ACM GIS 2011
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ACM GIS 2011

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Frank used these slides during his talk in Chicago....

Frank used these slides during his talk in Chicago.

Paper:
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 ACM GIS 2011 Presentation Transcript

  • 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
  • Problem: direction between regionsK. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  • Problem: direction between regionsK. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  • 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
  • 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
  • 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
  • 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
  • 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
  • Problem: direction between regions Matrix: ????????K. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  • Directional relations are subjective North or northwest?K. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  • 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
  • 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
  • Criteria for directional relations AreaK. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  • Criteria for directional relations Area AlignmentK. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  • Criteria for directional relations Area Alignment Removal directionK. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  • 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
  • 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
  • A splitting line for directional relationsK. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • Future Work • 8 directions vs many directionsK. Buchin, V. Kusters, B. Speckmann, F. Staals, B. Vasilescu A splitting line model for directional relations
  • 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
  • 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
  • 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