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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.
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Motivation: Why qualitative?
5
Motivation: Why qualitative?
5
Motivation: Why qualitative?
5
Motivation: Why qualitative?
“thirteen centimeters”
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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”
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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”
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Motivation: Why qualitative?
7
Motivation: Why qualitative?
• Formal izat ion required for GIS implementation.
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Motivation: Why qualitative?
• Formal izat ion required for GIS implementation.
• Interpretation of spatial relations expressed in natural language.
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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
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Method: An algebraic approach• Focus on not on directional relations
between points...
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Method: An algebraic approach• Focus on not on directional relations
between points... • Find rules for manipulating directional
symbols & operators.
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
