qualitative spatial reasoning: cardinal directions as an example

Qualitative Spatial Reasoning: Cardinal Directions as an Example

Andrew U. Frank1995

2

Outline

2

Outline• Introduction

2

Outline• Introduction

• Motivation: Why qualitative? Why cardinal?

2

Outline• Introduction

• Motivation: Why qualitative? Why cardinal?

• Method: An algebraic approach

2

Outline• Introduction

• Motivation: Why qualitative? Why cardinal?

• Method: An algebraic approach

• Two cardinal direction systems

2

Outline• Introduction

• Motivation: Why qualitative? Why cardinal?

• Method: An algebraic approach

• Two cardinal direction systems

- Cone-shaped directions

2

Outline• Introduction

• Motivation: Why qualitative? Why cardinal?

• Method: An algebraic approach

• Two cardinal direction systems

- Cone-shaped directions

- Projection-based directions

2

Outline• Introduction

• Motivation: Why qualitative? Why cardinal?

• Method: An algebraic approach

• Two cardinal direction systems

- Cone-shaped directions

- Projection-based directions

• Assessment

2

Outline• Introduction

• Motivation: Why qualitative? Why cardinal?

• Method: An algebraic approach

• Two cardinal direction systems

- Cone-shaped directions

- Projection-based directions

• Assessment

• Research envisioned

3

Introduction

Geography utilizes large scale spatial reasoning extensively.

•Formalized qualitative reasoning processes are

essential to GIS. •

An approach to spatial reasoning using qualitative cardinal directions.

4

Motivation: Why qualitative?Spatial relations are typically formalized in a

quant i tat ive manner with Car tes ian coordinates and vector algebra.

5

Motivation: Why qualitative?

5

Motivation: Why qualitative?

5

Motivation: Why qualitative?

5

Motivation: Why qualitative?

“thirteen centimeters”

6

Motivation: Why qualitative?Human spatial reasoning is based on qualitative

comparisons.

6

Motivation: Why qualitative?Human spatial reasoning is based on qualitative

comparisons.

6

Motivation: Why qualitative?Human spatial reasoning is based on qualitative

comparisons.

“longer”

6

Motivation: Why qualitative?Human spatial reasoning is based on qualitative

comparisons.

• precision is not always desirable

“longer”

6

Motivation: Why qualitative?Human spatial reasoning is based on qualitative

comparisons.

• precision is not always desirable

• precise data is not always available

“longer”

6

Motivation: Why qualitative?Human spatial reasoning is based on qualitative

comparisons.

• precision is not always desirable

• precise data is not always available

• numerical approximations do not account for uncertainty

“longer”

7

Motivation: Why qualitative?

7

Motivation: Why qualitative?

• Formal izat ion required for GIS implementation.

7

Motivation: Why qualitative?

• Formal izat ion required for GIS implementation.

• Interpretation of spatial relations expressed in natural language.

7

Motivation: Why qualitative?

• Formal izat ion required for GIS implementation.

• Interpretation of spatial relations expressed in natural language.

• Comparison of semantics of spatial terms in different languages.

Motivation: Why cardinal?

8

Pullar and Egenhofer’s geographical scale spatial relations (1988):

Motivation: Why cardinal?

8

Pullar and Egenhofer’s geographical scale spatial relations (1988):

• direction north, northwest

Motivation: Why cardinal?

8

Pullar and Egenhofer’s geographical scale spatial relations (1988):

• direction north, northwest• topological disjoint, touches

Motivation: Why cardinal?

8

Pullar and Egenhofer’s geographical scale spatial relations (1988):

• direction north, northwest• topological disjoint, touches• ordinal in, at

Motivation: Why cardinal?

8

Pullar and Egenhofer’s geographical scale spatial relations (1988):

• direction north, northwest• topological disjoint, touches• ordinal in, at• distance far, near

Motivation: Why cardinal?

8

Pullar and Egenhofer’s geographical scale spatial relations (1988):

• direction north, northwest• topological disjoint, touches• ordinal in, at• distance far, near• fuzzy next to, close

Motivation: Why cardinal?

8

Pullar and Egenhofer’s geographical scale spatial relations (1988):

• direction north, northwest• topological disjoint, touches• ordinal in, at• distance far, near• fuzzy next to, close

Motivation: Why cardinal?

8

Cardinal direction chosen as a major example.

Method: An algebraic approach

9

Method: An algebraic approach• Focus on not on directional relations

between points...

9

Method: An algebraic approach• Focus on not on directional relations

between points... • Find rules for manipulating directional

symbols & operators.

9

Method: An algebraic approach• Focus on not on directional relations

between points... • Find rules for manipulating directional

symbols & operators.

9

Directional symbols: N, S, E, W... NE, NW...

Operators: inv ∞ ()

Method: An algebraic approach• Focus on not on directional relations

between points... • Find rules for manipulating directional

symbols & operators.

9

Directional symbols: N, S, E, W... NE, NW...

Operators: inv ∞ ()• Operational meaning in a set of formal

axioms.

Method: An algebraic approach

10

Inverse

Composition

Identity

Method: An algebraic approach

10

P2

P1

Inverse

Composition

Identity

Method: An algebraic approach

10

P2

P1

dir(P1,P2)Inverse

Composition

Identity

Method: An algebraic approach

10

P2

P1

dir(P1,P2)inv(dir(P1,P2))

Inverse

Composition

Identity

Method: An algebraic approach

10

P2

P1

dir(P1,P2)inv(dir(P1,P2))

Inverse

P2

P1

Composition

P3

Identity

Method: An algebraic approach

10

P2

P1

dir(P1,P2)inv(dir(P1,P2))

Inverse

P2

P1

dir(P1,P2)Composition

P3

Identity

Method: An algebraic approach

10

P2

P1

dir(P1,P2)inv(dir(P1,P2))

Inverse

P2

P1

dir(P1,P2)Composition

P3

dir(P2,P3)

Identity

Method: An algebraic approach

10

P2

P1

dir(P1,P2)inv(dir(P1,P2))

Inverse

P2

P1

dir(P1,P2)

dir(P1,P2) ∞ dir(P2,P3)dir (P1,P3)

Composition

P3

dir(P2,P3)

Identity

Method: An algebraic approach

10

P2

P1

dir(P1,P2)inv(dir(P1,P2))

Inverse

P2

P1

dir(P1,P2)

dir(P1,P2) ∞ dir(P2,P3)dir (P1,P3)

Composition

P3

dir(P2,P3)

Identity P1

Method: An algebraic approach

10

P2

P1

dir(P1,P2)inv(dir(P1,P2))

Inverse

P2

P1

dir(P1,P2)

dir(P1,P2) ∞ dir(P2,P3)dir (P1,P3)

Composition

P3

dir(P2,P3)

Identity P1dir(P1,P1)=0

Method: Euclidean exact reasoning

11

Method: Euclidean exact reasoning

• Comparison between qualitative reasoning and quantitative reasoning using analytical geometry

11

Method: Euclidean exact reasoning

• Comparison between qualitative reasoning and quantitative reasoning using analytical geometry

• A qualitative rule is called Euclidean exact if the result of applying the rule is the same as that obtained by analytical geometry

11

Method: Euclidean exact reasoning

• Comparison between qualitative reasoning and quantitative reasoning using analytical geometry

• A qualitative rule is called Euclidean exact if the result of applying the rule is the same as that obtained by analytical geometry

• If the results differ, the rule is considered Euclidean approximate

11

Two cardinal system examples

12

NENW

S SE

W E

N

SW

Oc

Cone-shaped Projection-based

“going toward” “relative position of points on the Earth”

NENW

SSE

W E

N

SW

Directions in cones

13

NENW

SSE

W E

N

SW

Directions in cones

13

NENW

SSE

W E

N

SW

• Angle assigned to nearest named direction

• Area of acceptance increases with distance

Directions in cones

13

NENW

SSE

W E

N

SW

Directions in cones

14

NENW

SSE

W E

N

SW

Directions in cones

14

NENW

SSE

W E

N

SW

Algebraic operations can be performed with symbols:

Directions in cones

14

NENW

SSE

W E

N

SW

• 1/8 turn changes the symbol:e(N)=NE

Algebraic operations can be performed with symbols:

Directions in cones

14

NENW

SSE

W E

N

SW

• 1/8 turn changes the symbol:e(N)=NE

Algebraic operations can be performed with symbols:

Directions in cones

14

NENW

SSE

W E

N

SW

• 1/8 turn changes the symbol:e(N)=NE

• 4/8 turn gives the inverse symbol:e⁴(N)= inv(N) = S

Algebraic operations can be performed with symbols:

Directions in cones

14

NENW

SSE

W E

N

SW

• 1/8 turn changes the symbol:e(N)=NE

• 4/8 turn gives the inverse symbol:e⁴(N)= inv(N) = S

Algebraic operations can be performed with symbols:

Directions in cones

14

NENW

SSE

W E

N

SW

• 1/8 turn changes the symbol:e(N)=NE

• 4/8 turn gives the inverse symbol:e⁴(N)= inv(N) = S

Algebraic operations can be performed with symbols:

Directions in cones

14

NENW

SSE

W E

N

SW

• 1/8 turn changes the symbol:e(N)=NE

• 4/8 turn gives the inverse symbol:e⁴(N)= inv(N) = S

• 8/8 turn gives the identity symbol: e⁸(N)= N

Algebraic operations can be performed with symbols:

Directions in cones

15

NENW

SSE

W E

N

SW

• 1/8 turn changes the symbol:e(N)=NE

• 4/8 turn gives the inverse symbol:e⁴(N)= inv(N) = S

• 8/8 turn gives the identity symbol, 0: e⁸(N)= N = 0

Algebraic operations can be performed with symbols:

Directions in cones

15

NENW

SSE

W E

N

SW

• 1/8 turn changes the symbol:e(N)=NE

• 4/8 turn gives the inverse symbol:e⁴(N)= inv(N) = S

• 8/8 turn gives the identity symbol, 0: e⁸(N)= N = 0

Algebraic operations can be performed with symbols:

Directions in cones

15

NENW

SSE

W E

N

SW

• 1/8 turn changes the symbol:e(N)=NE

• 4/8 turn gives the inverse symbol:e⁴(N)= inv(N) = S

• 8/8 turn gives the identity symbol, 0: e⁸(N)= N = 0

Algebraic operations can be performed with symbols:

Directions in cones

15

NENW

SSE

W E

N

SW

• 1/8 turn changes the symbol:e(N)=NE

• 4/8 turn gives the inverse symbol:e⁴(N)= inv(N) = S

• 8/8 turn gives the identity symbol, 0: e⁸(N)= N = 0

Algebraic operations can be performed with symbols:

0

Directions in cones

16

NENW

SSE

W E

N

SW

• 1/8 turn changes the symbol:e(N)=NE

• 4/8 turn gives the inverse symbol:e⁴(N)= inv(N) = S

• 8/8 turn gives the identity symbol, 0: e⁸(N)= N = 0

Algebraic operations can be performed with symbols:

Directions in cones

16

NENW

SSE

W E

N

SW

• 1/8 turn changes the symbol:e(N)=NE

• 4/8 turn gives the inverse symbol:e⁴(N)= inv(N) = S

• 8/8 turn gives the identity symbol, 0: e⁸(N)= N = 0

• Composition can be computed with averaging rules:

Algebraic operations can be performed with symbols:

Directions in cones

16

NENW

SSE

W E

N

SW

• 1/8 turn changes the symbol:e(N)=NE

• 4/8 turn gives the inverse symbol:e⁴(N)= inv(N) = S

• 8/8 turn gives the identity symbol, 0: e⁸(N)= N = 0

• Composition can be computed with averaging rules:

Algebraic operations can be performed with symbols:

e(N) ∞ N = n

Directions in cones

16

NENW

SSE

W E

N

SW

• 1/8 turn changes the symbol:e(N)=NE

• 4/8 turn gives the inverse symbol:e⁴(N)= inv(N) = S

• 8/8 turn gives the identity symbol, 0: e⁸(N)= N = 0

• Composition can be computed with averaging rules:

Algebraic operations can be performed with symbols:

e(N) ∞ N = n

Directions in cones

16

NENW

SSE

W E

N

SW

• 1/8 turn changes the symbol:e(N)=NE

• 4/8 turn gives the inverse symbol:e⁴(N)= inv(N) = S

• 8/8 turn gives the identity symbol, 0: e⁸(N)= N = 0

• Composition can be computed with averaging rules:

Algebraic operations can be performed with symbols:

e(N) ∞ N = n

Directions in cones

16

NENW

SSE

W E

N

SW

• 1/8 turn changes the symbol:e(N)=NE

• 4/8 turn gives the inverse symbol:e⁴(N)= inv(N) = S

• 8/8 turn gives the identity symbol, 0: e⁸(N)= N = 0

• Composition can be computed with averaging rules:

Algebraic operations can be performed with symbols:

e(N) ∞ N = n

Directions in cones

16

NENW

SSE

W E

N

SW

• 1/8 turn changes the symbol:e(N)=NE

• 4/8 turn gives the inverse symbol:e⁴(N)= inv(N) = S

• 8/8 turn gives the identity symbol, 0: e⁸(N)= N = 0

• Composition can be computed with averaging rules:

Algebraic operations can be performed with symbols:

e(N) ∞ N = n

Directions in cones

16

NENW

SSE

W E

N

SW

• 1/8 turn changes the symbol:e(N)=NE

• 4/8 turn gives the inverse symbol:e⁴(N)= inv(N) = S

• 8/8 turn gives the identity symbol, 0: e⁸(N)= N = 0

• Composition can be computed with averaging rules:

Algebraic operations can be performed with symbols:

e(N) ∞ N = n e(N) ∞ inv (N)

Directions in cones

16

NENW

SSE

W E

N

SW

• 1/8 turn changes the symbol:e(N)=NE

• 4/8 turn gives the inverse symbol:e⁴(N)= inv(N) = S

• 8/8 turn gives the identity symbol, 0: e⁸(N)= N = 0

• Composition can be computed with averaging rules:

Algebraic operations can be performed with symbols:

e(N) ∞ N = n e(N) ∞ inv (N)

Directions in cones

16

NENW

SSE

W E

N

SW

• 1/8 turn changes the symbol:e(N)=NE

• 4/8 turn gives the inverse symbol:e⁴(N)= inv(N) = S

• 8/8 turn gives the identity symbol, 0: e⁸(N)= N = 0

• Composition can be computed with averaging rules:

Algebraic operations can be performed with symbols:

e(N) ∞ N = n e(N) ∞ inv (N)

Directions in cones

16

NENW

SSE

W E

N

SW

• 1/8 turn changes the symbol:e(N)=NE

• 4/8 turn gives the inverse symbol:e⁴(N)= inv(N) = S

• 8/8 turn gives the identity symbol, 0: e⁸(N)= N = 0

• Composition can be computed with averaging rules:

Algebraic operations can be performed with symbols:

e(N) ∞ N = n e(N) ∞ inv (N)

0

Cone direction composition table

17

Cone direction composition table

17

Cone direction composition table

17

Out of 64 combinations, only 10 are Euclidean exact.

Projection-based directions

18

Projection-based directions

18

EW

Projection-based directions

18

N

S

Projection-based directions

18

NENW

SESW

Projection-based directions

18

NENW

SESW

• With half-planes, only trivial cases can be resolved:NE ∞ NE = NE

Projection-based directions

19

NENW

S SE

W E

N

SW

Oc

Projection-based directions

19

NENW

S SE

W E

N

SW

Oc

• Assign neutral zone in the center of 9 regions

Projection-based directions

19

NENW

S SE

W E

N

SW

Oc

Algebraic operations can be performed with symbols:

Projection-based directions

19

NENW

S SE

W E

N

SW

Oc • The identity symbol, 0, resides in the neutral area.

Algebraic operations can be performed with symbols:

Projection-based directions

19

NENW

S SE

W E

N

SW

Oc

• Inverse gives the symbol opposite the neutral area:inv(N) = S

• The identity symbol, 0, resides in the neutral area.

Algebraic operations can be performed with symbols:

Projection-based directions

19

NENW

S SE

W E

N

SW

Oc

• Inverse gives the symbol opposite the neutral area:inv(N) = S

• The identity symbol, 0, resides in the neutral area.

Algebraic operations can be performed with symbols:

Projection-based directions

19

NENW

S SE

W E

N

SW

Oc

• Inverse gives the symbol opposite the neutral area:inv(N) = S

• The identity symbol, 0, resides in the neutral area.

Algebraic operations can be performed with symbols:

Projection-based directions

19

NENW

S SE

W E

N

SW

Oc

• Inverse gives the symbol opposite the neutral area:inv(N) = S

• The identity symbol, 0, resides in the neutral area.

• Composition combines each projection:

Algebraic operations can be performed with symbols:

Projection-based directions

19

NENW

S SE

W E

N

SW

Oc

• Inverse gives the symbol opposite the neutral area:inv(N) = S

• The identity symbol, 0, resides in the neutral area.

• Composition combines each projection:

Algebraic operations can be performed with symbols:

NE ∞ SW = 0

Projection-based directions

19

NENW

S SE

W E

N

SW

Oc

• Inverse gives the symbol opposite the neutral area:inv(N) = S

• The identity symbol, 0, resides in the neutral area.

• Composition combines each projection:

Algebraic operations can be performed with symbols:

NE ∞ SW = 0

Projection-based directions

19

NENW

S SE

W E

N

SW

Oc

• Inverse gives the symbol opposite the neutral area:inv(N) = S

• The identity symbol, 0, resides in the neutral area.

• Composition combines each projection:

Algebraic operations can be performed with symbols:

NE ∞ SW = 0 S ∞ E = SE

Projection-based directions

19

NENW

S SE

W E

N

SW

Oc

• Inverse gives the symbol opposite the neutral area:inv(N) = S

• The identity symbol, 0, resides in the neutral area.

• Composition combines each projection:

Algebraic operations can be performed with symbols:

NE ∞ SW = 0 S ∞ E = SE

Projection-based directions

19

NENW

S SE

W E

N

SW

Oc

• Inverse gives the symbol opposite the neutral area:inv(N) = S

• The identity symbol, 0, resides in the neutral area.

• Composition combines each projection:

Algebraic operations can be performed with symbols:

NE ∞ SW = 0 S ∞ E = SE

Projection composition table

20

Projection composition table

20

Projection composition table

20

Out of 64 combinations, 32 are Euclidean exact.

Assessment

21

Assessment

21

• Both systems use 9 directional symbols.

Assessment

21

• Both systems use 9 directional symbols.

• Cone-shaped system relies on averaging rules.

Assessment

21

• Both systems use 9 directional symbols.

• Cone-shaped system relies on averaging rules.

• Introducing the identity symbol 0 increases the number of deductions in both cases.

Assessment

21

• Both systems use 9 directional symbols.

• Cone-shaped system relies on averaging rules.

• Introducing the identity symbol 0 increases the number of deductions in both cases.

• There are fewer Euclidean approximations using projection-based directions:

Assessment

21

• Both systems use 9 directional symbols.

• Cone-shaped system relies on averaging rules.

• Introducing the identity symbol 0 increases the number of deductions in both cases.

• There are fewer Euclidean approximations using projection-based directions:

‣ 56 approximations using cones

Assessment

21

• Both systems use 9 directional symbols.

• Cone-shaped system relies on averaging rules.

• Introducing the identity symbol 0 increases the number of deductions in both cases.

• There are fewer Euclidean approximations using projection-based directions:

‣ 56 approximations using cones ‣ 32 approximations using projections

Assessment

22

Assessment

22

• Both theoretical systems were implemented and compared with actual results to assess accuracy:

Assessment

22

• Both theoretical systems were implemented and compared with actual results to assess accuracy:‣ Cone-shaped directions correct in 25% of cases.

Assessment

22

• Both theoretical systems were implemented and compared with actual results to assess accuracy:‣ Cone-shaped directions correct in 25% of cases.‣ Projection-based directions correct in 50% of

cases.

Assessment

22

• Both theoretical systems were implemented and compared with actual results to assess accuracy:‣ Cone-shaped directions correct in 25% of cases.‣ Projection-based directions correct in 50% of

cases. - 1/4 turn off in only 2% of cases

Assessment

22

• Both theoretical systems were implemented and compared with actual results to assess accuracy:‣ Cone-shaped directions correct in 25% of cases.‣ Projection-based directions correct in 50% of

cases. - 1/4 turn off in only 2% of cases- deviations in remaining 48% never greater

than 1/8 turn

Assessment

22

• Both theoretical systems were implemented and compared with actual results to assess accuracy:‣ Cone-shaped directions correct in 25% of cases.‣ Projection-based directions correct in 50% of

cases. - 1/4 turn off in only 2% of cases- deviations in remaining 48% never greater

than 1/8 turn• Projection-based directions produce a result that is

within 45˚ of actual values in 80% of cases.

Research envisioned

23

Research envisioned

Formalization of other large-scale spatial relations using similar methods:

23

Research envisioned

Formalization of other large-scale spatial relations using similar methods:

• Qualitative reasoning with distances

23

Research envisioned

Formalization of other large-scale spatial relations using similar methods:

• Qualitative reasoning with distances

• Integrated reasoning about distances and directions

23

Research envisioned

Formalization of other large-scale spatial relations using similar methods:

• Qualitative reasoning with distances

• Integrated reasoning about distances and directions

• Generalize distance and direction relations to extended objects

23

Conclusion

24

Conclusion• Qualitative spatial reasoning is crucial for

progress in GIS.

24

Conclusion• Qualitative spatial reasoning is crucial for

progress in GIS.

• A system of qualitative spatial reasoning with cardinal directions can be formalized using an algebraic approach.

24

Conclusion• Qualitative spatial reasoning is crucial for

progress in GIS.

• A system of qualitative spatial reasoning with cardinal directions can be formalized using an algebraic approach.

• Similar techniques should be applied to other types of spatial reasoning.

24

Conclusion• Qualitative spatial reasoning is crucial for

progress in GIS.

• A system of qualitative spatial reasoning with cardinal directions can be formalized using an algebraic approach.

• Similar techniques should be applied to other types of spatial reasoning.

• Accuracy cannot be found in a single method.

24

25

Subjective impact

A new sidewalk decal designed to help pedestrians find their way in New York City.

26

Questions?

Qualitative Spatial Reasoning: Cardinal Directions as an Example

Andrew U. Frank1995