Successfully reported this slideshow.
We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime.
Qualitative Spatial Reasoning:
Cardinal Directions as an Example

         Andrew U. Frank
              1995
Outline




          2
Outline
•   Introduction




                   2
Outline
•   Introduction
•   Motivation: Why qualitative? Why cardinal?




                                              ...
Outline
•   Introduction
•   Motivation: Why qualitative? Why cardinal?
•   Method: An algebraic approach




            ...
Outline
•   Introduction
•   Motivation: Why qualitative? Why cardinal?
•   Method: An algebraic approach
•   Two cardinal...
Outline
•   Introduction
•   Motivation: Why qualitative? Why cardinal?
•   Method: An algebraic approach
•   Two cardinal...
Outline
•   Introduction
•   Motivation: Why qualitative? Why cardinal?
•   Method: An algebraic approach
•   Two cardinal...
Outline
•   Introduction
•   Motivation: Why qualitative? Why cardinal?
•   Method: An algebraic approach
•   Two cardinal...
Outline
•   Introduction
•   Motivation: Why qualitative? Why cardinal?
•   Method: An algebraic approach
•   Two cardinal...
Introduction
Geography utilizes large scale spatial reasoning
  extensively.
                        •
Formalized qualitat...
Motivation: Why qualitative?
Spatial relations are typically formalized in a
  quantitative manner with Car tesian
  coord...
Motivation: Why qualitative?




                               5
Motivation: Why qualitative?




                               5
Motivation: Why qualitative?




                               5
Motivation: Why qualitative?


                       “thirteen centimeters”




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




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




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

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

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

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

                              ...
Motivation: Why qualitative?




                               7
Motivation: Why qualitative?
• For malization required for GIS
 implementation.




                                    7
Motivation: Why qualitative?
• For malization required for GIS
 implementation.

• Interpretation of spatial relations
 ex...
Motivation: Why qualitative?
• For malization required for GIS
 implementation.

• Interpretation of spatial relations
 ex...
Motivation: Why cardinal?




                            8
Motivation: Why cardinal?
Pullar and Egenhofer’s geographical scale
spatial relations (1988):




                        ...
Motivation: Why cardinal?
Pullar and Egenhofer’s geographical scale
spatial relations (1988):
   • direction north, northw...
Motivation: Why cardinal?
Pullar and Egenhofer’s geographical scale
spatial relations (1988):
   • direction north, northw...
Motivation: Why cardinal?
Pullar and Egenhofer’s geographical scale
spatial relations (1988):
   • direction north, northw...
Motivation: Why cardinal?
Pullar and Egenhofer’s geographical scale
spatial relations (1988):
   •   direction north, nort...
Motivation: Why cardinal?
Pullar and Egenhofer’s geographical scale
spatial relations (1988):
   •   direction north, nort...
Motivation: Why cardinal?
Pullar and Egenhofer’s geographical scale
spatial relations (1988):
   •   direction north, nort...
Method: An algebraic approach




                                9
Method: An algebraic approach
• Focus on not on directional relations
  between points...




                            ...
Method: An algebraic approach
• Focus on not on directional relations
  between points...
• Find rules for manipulating di...
Method: An algebraic approach
• Focus on not on directional relations
  between points...
• Find rules for manipulating di...
Method: An algebraic approach
• Focus on not on directional relations
  between points...
• Find rules for manipulating di...
Method: An algebraic approach
Inverse


Composition




Identity
                                10
Method: An algebraic approach
Inverse           P2


           P1

Composition




Identity
                             ...
Method: An algebraic approach
Inverse                  P2
            dir(P1,P2)

           P1

Composition




Identity
...
Method: An algebraic approach
Inverse                          P2
            dir(P1,P2)
                         inv(dir(...
Method: An algebraic approach
Inverse                          P2
            dir(P1,P2)
                         inv(dir(...
Method: An algebraic approach
Inverse                          P2
            dir(P1,P2)
                         inv(dir(...
Method: An algebraic approach
Inverse                          P2
            dir(P1,P2)
                         inv(dir(...
Method: An algebraic approach
Inverse                          P2
            dir(P1,P2)
                         inv(dir(...
Method: An algebraic approach
Inverse                           P2
            dir(P1,P2)
                          inv(di...
Method: An algebraic approach
Inverse                          P2
            dir(P1,P2)
                         inv(dir(...
Method: Euclidean exact reasoning




                                    11
Method: Euclidean exact reasoning
• Comparison between qualitative reasoning
  and quantitative reasoning using analytical...
Method: Euclidean exact reasoning
• Comparison between qualitative reasoning
  and quantitative reasoning using analytical...
Method: Euclidean exact reasoning
• Comparison between qualitative reasoning
  and quantitative reasoning using analytical...
Two cardinal system examples
    Cone-shaped              Projection-based
          N
 NW             NE            NW   ...
Directions in cones
       N
NW         NE

W               E


SW         SE
       S




                          13
Directions in cones
       N            • Angle assigned to nearest
NW         NE         named direction
                ...
Directions in cones
       N
NW         NE

W               E


SW         SE
       S




                          13
Directions in cones
       N
NW         NE

W               E


SW         SE
       S




                          14
Directions in cones
       N            Algebraic operations can be
NW         NE       performed with symbols:

W        ...
Directions in cones
       N            Algebraic operations can be
NW         NE       performed with symbols:
          ...
Directions in cones
       N            Algebraic operations can be
NW         NE       performed with symbols:
          ...
Directions in cones
       N            Algebraic operations can be
NW         NE       performed with symbols:
          ...
Directions in cones
       N            Algebraic operations can be
NW         NE       performed with symbols:
          ...
Directions in cones
       N            Algebraic operations can be
NW         NE       performed with symbols:
          ...
Directions in cones
       N            Algebraic operations can be
NW         NE       performed with symbols:
          ...
Directions in cones
       N            Algebraic operations can be
NW         NE       performed with symbols:
          ...
Directions in cones
       N            Algebraic operations can be
NW         NE       performed with symbols:
          ...
Directions in cones
       N            Algebraic operations can be
NW         NE       performed with symbols:
          ...
Directions in cones
       N            Algebraic operations can be
NW         NE       performed with symbols:
          ...
Directions in cones
       N            Algebraic operations can be
NW         NE       performed with symbols:
          ...
Directions in cones
       N             Algebraic operations can be
NW          NE       performed with symbols:
        ...
Directions in cones
       N             Algebraic operations can be
NW          NE       performed with symbols:
        ...
Directions in cones
       N             Algebraic operations can be
NW          NE       performed with symbols:
        ...
Directions in cones
       N             Algebraic operations can be
NW          NE       performed with symbols:
        ...
Directions in cones
       N             Algebraic operations can be
NW          NE       performed with symbols:
        ...
Directions in cones
       N             Algebraic operations can be
NW          NE       performed with symbols:
        ...
Directions in cones
       N             Algebraic operations can be
NW          NE       performed with symbols:
        ...
Directions in cones
       N             Algebraic operations can be
NW          NE       performed with symbols:
        ...
Directions in cones
       N             Algebraic operations can be
NW          NE       performed with symbols:
        ...
Directions in cones
       N             Algebraic operations can be
NW          NE       performed with symbols:
        ...
Cone direction composition table




                                   17
Cone direction composition table




                                   17
Cone direction composition table




Out of 64 combinations, only 10 are Euclidean exact.
                                ...
Projection-based directions




                              18
Projection-based directions

 W     E




                              18
Projection-based directions
    N


    S




                              18
Projection-based directions
NW    NE


 SW   SE




                              18
Projection-based directions
           • With half-planes, only trivial
NW    NE    cases can be resolved:
            NE ...
Projection-based directions
NW   N    NE

W    Oc   E

SW   S    SE




                              19
Projection-based directions
               • Assign neutral zone in the
NW   N    NE
                center of 9 regions
W...
Projection-based directions
               Algebraic operations can be
NW   N    NE
               performed with symbols:...
Projection-based directions
               Algebraic operations can be
NW   N    NE
               performed with symbols:...
Projection-based directions
               Algebraic operations can be
NW   N    NE
               performed with symbols:...
Projection-based directions
               Algebraic operations can be
NW   N    NE
               performed with symbols:...
Projection-based directions
               Algebraic operations can be
NW   N    NE
               performed with symbols:...
Projection-based directions
                 Algebraic operations can be
NW    N    NE
                 performed with sym...
Projection-based directions
                 Algebraic operations can be
NW    N    NE
                 performed with sym...
Projection-based directions
                 Algebraic operations can be
NW    N    NE
                 performed with sym...
Projection-based directions
                 Algebraic operations can be
NW    N    NE
                 performed with sym...
Projection-based directions
                 Algebraic operations can be
NW    N    NE
                 performed with sym...
Projection-based directions
                 Algebraic operations can be
NW    N    NE
                 performed with sym...
Projection composition table




                               20
Projection composition table




                               20
Projection composition table




Out of 64 combinations, 32 are Euclidean exact.
                                         ...
Assessment




             21
Assessment
• Both systems use 9 directional symbols.




                                            21
Assessment
• Both systems use 9 directional symbols.

• Cone-shaped system relies on averaging rules.




                ...
Assessment
• Both systems use 9 directional symbols.

• Cone-shaped system relies on averaging rules.

• Introducing the i...
Assessment
• Both systems use 9 directional symbols.

• Cone-shaped system relies on averaging rules.

• Introducing the i...
Assessment
• Both systems use 9 directional symbols.

• Cone-shaped system relies on averaging rules.

• Introducing the i...
Assessment
• Both systems use 9 directional symbols.

• Cone-shaped system relies on averaging rules.

• Introducing the i...
Assessment




             22
Assessment
• Both theoretical systems were implemented and
 compared with actual results to assess accuracy:




         ...
Assessment
• Both theoretical systems were implemented and
 compared with actual results to assess accuracy:
   ‣   Cone-s...
Assessment
• Both theoretical systems were implemented and
 compared with actual results to assess accuracy:
   ‣   Cone-s...
Assessment
• Both theoretical systems were implemented and
 compared with actual results to assess accuracy:
   ‣   Cone-s...
Assessment
• Both theoretical systems were implemented and
 compared with actual results to assess accuracy:
   ‣   Cone-s...
Assessment
• Both theoretical systems were implemented and
 compared with actual results to assess accuracy:
   ‣   Cone-s...
Research envisioned




                      23
Research envisioned
Formalization of other large-scale spatial
relations using similar methods:




                      ...
Research envisioned
Formalization of other large-scale spatial
relations using similar methods:
  • Qualitative reasoning ...
Research envisioned
Formalization of other large-scale spatial
relations using similar methods:
  • Qualitative reasoning ...
Research envisioned
Formalization of other large-scale spatial
relations using similar methods:
  • Qualitative reasoning ...
Conclusion




             24
Conclusion
• Qualitative spatial reasoning is crucial for
  progress in GIS.




                                         ...
Conclusion
• Qualitative spatial reasoning is crucial for
  progress in GIS.
• A system of qualitative spatial reasoning w...
Conclusion
• Qualitative spatial reasoning is crucial for
  progress in GIS.
• A system of qualitative spatial reasoning w...
Conclusion
• Qualitative spatial reasoning is crucial for
  progress in GIS.
• A system of qualitative spatial reasoning w...
Subjective impact




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

  Qualitative Spatial Reasoning:
Cardinal Directions as an Example

         Andrew U. Frank
              199...
Upcoming SlideShare
Loading in …5
×

Qualitative Spatial Reasoning: Cardinal Directions as an Example

1,756 views

Published on

My presentation of Dr. Frank's 1995 paper. Not suitable as a substitute for reading the original work, but the visualizations may be helpful.

Published in: Education, Technology
  • Be the first to comment

Qualitative Spatial Reasoning: Cardinal Directions as an Example

  1. 1. Qualitative Spatial Reasoning: Cardinal Directions as an Example Andrew U. Frank 1995
  2. 2. Outline 2
  3. 3. Outline • Introduction 2
  4. 4. Outline • Introduction • Motivation: Why qualitative? Why cardinal? 2
  5. 5. Outline • Introduction • Motivation: Why qualitative? Why cardinal? • Method: An algebraic approach 2
  6. 6. Outline • Introduction • Motivation: Why qualitative? Why cardinal? • Method: An algebraic approach • Two cardinal direction systems 2
  7. 7. Outline • Introduction • Motivation: Why qualitative? Why cardinal? • Method: An algebraic approach • Two cardinal direction systems - Cone-shaped directions 2
  8. 8. Outline • Introduction • Motivation: Why qualitative? Why cardinal? • Method: An algebraic approach • Two cardinal direction systems - Cone-shaped directions - Projection-based directions 2
  9. 9. Outline • Introduction • Motivation: Why qualitative? Why cardinal? • Method: An algebraic approach • Two cardinal direction systems - Cone-shaped directions - Projection-based directions • Assessment 2
  10. 10. Outline • Introduction • Motivation: Why qualitative? Why cardinal? • Method: An algebraic approach • Two cardinal direction systems - Cone-shaped directions - Projection-based directions • Assessment • Research envisioned 2
  11. 11. 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. 3
  12. 12. Motivation: Why qualitative? Spatial relations are typically formalized in a quantitative manner with Car tesian coordinates and vector algebra. 4
  13. 13. Motivation: Why qualitative? 5
  14. 14. Motivation: Why qualitative? 5
  15. 15. Motivation: Why qualitative? 5
  16. 16. Motivation: Why qualitative? “thirteen centimeters” 5
  17. 17. Motivation: Why qualitative? Human spatial reasoning is based on qualitative comparisons. 6
  18. 18. Motivation: Why qualitative? Human spatial reasoning is based on qualitative comparisons. 6
  19. 19. Motivation: Why qualitative? Human spatial reasoning is based on qualitative comparisons. “longer” 6
  20. 20. Motivation: Why qualitative? Human spatial reasoning is based on qualitative comparisons. “longer” • precision is not always desirable 6
  21. 21. Motivation: Why qualitative? Human spatial reasoning is based on qualitative comparisons. “longer” • precision is not always desirable • precise data is not always available 6
  22. 22. Motivation: Why qualitative? Human spatial reasoning is based on qualitative comparisons. “longer” • precision is not always desirable • precise data is not always available • numerical approximations do not account for uncertainty 6
  23. 23. Motivation: Why qualitative? 7
  24. 24. Motivation: Why qualitative? • For malization required for GIS implementation. 7
  25. 25. Motivation: Why qualitative? • For malization required for GIS implementation. • Interpretation of spatial relations expressed in natural language. 7
  26. 26. Motivation: Why qualitative? • For malization required for GIS implementation. • Interpretation of spatial relations expressed in natural language. • Comparison of semantics of spatial terms in different languages. 7
  27. 27. Motivation: Why cardinal? 8
  28. 28. Motivation: Why cardinal? Pullar and Egenhofer’s geographical scale spatial relations (1988): 8
  29. 29. Motivation: Why cardinal? Pullar and Egenhofer’s geographical scale spatial relations (1988): • direction north, northwest 8
  30. 30. Motivation: Why cardinal? Pullar and Egenhofer’s geographical scale spatial relations (1988): • direction north, northwest • topological disjoint, touches 8
  31. 31. Motivation: Why cardinal? Pullar and Egenhofer’s geographical scale spatial relations (1988): • direction north, northwest • topological disjoint, touches • ordinal in, at 8
  32. 32. Motivation: Why cardinal? Pullar and Egenhofer’s geographical scale spatial relations (1988): • direction north, northwest • topological disjoint, touches • ordinal in, at • distance far, near 8
  33. 33. Motivation: Why cardinal? 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 8
  34. 34. Motivation: Why cardinal? 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 Cardinal direction chosen as a major example. 8
  35. 35. Method: An algebraic approach 9
  36. 36. Method: An algebraic approach • Focus on not on directional relations between points... 9
  37. 37. Method: An algebraic approach • Focus on not on directional relations between points... • Find rules for manipulating directional symbols & operators. 9
  38. 38. Method: An algebraic approach • Focus on not on directional relations between points... • Find rules for manipulating directional symbols & operators. Directional symbols: N, S, E, W... NE, NW... Operators: inv ∞ () 9
  39. 39. Method: An algebraic approach • Focus on not on directional relations between points... • Find rules for manipulating directional symbols & operators. Directional symbols: N, S, E, W... NE, NW... Operators: inv ∞ () • Operational meaning in a set of formal axioms. 9
  40. 40. Method: An algebraic approach Inverse Composition Identity 10
  41. 41. Method: An algebraic approach Inverse P2 P1 Composition Identity 10
  42. 42. Method: An algebraic approach Inverse P2 dir(P1,P2) P1 Composition Identity 10
  43. 43. Method: An algebraic approach Inverse P2 dir(P1,P2) inv(dir(P1,P2)) P1 Composition Identity 10
  44. 44. Method: An algebraic approach Inverse P2 dir(P1,P2) inv(dir(P1,P2)) P1 Composition P2 P1 P3 Identity 10
  45. 45. Method: An algebraic approach Inverse P2 dir(P1,P2) inv(dir(P1,P2)) P1 Composition P2 dir(P1,P2) P1 P3 Identity 10
  46. 46. Method: An algebraic approach Inverse P2 dir(P1,P2) inv(dir(P1,P2)) P1 Composition P2 dir(P1,P2) dir(P2,P3) P1 P3 Identity 10
  47. 47. Method: An algebraic approach Inverse P2 dir(P1,P2) inv(dir(P1,P2)) P1 Composition P2 dir(P1,P2) dir(P2,P3) P1 P3 dir(P1,P2) ∞ dir(P2,P3) dir (P1,P3) Identity 10
  48. 48. Method: An algebraic approach Inverse P2 dir(P1,P2) inv(dir(P1,P2)) P1 Composition P2 dir(P1,P2) dir(P2,P3) P1 P3 dir(P1,P2) ∞ dir(P2,P3) dir (P1,P3) Identity P1 10
  49. 49. Method: An algebraic approach Inverse P2 dir(P1,P2) inv(dir(P1,P2)) P1 Composition P2 dir(P1,P2) dir(P2,P3) P1 P3 dir(P1,P2) ∞ dir(P2,P3) dir (P1,P3) Identity P1 dir(P1,P1)=0 10
  50. 50. Method: Euclidean exact reasoning 11
  51. 51. Method: Euclidean exact reasoning • Comparison between qualitative reasoning and quantitative reasoning using analytical geometry 11
  52. 52. 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
  53. 53. 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
  54. 54. Two cardinal system examples Cone-shaped Projection-based N NW NE NW N NE W E W Oc E SW SE SW S SE S “relative position of points “going toward” on the Earth” 12
  55. 55. Directions in cones N NW NE W E SW SE S 13
  56. 56. Directions in cones N • Angle assigned to nearest NW NE named direction • Area of acceptance increases W E with distance SW SE S 13
  57. 57. Directions in cones N NW NE W E SW SE S 13
  58. 58. Directions in cones N NW NE W E SW SE S 14
  59. 59. Directions in cones N Algebraic operations can be NW NE performed with symbols: W E SW SE S 14
  60. 60. Directions in cones N Algebraic operations can be NW NE performed with symbols: • 1/8 turn changes the symbol: W E e(N)=NE SW SE S 14
  61. 61. Directions in cones N Algebraic operations can be NW NE performed with symbols: • 1/8 turn changes the symbol: W E e(N)=NE SW SE S 14
  62. 62. Directions in cones N Algebraic operations can be NW NE performed with symbols: • 1/8 turn changes the symbol: W E e(N)=NE • 4/8 turn gives the inverse symbol: SW SE e⁴(N)= inv(N) = S S 14
  63. 63. Directions in cones N Algebraic operations can be NW NE performed with symbols: • 1/8 turn changes the symbol: W E e(N)=NE • 4/8 turn gives the inverse symbol: SW SE e⁴(N)= inv(N) = S S 14
  64. 64. Directions in cones N Algebraic operations can be NW NE performed with symbols: • 1/8 turn changes the symbol: W E e(N)=NE • 4/8 turn gives the inverse symbol: SW SE e⁴(N)= inv(N) = S S 14
  65. 65. Directions in cones N Algebraic operations can be NW NE performed with symbols: • 1/8 turn changes the symbol: W E e(N)=NE • 4/8 turn gives the inverse symbol: SW SE e⁴(N)= inv(N) = S S • 8/8 turn gives the identity symbol: e⁸(N)= N 14
  66. 66. Directions in cones N Algebraic operations can be NW NE performed with symbols: • 1/8 turn changes the symbol: W E e(N)=NE • 4/8 turn gives the inverse symbol: SW SE e⁴(N)= inv(N) = S S • 8/8 turn gives the identity symbol, 0: e⁸(N)= N = 0 15
  67. 67. Directions in cones N Algebraic operations can be NW NE performed with symbols: • 1/8 turn changes the symbol: W E e(N)=NE • 4/8 turn gives the inverse symbol: SW SE e⁴(N)= inv(N) = S S • 8/8 turn gives the identity symbol, 0: e⁸(N)= N = 0 15
  68. 68. Directions in cones N Algebraic operations can be NW NE performed with symbols: • 1/8 turn changes the symbol: W E e(N)=NE • 4/8 turn gives the inverse symbol: SW SE e⁴(N)= inv(N) = S S • 8/8 turn gives the identity symbol, 0: e⁸(N)= N = 0 15
  69. 69. Directions in cones N Algebraic operations can be NW NE performed with symbols: • 1/8 turn changes the symbol: W 0 E e(N)=NE • 4/8 turn gives the inverse symbol: SW SE e⁴(N)= inv(N) = S S • 8/8 turn gives the identity symbol, 0: e⁸(N)= N = 0 15
  70. 70. Directions in cones N Algebraic operations can be NW NE performed with symbols: • 1/8 turn changes the symbol: W E e(N)=NE • 4/8 turn gives the inverse symbol: SW SE e⁴(N)= inv(N) = S S • 8/8 turn gives the identity symbol, 0: e⁸(N)= N = 0 16
  71. 71. Directions in cones N Algebraic operations can be NW NE performed with symbols: • 1/8 turn changes the symbol: W E e(N)=NE • 4/8 turn gives the inverse symbol: SW SE e⁴(N)= inv(N) = S S • 8/8 turn gives the identity symbol, 0: e⁸(N)= N = 0 • Composition can be computed with averaging rules: 16
  72. 72. Directions in cones N Algebraic operations can be NW NE performed with symbols: • 1/8 turn changes the symbol: W E e(N)=NE • 4/8 turn gives the inverse symbol: SW SE e⁴(N)= inv(N) = S S • 8/8 turn gives the identity symbol, 0: e⁸(N)= N = 0 • Composition can be computed with averaging rules: e(N) ∞ N = n 16
  73. 73. Directions in cones N Algebraic operations can be NW NE performed with symbols: • 1/8 turn changes the symbol: W E e(N)=NE • 4/8 turn gives the inverse symbol: SW SE e⁴(N)= inv(N) = S S • 8/8 turn gives the identity symbol, 0: e⁸(N)= N = 0 • Composition can be computed with averaging rules: e(N) ∞ N = n 16
  74. 74. Directions in cones N Algebraic operations can be NW NE performed with symbols: • 1/8 turn changes the symbol: W E e(N)=NE • 4/8 turn gives the inverse symbol: SW SE e⁴(N)= inv(N) = S S • 8/8 turn gives the identity symbol, 0: e⁸(N)= N = 0 • Composition can be computed with averaging rules: e(N) ∞ N = n 16
  75. 75. Directions in cones N Algebraic operations can be NW NE performed with symbols: • 1/8 turn changes the symbol: W E e(N)=NE • 4/8 turn gives the inverse symbol: SW SE e⁴(N)= inv(N) = S S • 8/8 turn gives the identity symbol, 0: e⁸(N)= N = 0 • Composition can be computed with averaging rules: e(N) ∞ N = n 16
  76. 76. Directions in cones N Algebraic operations can be NW NE performed with symbols: • 1/8 turn changes the symbol: W E e(N)=NE • 4/8 turn gives the inverse symbol: SW SE e⁴(N)= inv(N) = S S • 8/8 turn gives the identity symbol, 0: e⁸(N)= N = 0 • Composition can be computed with averaging rules: e(N) ∞ N = n 16
  77. 77. Directions in cones N Algebraic operations can be NW NE performed with symbols: • 1/8 turn changes the symbol: W E e(N)=NE • 4/8 turn gives the inverse symbol: SW SE e⁴(N)= inv(N) = S S • 8/8 turn gives the identity symbol, 0: e⁸(N)= N = 0 • Composition can be computed with averaging rules: e(N) ∞ N = n e(N) ∞ inv (N) 16
  78. 78. Directions in cones N Algebraic operations can be NW NE performed with symbols: • 1/8 turn changes the symbol: W E e(N)=NE • 4/8 turn gives the inverse symbol: SW SE e⁴(N)= inv(N) = S S • 8/8 turn gives the identity symbol, 0: e⁸(N)= N = 0 • Composition can be computed with averaging rules: e(N) ∞ N = n e(N) ∞ inv (N) 16
  79. 79. Directions in cones N Algebraic operations can be NW NE performed with symbols: • 1/8 turn changes the symbol: W E e(N)=NE • 4/8 turn gives the inverse symbol: SW SE e⁴(N)= inv(N) = S S • 8/8 turn gives the identity symbol, 0: e⁸(N)= N = 0 • Composition can be computed with averaging rules: e(N) ∞ N = n e(N) ∞ inv (N) 16
  80. 80. Directions in cones N Algebraic operations can be NW NE performed with symbols: • 1/8 turn changes the symbol: W 0 E e(N)=NE • 4/8 turn gives the inverse symbol: SW SE e⁴(N)= inv(N) = S S • 8/8 turn gives the identity symbol, 0: e⁸(N)= N = 0 • Composition can be computed with averaging rules: e(N) ∞ N = n e(N) ∞ inv (N) 16
  81. 81. Cone direction composition table 17
  82. 82. Cone direction composition table 17
  83. 83. Cone direction composition table Out of 64 combinations, only 10 are Euclidean exact. 17
  84. 84. Projection-based directions 18
  85. 85. Projection-based directions W E 18
  86. 86. Projection-based directions N S 18
  87. 87. Projection-based directions NW NE SW SE 18
  88. 88. Projection-based directions • With half-planes, only trivial NW NE cases can be resolved: NE ∞ NE = NE SW SE 18
  89. 89. Projection-based directions NW N NE W Oc E SW S SE 19
  90. 90. Projection-based directions • Assign neutral zone in the NW N NE center of 9 regions W Oc E SW S SE 19
  91. 91. Projection-based directions Algebraic operations can be NW N NE performed with symbols: W Oc E SW S SE 19
  92. 92. Projection-based directions Algebraic operations can be NW N NE performed with symbols: W Oc E • The identity symbol, 0, resides in the neutral area. SW S SE 19
  93. 93. Projection-based directions Algebraic operations can be NW N NE performed with symbols: W Oc E • The identity symbol, 0, resides in the neutral area. SW S SE • Inverse gives the symbol opposite the neutral area: inv(N) = S 19
  94. 94. Projection-based directions Algebraic operations can be NW N NE performed with symbols: W Oc E • The identity symbol, 0, resides in the neutral area. SW S SE • Inverse gives the symbol opposite the neutral area: inv(N) = S 19
  95. 95. Projection-based directions Algebraic operations can be NW N NE performed with symbols: W Oc E • The identity symbol, 0, resides in the neutral area. SW S SE • Inverse gives the symbol opposite the neutral area: inv(N) = S 19
  96. 96. Projection-based directions Algebraic operations can be NW N NE performed with symbols: W Oc E • The identity symbol, 0, resides in the neutral area. SW S SE • Inverse gives the symbol opposite the neutral area: inv(N) = S • Composition combines each projection: 19
  97. 97. Projection-based directions Algebraic operations can be NW N NE performed with symbols: W Oc E • The identity symbol, 0, resides in the neutral area. SW S SE • Inverse gives the symbol opposite the neutral area: inv(N) = S • Composition combines each projection: NE ∞ SW = 0 19
  98. 98. Projection-based directions Algebraic operations can be NW N NE performed with symbols: W Oc E • The identity symbol, 0, resides in the neutral area. SW S SE • Inverse gives the symbol opposite the neutral area: inv(N) = S • Composition combines each projection: NE ∞ SW = 0 19
  99. 99. Projection-based directions Algebraic operations can be NW N NE performed with symbols: W Oc E • The identity symbol, 0, resides in the neutral area. SW S SE • Inverse gives the symbol opposite the neutral area: inv(N) = S • Composition combines each projection: NE ∞ SW = 0 S ∞ E = SE 19
  100. 100. Projection-based directions Algebraic operations can be NW N NE performed with symbols: W Oc E • The identity symbol, 0, resides in the neutral area. SW S SE • Inverse gives the symbol opposite the neutral area: inv(N) = S • Composition combines each projection: NE ∞ SW = 0 S ∞ E = SE 19
  101. 101. Projection-based directions Algebraic operations can be NW N NE performed with symbols: W Oc E • The identity symbol, 0, resides in the neutral area. SW S SE • Inverse gives the symbol opposite the neutral area: inv(N) = S • Composition combines each projection: NE ∞ SW = 0 S ∞ E = SE 19
  102. 102. Projection composition table 20
  103. 103. Projection composition table 20
  104. 104. Projection composition table Out of 64 combinations, 32 are Euclidean exact. 20
  105. 105. Assessment 21
  106. 106. Assessment • Both systems use 9 directional symbols. 21
  107. 107. Assessment • Both systems use 9 directional symbols. • Cone-shaped system relies on averaging rules. 21
  108. 108. Assessment • 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. 21
  109. 109. Assessment • 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: 21
  110. 110. Assessment • 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 21
  111. 111. Assessment • 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 21
  112. 112. Assessment 22
  113. 113. Assessment • Both theoretical systems were implemented and compared with actual results to assess accuracy: 22
  114. 114. Assessment • Both theoretical systems were implemented and compared with actual results to assess accuracy: ‣ Cone-shaped directions correct in 25% of cases. 22
  115. 115. Assessment • 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. 22
  116. 116. Assessment • 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 22
  117. 117. Assessment • 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 22
  118. 118. Assessment • 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. 22
  119. 119. Research envisioned 23
  120. 120. Research envisioned Formalization of other large-scale spatial relations using similar methods: 23
  121. 121. Research envisioned Formalization of other large-scale spatial relations using similar methods: • Qualitative reasoning with distances 23
  122. 122. Research envisioned Formalization of other large-scale spatial relations using similar methods: • Qualitative reasoning with distances • Integrated reasoning about distances and directions 23
  123. 123. 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
  124. 124. Conclusion 24
  125. 125. Conclusion • Qualitative spatial reasoning is crucial for progress in GIS. 24
  126. 126. 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
  127. 127. 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
  128. 128. 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
  129. 129. Subjective impact A new sidewalk decal designed to help pedestrians find their way in New York City. 25
  130. 130. Questions? Qualitative Spatial Reasoning: Cardinal Directions as an Example Andrew U. Frank 1995 26

×