ACM GIS 2011

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Frank used these slides during his talk in Chicago.

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Buchin K, Kusters V, Speckmann B, Staals F and Vasilescu B (2011), "A splitting line model for directional relations", In Proceedings of the 19th SIGSPATIAL International Conference on Advances in Geographic Information Systems. New York, NY, USA ACM.

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ACM GIS 2011

  1. 1. A splitting line model for directional relations Kevin Buchin1 Vincent Kusters2 Bettina Speckmann1 Frank Staals3 Bogdan Vasilescu1 1 Eindhoven University of Technology 2 ETH Z¨rich u 3 Utrecht University November 3, 2011K. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  2. 2. Problem: direction between regionsK. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  3. 3. Problem: direction between regionsK. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  4. 4. Problem: direction between regions Where is New York with respect to Illinois?K. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  5. 5. Problem: direction between regions Where is target polygon B with respect to reference polygon A?K. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  6. 6. Problem: direction between regions Centroids: B is northeast of A.K. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  7. 7. Problem: direction between regions Centroids: B is southwest of A.K. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  8. 8. Problem: direction between regions Matrix: B is south of A.K. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  9. 9. Problem: direction between regions Matrix: ????????K. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  10. 10. Directional relations are subjective North or northwest?K. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  11. 11. Directional relations are subjective North or northwest? North, northeast, or east?K. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  12. 12. Directional relations are subjective North or northwest? North, northeast, or east? ????K. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  13. 13. Criteria for directional relations AreaK. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  14. 14. Criteria for directional relations Area AlignmentK. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  15. 15. Criteria for directional relations Area Alignment Removal directionK. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  16. 16. Criteria for directional relations Area Alignment Removal direction RobustnessK. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  17. 17. Criteria for directional relations Area Alignment Removal direction Robustness Affine transformationK. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  18. 18. A splitting line for directional relationsK. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  19. 19. A splitting line for directional relations Our Approach: Compute the best splitting line that separates A and B.K. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  20. 20. A splitting line for directional relations Our Approach: Compute the best splitting line that separates A and B. Question: What does it mean to be the best splitting line?K. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  21. 21. Measuring the quality of a splitting line 1 Divide the scene in slabs A 2 Compute the quality of each slab. BK. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  22. 22. Measuring the quality of a splitting line 1 Divide the scene in slabs A 2 Compute the quality of each slab. B ∞ Mline (y ) = M(x, y )dx −∞K. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  23. 23. The slab measure M User definable parameters. M = ρ1 · f (B) · GoodA + ρ1 · g (A) · GoodB − ρ2 · h(B) · AlignmentA − ρ3 · ObstructA − ρ3 · ObstructBK. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  24. 24. The slab measure M Measures the amount of A on M = ρ1 · f (B) · GoodA + the A-side. ρ1 · g (A) · GoodB − ρ2 · h(B) · AlignmentA − A ρ3 · ObstructA − ρ3 · ObstructB B criterion: Area height(a) GoodA = area(A) a∈TopAK. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  25. 25. The slab measure M Measures the amount of A on M = ρ1 · f (B) · GoodA + the A-side. ρ1 · g (A) · GoodB − ρ2 · h(B) · AlignmentA − ρ3 · ObstructA − ρ3 · ObstructB criterion: Area height(a) GoodA = area(A) a∈TopAK. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  26. 26. The slab measure M Measures the amount of B on M = ρ1 · f (B) · GoodA + the B-side. ρ1 · g (A) · GoodB − ρ2 · h(B) · AlignmentA − ρ3 · ObstructA − ρ3 · ObstructB criterion: AreaK. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  27. 27. The slab measure M Measures the alignment of M = ρ1 · f (B) · GoodA + with A. ρ1 · g (A) · GoodB − ρ2 · h(B) · AlignmentA − ρ3 · ObstructA − ρ3 · ObstructB criterion: AlignmentK. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  28. 28. The slab measure M Measures the ease with which M = ρ1 · f (B) · GoodA + A can be moved away from B. ρ1 · g (A) · GoodB − ρ2 · h(B) · AlignmentA − ρ3 · ObstructA − ρ3 · ObstructB criterion: Removal direction.K. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  29. 29. The slab measure M Measures the ease with which M = ρ1 · f (B) · GoodA + B can be moved away from A. ρ1 · g (A) · GoodB − ρ2 · h(B) · AlignmentA − ρ3 · ObstructA − ρ3 · ObstructB criterion: Removal direction.K. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  30. 30. The slab measure M Weighing functions: M = ρ1 · f (B) · GoodA + ρ1 · g (A) · GoodB − ρ2 · h(B) · AlignmentA − ρ3 · ObstructA − ρ3 · ObstructB criterion: AreaK. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  31. 31. The slab measure M Weighing functions: M = ρ1 · f (B) · GoodA + ρ1 · g (A) · GoodB − ρ2 · h(B) · AlignmentA − ρ3 · ObstructA − ρ3 · ObstructB criterion: AreaK. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  32. 32. The slab measure M Weighing functions: M = ρ1 · f (B) · GoodA + ρ1 · g (A) · GoodB − ρ2 · h(B) · AlignmentA − ρ3 · ObstructA − ρ3 · ObstructB criterion: AreaK. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  33. 33. The slab measure M M = ρ1 · f (B) · GoodA + ρ1 · g (A) · GoodB − ρ2 · h(B) · AlignmentA − ρ3 · ObstructA − ρ3 · ObstructB How about Robustness and Affine transformation?K. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  34. 34. ObstructB A B height(a) height(b) a∈{s|s∈TopA∧b>s} ObstructB = · area(B) height(a) b∈TopB a∈TopAK. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  35. 35. ObstructB A B height(a) height(b) a∈{s|s∈TopA∧b>s} ObstructB = · area(B) height(a) b∈TopB a∈TopAK. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  36. 36. AlignmentA A B height(a) height(w ) a∈{s|s∈TopA∧w <s} AlignmentA = · height(s) area(A) w ∈TopW ∪TopB s∈StripK. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  37. 37. AlignmentA A B height(a) height(w ) a∈{s|s∈TopA∧w <s} AlignmentA = · height(s) area(A) w ∈TopW ∪TopB s∈StripK. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  38. 38. An algorithm for Mline d Observation: Mline (y ) maximal if dy Mline (y ) = 0. Algorithm Sweep downwards and compute a description of Mline and its derivative.K. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  39. 39. Computing GoodA GoodA(y , x)dx is a piecewise quadratic function in y : y0 y1K. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  40. 40. Computing GoodA GoodA(y , x)dx is a piecewise quadratic function in y : y0 y1 Maintain set P of trapezoids intersected by . P changes at most n times =⇒ O(n log n) to compute GoodA.K. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  41. 41. Computing ObstructB ObstructB(y , x)dx is the sum of rational functions in y : hB (x) y0 y1 Maintain hB (x): the amount of B above the sweep line.K. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  42. 42. Computing ObstructB ObstructB(y , x)dx is the sum of rational functions in y : hB (x) y0 y1 Maintain hB (x): the amount of B above the sweep line. O(n2 ) events =⇒ O(n2 log n) to compute ObstructB.K. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  43. 43. An algorithm for directional relationsCompute the optimal splitting line for eachdirection and pick the best one.K. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  44. 44. An algorithm for directional relationsCompute the optimal splitting line for eachdirection and pick the best one.Question: Which directions should we use?K. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  45. 45. An algorithm for directional relationsCompute the optimal splitting line for eachdirection and pick the best one.Question: Which directions should we use? • Use the 8 compass directions N,NE,E,..,NWK. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  46. 46. An algorithm for directional relationsCompute the optimal splitting line for eachdirection and pick the best one.Question: Which directions should we use? • Use the 8 compass directions N,NE,E,..,NW • Use k > 8 directions (for example k = 360)K. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  47. 47. An algorithm for directional relationsCompute the optimal splitting line for eachdirection and pick the best one.Question: Which directions should we use? B • Use the 8 compass directions N,NE,E,..,NW • Use k > 8 directions (for example k = 360) AK. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  48. 48. An algorithm for directional relationsCompute the optimal splitting line for eachdirection and pick the best one.Question: Which directions should we use? B • Use the 8 compass directions N,NE,E,..,NW • Use k > 8 directions (for example k = 360) AK. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  49. 49. An algorithm for directional relationsCompute the optimal splitting line for eachdirection and pick the best one.Question: Which directions should we use? B • Use the 8 compass directions N,NE,E,..,NW • Use k > 8 directions (for example k = 360) AK. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  50. 50. Optimal splitting lines using 360 directions AL → MK BA → ME BG → MK CZ → PL DE → PL FR → BE LV → LT MD → RO NL → BE NL → DE RO → RS RS → BA SI → AT SK → CZ SK → HUK. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  51. 51. Splitting lines for directional relationsSplittingLine Matrix Centroids SplittingLine Matrix Centroids no result E SW W W NW no result NW N N NW SEK. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  52. 52. Splitting lines for directional relations AT → DE SK → CZ NL → BE LT → LV BE → LUSplittingLine NW NW S N SE Centroids NW W SW NE SE Matrix N NW S N - BY → LV BY → UA GR → MK RO → HU CZ → DESplittingLine NW S N NW W Centroids NW SE NW W W Matrix N SE - W NW K. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  53. 53. Future Work • 8 directions vs many directionsK. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  54. 54. Future Work • 8 directions vs many directions • Non-linear separator (e.g. curve or polyline)K. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  55. 55. Future Work • 8 directions vs many directions • Non-linear separator (e.g. curve or polyline) • Are directional relations (a)symmetric?K. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  56. 56. Future Work • 8 directions vs many directions • Non-linear separator (e.g. curve or polyline) • Are directional relations (a)symmetric? Thank you! Questions?K. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations

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