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A Qualitative Representation of Trajectory Pairs

Nico Van de Weghe1, Anthony G. Cohn2, and Philippe De Maeyer1

There has been much research in temporal and spatial reasoning,

both quantitative and qualitative, the latter being of particular

interest from a cognitive viewpoint [2]. Attempts to combine both

spatial and temporal relationships include [1,7,9]. A database

approach to specifying spatial relationships that hold between

moving objects during a particular interval is [10]. An approach

that combines topological relationships between regions in 2D

space with temporal relationships between convex intervals is [1].

However, we believe the question remains of how to describe

motion adequately within a qualitative calculus. Motion can be

divided into change of location, i.e. translation, and change in

orientation, i.e. rotation. We focus on the former aspect here. A

thorough investigation into mereotopological spatio-temporal

continuous change has been conducted in [6], though there has

been little work on describing relative motion of disconnected

objects. However, it is clear that mobile disconnected objects

(animals, vehicles,…) are prevalent in many domains and it would

be highly desirable to be able to describe their motion in a

qualitative manner. One move in this direction is the extension of

qualitative physics to handle relative positions of objects in 2D

[11], but this relies on projecting positions to x and y axes and does

not provide a calculus with a set of jointly exhaustive and pairwise

disjoint (JEPD) relationships. A simple calculus for describing

traffic events is [4]. The work presented in this paper can be

viewed as a continuation of these strands of previous research; i.e.

it is an exploration of trajectories of moving (point like) objects.

If objects do not change their form during the movement and

we focus on the representation of spatially disjoint objects, then we

can take an arbitrary point (e.g. the centroid) as the spatial location

of an object. Therefore in this paper objects are represented simply

as points. Moving objects can be partitioned in those having a free

trajectory and those with a constrained trajectory [8]. A free

trajectory means that there are no significant restrictions on the

movement of a point in an nD space, such as an airplane traveling

through the sky. A constrained trajectory means that the movement

of an object in space is strongly restricted. A 1D representation can

provide a useful abstraction for many free trajectory applications;

e.g. even though a prey and a predator move in nD, the vital

question is whether or not the predator catches the prey

(represented by their Eucledian distance apart).

Positional information is determined by the orientation and the

distance relation [3]. Based on this and the notion of mode space

(in which a space is subdivided in homogenous clusters) [6], the

movement or transition between two objects at an instant can be

qualitatively represented using three functions:

i. movement of the 1st object wrt the 2nd object’s position

ii. movement of the 2nd object wrt the 1st object’s position

iii. relative speed of the 1st object wrt the 2nd object

Since we are interested in a qualitative calculus, we can represent

the values of each of these functions by “+”, “0” or “−” (cf [13]).

1

Department of Geography, Ghent University, Ghent, Belgium,

{nico.vandeweghe,philippe.demaeyer}@ugent.be

2

School of Computing, University of Leeds, Leeds, United Kingdom,

agc@comp.leeds.ac.uk

For the first two functions, we take “−” to mean motion towards

the other object, “+” to mean motion away, and “0” to mean an

absence of motion to/from the other object. In (iii), “+/0/−” mean a

greater/same/lower speed respectively. This triple function forms

the basis of our Qualitative Trajectory Calculus (QTC).

In order to clarify what we mean by motion towards/away

another object, consider Fig. 1. In each case the position of l at

time t is indicated with a black dot. The trajectory of k is indicated

with an arrow such that at time t it is exactly at the intersection of

the circle centered at l. In a. we say that k is moving toward l at t

(−); in b. that k is moving away (+), and in c. and d., that k is

(instantaneously) stationary (0) with respect to the position of l at t.

We can represent a trajectory by a label consisting of 3

characters, each one giving a value for (i)-(iii) above. Thus

there are 27 (3³) potential trajectory pairs. When only the

two movement constraints are considered, there are 9

trajectories – see Fig. 2.

Figure 1. Qualitatively different cases of motion of k wrt l at t.

Figure 2. Visualization of Qualitative Trajectory Pairs in a conceptual

neighborhood diagram. The dots represent the positions of the objects

(solid: object can be stationary, open: object cannot be stationary). The

lines and crescents represent the potential object movements (if a crescent

is used, then the movements start in the dot and ends somewhere on the

curved side of the crescent).

inverse (– 0 inverse to 0 –, – + inverse to + –, and + 0 inverse to

0 +) and the other are self-inverse (– –, + +, and 0 0).

There are 6 basic trajectory pairs of which three have an

neighborhood:‘Two relations between pairs of events are

conceptual neighbors, if they can be directly transformed into one

another by continuously deforming (i.e. shortening, lengthening,

moving) the events in a topological sense’. In QTC, two trajectory

pairs are conceptual neighbors if they can directly follow each

other during a continuous movement. We can analyze this in terms

of the continuity of each component of a QTC triple. It is clear that

“–” and “0” are neighbors and “0” and “+” are also neighbors

whilst “–” and “+” are not, since a change of value from “+” to “–”

has to go through 0 on the real number line if changes are assumed

to be continuous. Thus the trajectory pairs “– +” and “0 +” are

Freksa [5] introduced the important idea of conceptual

-+

0+

00

--

0-

+-

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l

k

l

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l

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k

k

a

b

c

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Fig 1.

Fig. 2

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conceptual neighbors since the second characters are identical (+)

and the first characters are neighbors (–, 0). On the other hand

“– +” and “+ +” are not since they can only be linked via the “0 +”

transition. Together with the theory of dominance space [6], this

results in the conceptual neighborhood presented in Fig. 2.

To clarify the way in which trajectories are represented within

QTC it may be helpful to consider some examples. A particularly

interesting case is that of circular motion. Consider the situations

depicted in Fig. 3 where two objects are traveling along the same

circular path (shown with a thin continuous line). In Fig. 3a k and l

are diametrically opposite at time t. If k moves anywhere below the

dashed line (which is perpendicular to the segment joining k and l),

then it will be moving closer to where l was at time t. However just

before t, k was moving away from where l was at time t.

Therefore the appropriate qualitative value representing the

relationship between k and l is “0” at time t, “–” just before t, and

“+” just after t. Dual reasoning applies for the movement of l with

respect to the position of k at t; so we have 0 0 at t. Note that it is

irrelevant whether the objects are moving clockwise or

anticlockwise. Now assume that both objects from Fig 3b are

traveling clockwise. It can be seen that k is moving away from l

and l is moving towards k, so the description is + –. If the motion

were anticlockwise, then the description would be – +.

l

Figure 3. Circular Trajectories

An important task in qualitative dynamics is to be able to represent

the relationship between specified individuals over an interval. We

illustrate this task with an example consisting of the evolution of

the interaction between a carnivore and its prey. When a carnivore

hunts a prey, the mereotopological relationship is typically that of

disjointness until the time that the prey is caught. We now describe

a hunt, both informally in English, and with annotations in QTC:

(1) A resting lion sees a resting zebra and starts stalking the zebra.

Conceptual path of qualitative trajectory pairs (CPT 0 0 , – 0)

(2) All of a sudden the zebra gets a glimpse of the lion and tries to

escape. (CPT – 0 , – + + , – + 0 , – + – )

(3) The lion reacts, and starts following the zebra with a higher

velocity. (CPT – + – , – + 0 , – + + )

(4) After a while the lion gets tired and is not able to run as fast.

(CPT – + + , – + 0 , – + – )

(5) The lion realizes that he will have to do it without food, stops

chasing the zebra and takes a rest. (CPT – + – , 0 + )

(6) After a while, the zebra is certain that he has got rid of the lion,

stops running and continues with grazing. (CPT 0 + , 0 0 )

The composition of these qualitative trajectory pairs can be

visualized using the conceptual neighborhood diagram (see Fig. 4).

In many environments, moving objects such as cars tend to

follow predefined spatial paths namely roads, highways, etc. In

[12] we outline how the calculus might be specialized to this case.

We believe that the proposed approach may be a useful starting

point for analyzing the complex interaction between moving

objects. Although the 1D case can be seen as a solid basis,

extensions have to be made to 2D and 3D. Another important issue

for future research is to investigate the composition tables and the

complexity of reasoning in the calculus. Our initial investigations

have shown that unless restricted to a pure 1D domain, the

composition table is rather weak; however we are also working on

extending the relational calculus with orientation knowledge (e.g.

14] which will allow a more useful composition table).

Figure 4. Carnivore-prey example. Note that this figure extends the

conceptual neighbourhood diagram displayed in Fig 2 by representing also

the third element of a QTC-triple.

[1] Claramunt, Ch., Jiang, B., An Integrated Representation of

Spatial and Temporal Relationships between Evolving

Regions, Geographical Systems 3 (4) (2001) 411-428

[2] Cohn, A.G., Hazarika, S.M., Qualitative Spatial Representation

and Reasoning: An Overview, Fund. Inf. 46 (1-2)(2001) 1-29

[3] Downs, R.M., Stea, D., Downs, R.M., Stea, D., Image and

Environment: Cognitive Mapping and Spatial Behavior,

Aldine, Chicago, 1973

[4] Fernyhough, J., Cohn, A.G., Hogg, D., Constructing

Qualitative Event Models Automatically from Video Input,

Image and Vision Computing 18 (2) (2000) 81-103

[5] Freksa, C., Temporal Reasoning Based on Semi-Intervals,

Artificial intelligence 54 (1992) 199-227

[6] Galton, A., Qualitative Spatial Change, Oxford Univ. Press, ,

2000

[7] Hazarika, S.M., Cohn A.G., Abducing Qualitative Spatio-

Temporal Histories from Partial Observations, In: Proc. of

Eight KR Conf. edited by Fensel, D., et al., M.-A., Morgan

Kaufmann (2002) 14-25

[8] Moreira, J., Ribeiro, C., Saglio, J.-M., Representation and

Manipulation of Moving Points: an Extended Data Model for

Location Estimation, Cartography

Information Systems 26 (2) (1999) 109-123

[9] Muller, P., A Qualitative Theory of Motion Based on

Spatiotemporal Primitives, In: Cohn, A.G., Schubert, L.,

Shapiro, S., Proc. of 6th KR (1998) 131-142

[10] Nabil, M., et al., Modeling and Retrieval of Moving Objects,

Multimedia Tools and Applications 13 (1) (2001) 35-71

[11] Rajagopalan, R., Qualitative Reasoning about Dynamic

Change in the Spatial Properties of a Physical System, Ph.D,

Dept of Computer Sciences, University of Texas (1995)

[12] Van de Weghe N., Cohn A.G., Bogaert P., De Maeyer,

Ph., Representation of Moving Objects along a Road

Network, In: Geoinformatics (2004).

[13] Weld, D.S., de Kleer, J., editors. Readings in Qualitative

Reasoning About Physical Systems. Morgan Kaufmann

Publishers Inc., San Mateo, California, 1990

[14] Zimmermann, K., Freksa, Ch., Qualitative Spatial Reasoning

Using Orientation, Distance, and Path Knowledge, Applied

Intelligence 6 (1) (1996) 49-58

and Geographic

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k

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+ -+

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-+ -

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-+ + -+ +

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-+ --+ --+ -

-+ +-+ +

-+ 0-+ 0-+ 0

-+ --+ --+ -

(1)(2)(3)

(4)

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