� �1

p � m

� �Xmi¼1

Xnþp

nþ1

tileðai=bjÞ

¼ nnþ p

� �tileðA=BÞ þ p

nþ p

� �tileðA=B0Þ �

Theorem 2. Assume A, A0 and B0 are C-regions where A ¼ fa1; a2; . . . ; amg,A0 ¼ famþ1; amþ2; . . . ; amþng and B ¼ fb1; b2; . . . ; bpg, A and A0 being disjoint non-empty C-regions. The cardinalities of A, A0 and B0 are respectively m, n and p, andtile 2 TILE, hence

tileðA [ A0=BÞ ¼ mmþ n

� �tileðA=BÞ þ n

mþ n

� �tileðA0=BÞ ð6Þ

The proof of Theorem 2 is similar to the one given for Theorem 1. From

Theorems 1 and 2, the following corollaries derive the fuzzy matrix of direction

relations of one reference C-region (resp. two reference C-regions) vs. twotarget C-regions (resp. one target C-region).


Corollary 1. Assume A, B and B0 are C-regions where A ¼ fa1; a2; . . . ; amg,B ¼ fb1; b2; . . . ; bng and B0 ¼ fbnþ1; bnþ2; . . . ; bnþpg, B and B0 being disjoint non-empty C-regions. The cardinalities of A, B and B0 are respectively m, n and p, andtile 2 TILE then

DirbðA=B [ B0Þ ¼ nnþ p

� �DirbðA=BÞ þ

pnþ p

� �DirbðA=B0Þ ð7Þ

Corollary 2. Assume A, A0 and B are C-regions where A ¼ fa1; a2; . . . ; amg,A0 ¼ famþ1; amþ2; . . . ; amþng and B ¼ fb1; b2; . . . ; bpg, A and A0 are disjoint non-empty C-regions. The cardinalities of A, A0 and B are respectively m, n and p, andtile 2 TILE, hence

DirbðA [ A0=BÞ ¼ mmþ n


nmþ n

� �DirbðA0=BÞ ð8Þ

Let use introduce a second example of spatial configuration where

A ¼ fa1; a2; a3; a4g, B ¼ fb1; b2g and C ¼ fc1; c2; c3g, with B and C disjoint non-

empty C-regions (Fig. 3).Fuzzy matrices of direction relations for this spatial configuration are as

follows:

DirbðA=BÞ ¼0 0:25 0:620 0:12 0:12

0:12 0:12 0:25

24

35 DirbðA=CÞ ¼

0 0 0:660 0 0:08

0:25 0 0:08

24

35

An example of manipulation is given by the evaluation of the fuzzy matrix ofdirection relations between the union of the C-regions B and C relative to a

a1

a2

a3

b1

b2

a4

c1

c2

c3

Fig. 3. Spatial configuration example.


third C-region A. The cardinalities of A, B and C are respectively m ¼ 4, n ¼ 2

and p ¼ 3, this gives

DirbðA=B [ CÞ ¼ nnþ p


pnþ p

� �DirbðA=CÞ

Applying the bounded addition and scalar product of two fuzzy membershipfunction give

DirbðA=B [ CÞ 2

5

� � 0 0:25 0:620 0:12 0:12

0:12 0:12 0:25

24

35þ 3

5

� � 0 0 0:660 0 0:08

0:25 0 0:08

24

35

¼0 0:1 0:650 0:05 0:10:2 0:05 0:15

24

35

Additional properties are defined by applying basic operations on fuzzy sub-

sets. Those will give a definition for the union and the intersection of fuzzy

matrices of direction relations, i.e., union and intersection of fuzzy subsets.Although many possible ways have been proposed to define these operations

we consider the default definitions most usually admitted [37]. This leads to

definitions where the union and intersection of fuzzy subsets are represented by

Max and Min operators, respectively.

Definition 2 (union vs. intersection of two fuzzy matrices of direction relations).Assume DirbðA=BÞ and DirbðC=DÞ are fuzzy subsets of direction relations be-

tween C-regions where

DirbðA=BÞ ¼NW ðA=BÞ NðA=BÞ NEðA=BÞW ðA=BÞ OðA=BÞ EðA=BÞSW ðA=BÞ SðA=BÞ SEðA=BÞ

24

35

DirbðC=DÞ ¼NW ðC=DÞ NðC=DÞ NEðC=DÞW ðC=DÞ OðC=DÞ EðC=DÞSW ðC=DÞ SðC=DÞ SEðC=DÞ

24

35

then

DirbðA=BÞ[DirbðC=DÞ

¼MaxðDirbðA=BÞ;DirbðC=DÞÞ

¼

MaxðNW ðA=BÞ;NW ðC=DÞÞ MaxðNðA=BÞ;NðC=DÞÞ MaxðNEðA=BÞ;NEðC=DÞÞ

MaxðW ðA=BÞ;W ðC=DÞÞ MaxðOðA=BÞ;OðC=DÞÞ MaxðEðA=BÞ;EðC=DÞÞ

MaxðSW ðA=BÞ;SW ðC=DÞÞ MaxðSðA=BÞ;SðC=DÞÞ MaxðSEðA=BÞ;SEðC=DÞÞ

2664

3775

ð9Þ


and

DirbðA=BÞ\DirbðC=DÞ¼MinðDirbðA=BÞ;DirbðC=DÞÞ

¼MinðNW ðA=BÞ;NW ðC=DÞÞ MinðNðA=BÞ;NðC=DÞÞ MinðNEðA=BÞ;NEðC=DÞÞMinðW ðA=BÞ;W ðC=DÞÞ MinðOðA=BÞ;OðC=DÞÞ MinðEðA=BÞ;EðC=DÞÞ

MinðSW ðA=BÞ;SW ðC=DÞÞ MinðSðA=BÞ;SðC=DÞÞ MinðSEðA=BÞ;SEðC=DÞÞ

264

375

ð10Þ

One should note that the union of two fuzzy matrices (DirbðA=BÞ [ DirbðC=DÞ)maximizes while the intersection DirbðA=BÞ \ DirbðC=DÞ minimizes member-

ship values. As for the union and intersection of fuzzy subsets, intersection andunion of fuzzy matrices of direction relations include additional basic prop-

erties that facilitate further manipulations (commutativity, associativity, dis-

tributivity). Those set operations can be used to explore how two classes of

composite regions are related through different spatial scenes, and to identify

the maximum and minimum occurrences of each direction relation.

4. Measures of similarities

Measuring the similarity between different configurations of direction rela-tions should be analysed by comparing fuzzy matrices. A similarity function

usually maps pairs of entities towards a unique degree of similarity between 0

and 1. We take the usual approach in which the value 1 corresponds to the

maximum similarity and the value 0 the maximum dissimilarity. Many ap-

proaches have been proposed to derive similarities between two fuzzy subsets

[41] such as identifying the maximum value of the intersection between the

membership functions [17] or weighing the average distance between the mem-

bership function values. We introduce a two-step technique that first derives thea-level sets of a fuzzy direction matrix [38], and secondly apply Zadeh’s measure

of similarity [39]. a-level sets present the advantage of giving the position of the

fuzzy subset and its coverage in the crisp Universe of Discourse, favouring thus

further computation and analysis. An a-level set is defined as follows.

Definition 3. Let DirbðA=BÞ be a fuzzy matrix of direction relations; the a-levelset of DirbðA=BÞ, denoted a-DirbðA=BÞ, is the crisp subset of TILE consisting of

all the elements of TILE for which tile ðA=BÞP a, it is given by

a-DirbðA=BÞ ¼ ftilejtileðA=BÞP a; tile 2 TILEg ð11Þ

An a-level set of a fuzzy matrix retains the significant values of the fuzzy matrix,that is, direction relations whose fuzzy values are higher or equal to the given


threshold a. The threshold value is user-defined and should reflect the degree of

membership which is of interest with respect to a given context. The value of the

threshold can be fixed considering the frequency distribution of direction

relations on several spatial configurations under analysis. For instance, whenmaking comparison of many C-regions, thresholds may be chosen using per-

centile values based on empirical data and their efficiency assessed using sen-

sitivity analysis [20]. High values of the threshold have a high level of selectivity

while low values have a low selectivity. It is straightforward to note that a level

sets are nested, i.e., a1-DirbðA=BÞ � a2-DirbðA=BÞ for a1 < a2. A measure of

similarity defined on two distinct a-level sets should evaluate to which degree

those two sets share some common elements or not. Accordingly, we define the

measure of similarity among fuzzy matrices of direction relations as follows.

Definition 4. Assume DirbðA=BÞ and DirbðC=DÞ are fuzzy matrices of direction

relations; let a-DirbðA=BÞ and a-DirbðC=DÞ be the a-level sets of DirbðA=BÞ andDirbðC=DÞ, respectively. The a-measure of similarity between DirbðA=BÞ and

DirbðC=DÞ is given by

a-Sim½DirbðA=BÞ;DirbðC=DÞ� ¼ Minkm;kn

� �ð12Þ

where m is the number of elements of a-DirbðA=BÞ, n the number of elements of

a-DirbðC=DÞ and k is the number of elements of the intersection of a-DirbðA=BÞwith a-DirbðC=DÞ.

It is easy to show that the a-measure of similarity fulfils some basic prop-erties of similarity measures, that is, for all A, B, C, DC-regions:

a-Sim½DirbðA=BÞ;DirbðC=DÞ� ¼ a-Sim½DirbðC=DÞ;DirbðA=BÞ� symmetry

a-Sim½DirbðA=BÞ;DirbðA=BÞ�P a-Sim½DirbðC=DÞ;DirbðA=BÞ�

The a-measure gives an overall measure of similarity, still bounded by the unit

interval, and flexible as a-level sets are user-defined (i.e., level of significance inselecting the tiles). One can remark that when k ¼ m (alternatively k ¼ n) thena-DirbðA=BÞ � DirbðC=DÞ (alternatively a-DirbðC=DÞ � DirbðA=BÞ). In order to

illustrate the similarity concept let us consider the spatial configurations given

by the matrices of fuzzy direction relations DirbðA=BÞ and DirbðA=CÞ. In order

to analyse different levels of direction relation similarities we give a1 ¼ 0:3 and

a2 ¼ 0:15, hence

a1-DirbðA=BÞ ¼ ftilejtileðA=BÞP 0:3; tile 2 TILEg ¼ fNEga2-DirbðA=BÞ ¼ ftilejtileðA=BÞP 0:15; tile 2 TILEg ¼ fNE;Nga1-DirbðA=CÞ ¼ ftilejtileðA=CÞP 0:3; tile 2 TILEg ¼ fNEga2-DirbðA=CÞ ¼ ftilejtileðA=CÞP 0:15; tile 2 TILEg ¼ fNE; SW g


The measures of similarity between these a-level sets are as follows:

a1-Sim½DirbðA=BÞ;DirbðA=CÞ� ¼ Min1

1;1

1

� �¼ 1


2;1

2

� �¼ 1

2

These results show the flexibility of these similarity measures and how,depending on a-values, two given configuration scenes can be compared and

qualified according to a degree of membership which is of interest for the

application. This approach offers a flexible tool by giving the user the choice of

defining what degrees of direction relations should be considered in analysing

two spatial scenes.

5. Application to an illustrative case study

Maritime navigation is an example of an application where relative positionsand direction relations between fleets of concurrent teams is of strategic interest

during a race for tactic decisions (e.g. to compare respective routes with respect

to wind and meteorological conditions), and after a race for debriefing in order

to analyse respective performances of the teams during the race. The sketch

example presented in this section reports on the respective positions of mul-

tihulls and monohulls sailing ships during the ‘‘Course du Rhum’’, an inter-

national sailor ship race that took place in 2002 between St Malo in France and

Guadeloupe. Fig. 4 reports on those positions on the 19th (top) and 20th ofNovember (bottom). For the purpose of the case study MBRs are defined as

relatively large areas around each reference ship.

Fuzzy matrices of direction relations for those two configurations denote the

following trends. The map on the 19th of November shows that monohulls

have chosen a route at the South while multihull have chosen a route at the

North at one exception. This is illustrated by patterns of North-East (0.4) and

North-West (0.25) directions of the multihulls relative to the monohulls, al-

though the fact that one multihull took a South route is reflected by thepresence of a South-East direction relation (0.2). Between the 19th and the 20th

of November one monohull and one multihull withdrew this reinforcing sig-

nificantly the North-East direction of the multihulls (0.58) relative to the

monohulls. Those direction patterns stress the good performance of monohulls

with respect to multihulls as the race target (Guadeloupe) is oriented South-

West. The local pattern identified at the North-West reflects the fact that one of

the multihull competitors took a different course to the North. Fig. 5 gives

another example of direction relations but this time making the differencebetween small multihulls relative to monohulls (i.e., DirbðA=BÞ), with large

Fig. 4. Case study first configuration example.

Fig. 5. Case study second configuration example.


multihulls relative to monohulls. An application of Theorem 1 derives theoverall direction relation of multihulls relative to monohulls (DirbðA=B [ CÞ),applied to the night between the 19th and the 20th of November.


The configuration above is more contrasted than the previous ones.

Monohulls and small multihulls took similar routes although small multihulls

are mostly behind monohulls. Large multihulls still took a route at the North,

one of the two large multihulls leading the race at the time of the snapshot.This pattern is reflected by the application of the intersection of fuzzy matrices

where minimal values across the two configurations are retained.

DirbðA=BÞ \ DirbðA=CÞ ¼ MinðDirbðA=BÞ;DirbðC=DÞÞ

¼0:25 0 0:250 0 0

0 0 0

24

35

We go further and apply a similarity measure to the matrices of fuzzydirection relations given in Fig. 5, i.e., DirbðA=BÞ, DirbðA=CÞ and

DirbðA=B [ CÞ. In order to retain significant direction relations we give

a1 ¼ 0:15 (which is intuitively a relevant threshold to keep the most important

directions in this case), hence

a1-DirbðA=BÞ ¼ ftilejtileðA=BÞP 0:15; tile 2 TILEg ¼ fNE;NW g

a1-DirbðA=CÞ ¼ ftilejtileðA=CÞP 0:15; tile 2 TILEg ¼ fNE;NW ;W g

a1-DirbðA=B [ CÞ ¼ ftilejtileðA=CÞP 0:15; tile 2 TILEg ¼ fNE;NW ;W g

This measure of similarity stresses the relatively high influence of North-East

and West direction relations of B relative to A and C relative to A, respectively;and the fact that the predominance of the direction North-East is increased

when considering the multihulls all together (i.e., B [ C).


2;2

3

� �¼ 2

3

a1-Sim½DirbðA=BÞ;DirbðA=B [ CÞ� ¼ Min3

3;3

3

� �¼ 2

3

a1-Sim½DirbðA=CÞ;DirbðA=B [ CÞ� ¼ Min2

2;2

3

� �¼ 1

These figures might help the different teams to adapt in real-time their course

depending on the concurrent positions, directions and navigation conditions,

and to analyse respective performances during debriefing discussions. Those

examples also illustrate the adaptability of the approach as composite regionsare derived from local regions on demand according to some user-defined

criteria.


6. Conclusion

Evaluation of spatial relations and derivation of measures of similarity in

spatial configurations are still an open challenge for qualitative spatial rea-soning. This paper introduces a fuzzy modelling technique that computes

direction relations in a spatial scene. The model combines a spatial reasoning

approach with fuzzy semantic typing to evaluate the configuration of direction

relations between two given composite regions. Fuzzy semantics typing pro-

vides a flexible and computable mean to assess direction relations between

composite regions. Although it is based on Goyal and Egenhofer’s direction

model, it should be also applicable to other direction models with some minor

adaptations.The fuzzy principles of the approach can be extended and applied at

different levels of the model to integrate further variability and flexibility in

the properties represented. Degrees of membership of S-regions with respect

to a C-region, part of a target S-region that intersects a tile, distance between

S-regions (as suggested in [9]) or their relative sizes can be also modelled as

fuzzy variables. Those fuzzy variables taken independently or aggregated can

give a fuzzy domain to tileða=bÞ thus augmenting the semantics of the ap-

proach.Similarity measures are introduced on top of the model. They give a flexible

tool in assessing to which degree configuration scenes reveal, or not, some

common patterns. Different measures of similarities can be applied to compare

fuzzy directions between spatial scenes. The approach retained in this paper

expresses similarities in a crisp Universe of Discourse but alternatives in the

fuzzy domains might also be of interest.

This approach should be of interest for a wide range of applications for

environmental, biological and epidemiological studies, emergency planningand coordinated navigation. We also believe that addressing these new issues

may contribute to open a bridge between formal reasoning approaches and

spatial data analysis, this for the entire benefits of the GIS community. Further

work concerns the integration of additional fuzzy variables in the model and

the analysis of direction relations in evolving scenes. Another direction to

explore is an evaluation experiment involving typical user groups in order to

compare the relevance of fuzzy direction relations identified by our model with

respect to cognitive interpretations.

Acknowledgements

The authors thank the referees for their valuable comments and suggestionwhich have significantly improved the quality of the paper.


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Fuzzy semantics for direction relations between composite regions

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