Peter BogaertGeos 2005

A Qualitative Trajectory Calculus

and the Composition of its Relations

Authors: Nico Van de Weghe, Bart Kuijpers,

Peter Bogaert and Philippe De Maeyer

Department of Geography, Ghent University, Belgium{nico.vandeweghe, peter.bogaert, philippe.demaeyer}@ugent.be

Theoretical Computer Science, Hasselt University, Belgiumbart.kuijpers@uhasselt.be

Peter BogaertGeos 2005

Overview

Problem statement

The Qualitative Trajectory Calculus –Double Cross (QTCC)

Composition Table

Composition Rules Table

Concluding Remarks

Peter BogaertGeos 2005

Problem Statement

• Qualitative Reasoning: Region Connection Calculus (RCC)

• Database: 9-Intersection Model

Randell, D., Cui, Z, and Cohn, A.G., 1992. A Spatial Logic Based on Regions and Connection, In: Nebel, B., Swartout, W., and Rich, C. (Eds.), Proc. of the 3rd Int. Conf. on Knowledge Representation and Reasoning (KR), Morgan Kaufmann, San Mateo, USA, 165‑176.

Egenhofer, M. and Franzosa, R., 1991. Point‑Set Topological Spatial Relations, International Journal of Geographical Information Systems, 5 (2), 161‑174.

• a lot of work has been done in stating the dyadic topological relations between two polygons

• Two approaches

Peter BogaertGeos 2005

Problem Statement

DC(k,l) EC(k,l) PO(k,l)

TPP(k,l) NTPP(k,l)

TPPI(k,l) NTPPI(k,l)

EQ(k,l)

• 8 possible topologic relations

• Constraints upon the ways these relations change

Peter BogaertGeos 2005

Problem Statement

Developing a calculus for representing and reasoning about movements of objects in a qualitative framework.

continuously moving objects: often only “disconnected from”

“How do we handle changes in movement between moving objects, if there is no change in their topological relationship?”

DC(k,l)

Peter BogaertGeos 2005

QTC

(Van de Weghe, N. (2004) Representing and Reasoning about Moving Objects:A Qualitative Approach, PhD thesis, Belgium, Ghent University,

Faculty of Sciences, Department of Geography, 268 pp.)

QTCB

1D2D

Distance, speed

QTCC

2D

Double Cross Concept

A calculus for representing and reasoning about movements of two disconnected point-like objects

in a qualitative framework.

Peter BogaertGeos 2005

QTC – Simplification

Peter BogaertGeos 2005

Freksa, Ch., 1992. Using Orientation Information for Qualitative Spatial reasoning, In: Frank, A.U., Campari, I., and Formentini, U. (Eds.), Proc. of the Int. Conf. on Theories and Methods of Spatio‑Temporal Reasoning in Geographic Space, Pisa, Italy, Lecture Notes in Computer Science, Springer‑Verlag, (639), 162‑178.

Double-Cross Concept

Peter BogaertGeos 2005

QTCCDouble-Cross Calculus (Freksa)

QTC – DOUBLE-CROSS (QTCC)

l

k

Peter BogaertGeos 2005

Double-Cross Calculus (Freksa)

QTCC

l

k

l

k

QTC – DOUBLE-CROSS (QTCC)

Peter BogaertGeos 2005

Double-Cross Calculus (Freksa)

QTCC

l

k

l

k

l

k

QTC – DOUBLE-CROSS (QTCC)

Peter BogaertGeos 2005

QTC – DOUBLE-CROSS (QTCC)

l

k

Peter BogaertGeos 2005

0

–+

–

QTC – DOUBLE-CROSS (QTCC)

l

k

Peter BogaertGeos 2005

– +

– –

0

QTC – DOUBLE-CROSS (QTCC)

l

k

Peter BogaertGeos 2005

0–

– – –

+

QTC – DOUBLE-CROSS (QTCC)

k

l

Peter BogaertGeos 2005

0–

+

– – – –

–

+

QTC – DOUBLE-CROSS (QTCC)

l

k

Peter BogaertGeos 2005

Qualitative Trajectory Calculus (QTC)QTCB2D

QTCC

Peter BogaertGeos 2005

Qualitative Trajectory Calculus (QTC)QTCB2D

QTCC

Peter BogaertGeos 2005

Qualitative Trajectory Calculus (QTC)QTCB2D

QTCC

Peter BogaertGeos 2005

R1(k,l) R2(l,m) = R3(k,m)

originate from temporal reasoning (Allen 1983)

encodes all possible compositions of relations

simple table look‑up

useful from a computational point of view

Composition Table

Idea: to compose a finite set of new facts andrules from existing ones

Peter BogaertGeos 2005

Composition Table

Peter BogaertGeos 2005

Composition Table

QTCC defines 81 different relations

Difficult visual presentation

A composition table would have 81*81 (6561) entries

Peter BogaertGeos 2005

Useful for visual presentation

Instrument to help generate the composition table

Composition Rules Table

Reduction of the number of entrances

Peter BogaertGeos 2005

Composition Rules Table

Based on 2 rules

Which rotation do we need,such that l of R2 matches l of R1?

How is k moving with respect to l in R1, and how is m moving with respect to l in R2?

Peter BogaertGeos 2005

Which rotation do we need,such that l of R2 matches l of R1?

Composition Rules Table

(k,l)(– + + –)C (l,m)(– – – +)C

k l

l m

(k,m)

Peter BogaertGeos 2005

(case 1 of n)(k,l)(– + + –)C (l,m)(– – – +)C

k l

l m

l

(k,m)

k

m

Composition Rules Table

Peter BogaertGeos 2005

(k,l)(– + + –)C (l,m)(– – – +)C

(case 1 of n)(k,m)

k

m

k ll m

Composition Rules Table

Peter BogaertGeos 2005

m

k

–

(k,m)(– )C

(k,l)(– + + –)C (l,m)(– – – +)C

(case 1 of n)(k,m)

k ll m

Composition Rules Table

Peter BogaertGeos 2005

k ll m

m

k

+

(k,m)(– + )C

(k,m)(– )C

(k,l)(– + + –)C (l,m)(– – – +)C

(case 1 of n)(k,m)

Composition Rules Table

Peter BogaertGeos 2005

m

k–

(k,m)(– + )C

(k,m)(– )C

(k,m)(– + – )C

(k,l)(– + + –)C (l,m)(– – – +)C

(case 1 of n)(k,m)

k ll m

Composition Rules Table

Peter BogaertGeos 2005

k ll m

m

k

+(k,m)(– + )C

(k,m)(– )C

(k,m)(– + – )C

(k,m)(– + – + )C

(k,l)(– + + –)C (l,m)(– – – +)C

(case 1 of n)(k,m)

Composition Rules Table

Peter BogaertGeos 2005

k ll m

m

k l

(k,l)(– + + –)C (l,m)(– – – +)C

(case 2 of n)(k,m)

Composition Rules Table

Peter BogaertGeos 2005

k ll m

k

(k,l)(– + + –)C (l,m)(– – – +)C

(case 2 of n)(k,m)

m

Composition Rules Table

Peter BogaertGeos 2005

k ll m

k

(k,l)(– + + –)C (l,m)(– – – +)C

(case 2 of n)(k,m)

m (k,m)(– – + +)C

Composition Rules Table

Peter BogaertGeos 2005

(k,l)(– + + –)C (l,m)(– – – +)C

> 180°

k l

ml

(k,m)

l

(n of n cases)

Composition Rules Table

Peter BogaertGeos 2005

(k,l)(– + + –)C (l,m)(– – – +)C

> 180°

< 360°

k l

m

l

l

k l

m

(n of n cases)(k,m)

Composition Rules Table

Peter BogaertGeos 2005

> 180° and < 360°

(n of n cases) (k,l)(– + + –)C (l,m)(– – – +)C(k,m)

Composition Rules Table

Peter BogaertGeos 2005

Which rotation do we need, such that l of R2 matches l of R1

> 180° and < 360°

(n of n cases) (k,l)(– + + –)C (l,m)(– – – +)C(k,m)

Composition Rules Table

Peter BogaertGeos 2005

How is k moving with respect to l in R1, and how is m moving with respect to l in R2

(n of n cases) (k,l)(– + + –)C (l,m)(– – – +)C(k,m)

Composition Rules Table

Peter BogaertGeos 2005

How is k moving with respect to l in R1, and how is m moving with respect to l in R2

(n of n cases) (k,l)(– + + –)C (l,m)(– – – +)C(k,m)

Composition Rules Table

Peter BogaertGeos 2005

Composition Rules Table

Peter BogaertGeos 2005

independent

goal: movement of object k with respect to object n (R3)

incomplete data

incomplete answer

two scientific teams

'Puzzling the Past'

(k,n)?

Peter BogaertGeos 2005

(k,l)(– + + –)C (l,n)(– – – +)C

Team I

Team II(k,m)(– + – +)C (m,n)(– + – +)C

k l

l

mm

n

nk

(k,n)

(k,n)

'Puzzling the Past'

Peter BogaertGeos 2005

Team III Team IV

Team I Team II

Team III

Team IV

(k,n)

'Puzzling the Past'

Peter BogaertGeos 2005

Team V (k,n)(– + – +)C

Team III Team IV

'Puzzling the Past'

Peter BogaertGeos 2005

Complex researches (huge number of anchor points, teams, measurements per team, updates, etc.)

Cuncluding Remarks

IMPLEMENTATION OF QTCC

The composition-rules table forms a basis for reasoning about

incomplete spatio-temporal knowledge in information systems.

Can be used in a variety of research domains: geomorphology, geology, archaeology, and biology.

Peter BogaertGeos 2005

A Qualitative Trajectory Calculus

and the Composition of its Relations

Authors: Nico Van de Weghe, Bart Kuijpers,

Peter Bogaert and Philippe De Maeyer

Department of Geography, Ghent University, Belgium{nico.vandeweghe, peter.bogaert, philippe.demaeyer}@ugent.be

Theoretical Computer Science, Hasselt University, Belgiumbart.kuijpers@uhasselt.be