The Topology of Spatial Scenes in ℝ 2

1. The Topology of Spatial Scenes in ℝ2 Beyond Sets of Binary Relations • School of Computing and Information Science • National Center for Geographic Information and Analysis • University of Maine Joshua A. Lewis Matthew P. Dube Max J. Egenhofer

2. The Topology of Spatial Scenes in ℝ2 Beyond Sets of Binary Relations • School of Computing and Information Science • National Center for Geographic Information and Analysis • University of Maine

3. DC EC PO EQ disjoint meet overlap equal Egenhofer Franzosa TPP NTPP TPPi NTPPi Randell Cui Cohn coveredBy inside covers contains Topological Relations Between Two Regions in ℝ2 B A 4-intersection RCC-8 3

4. Li Schneider Behr A simple region holed region holed and separated region separated region Objects of Interest

5. in sensor networks in spatial databases in sketches Collection of Regions

6. Collection of Regions … or a united region with a hole?

7. The Topological Relations surroundedBy vs. outside in ℝ2 Vasardani Egenhofer

8. DE CH IT SM VA Part of Europe

9. Part of Europe

10. DE LI AU FR CH IT Part of Europe

11. DE LI AU FR CH IT Border Table neighbors

12. Border Table DE LI AU FR CH IT separated

13. Border Table DE LI AU FR CH IT equal

14. DE LI AU FR CH IT Border Table equal separated neighbors

15. mt mt di di mt DE mt eq mt di mt di mt mt eq mt mt mt LI AU FR CH di di mt eq mt di IT di mt mt mt eq mt mt di mt di mt eq Scene Description with 4-Intersection equal disjoint meet

16. mt mt di di mt DE DE LI mt eq mt di mt di FR AU mt mt eq mt mt mt LI AU FR CH di di mt eq mt di IT CH IT di mt mt mt eq mt mt di mt di mt eq Scene Description produces produces but also produces

17. To Capture a Spatial Scene Completely • Account for placing objects within or outside of a collection of objects • Account for the boundary sequence interactions

18. MapTree Model side of the exterior side of the interior sequence of boundary parts, but no interactions Side of the boundary Interactions of boundaries Worboys

19. Modes of Overlap • objects have intersected (x > 0) crossing has occurred (a > 1 and b ≥ 1 or a ≥ 1 and b > 1) boundaries of a pair of objects segment their common exterior (o > 1), either touch or cross has occurred despite being able to show that interactions have happened at some point, the overlap matrix cannot discern the order in which types of boundary interactions occur Galton

20. Modes of Overlap • objects have intersected • crossings • exterior segmentation • no order in which types of boundary interactions occur Galton

21. To Capture a Spatial Scene Completely • Account for placing objects within or outside of a collection of objects • Account for the boundary sequence interactions Hull+o

22. [A], the topological hull of A closed set, with path-connected interior in ℝ2 A: [A]: • the smallest closed set, homeomorphic to a 2-disk such that A ⊆ [A]

23. [A], the topological hull of A closed set, with path-connected interior in ℝ2 A: [A]: • the smallest closed set, homeomorphic to a 2-disk such that A ⊆ [A]

24. [A], the topological hull of A closed set, with path-connected interior in ℝ2 A: [A]: • the smallest closed set, homeomorphic to a 2-disk such that A ⊆ [A]

25. closed set in ℝ2 path-connected subsets of B such that ⋃p∈P = B° and |P| is minimized ⋃p∈P [p] B: [B]: P: [B], the aggregate topological hull of B

26. Considering Boundary: Dimension ND SD CO AZ

27. Considering Boundary: Touch vs. Cross cv cv Arizona vs. Colorado N/S Highway vs. E/W Highway Maine vs. New Hampshire I-95 vs. ME-15 33 Egenhofer Franzosa

28. Clementini Di Felice Considering Boundary: Touch vs. Cross 0-touch 0-cross 1-touch 1-cross 34

29. Considering Boundary: Sequence cross cross touch cross cross cross touch cross cross cross touch cross cross cross cross cross cross cross cross touch cross cross cross touch cross Egenhofer Franzosa

30. B C A Spatial Scene

31. B C A o-Notation o{B,C}(0,ø,{B}) o{C}(1,{C},ø) o{C}(0,{C},{B}) o{B,C}(1,{C},ø) o{A,C}(0,ø,{C}) o{A}(0,ø,{A,C}) o{C}(0,ø,{A}) o{A,B}(1,{A},ø) o{A,B}(0,{A},{B}) o{A}(1,{A},ø) o{A}(0,ø,{B})

32. o-Notation for Binary Relations

33. A A • C • C • B • B Insufficiency of o-Notation o-notation does not account for diverging boundary on a touch configuration renders these scenes identical

34. A A • C • C • B • B Insufficiency of o-Notation A ∪ B [A ∪ B] [A ∪ B] A ∪ B coveredBy meet meet [p1] [A ∪ B]/(A ∪ B) [p2] [A ∪ B]/(A ∪ B)

35. Insufficiency of Hull o{B,C}(1,{B},ø) o{B,C}(0,{C},ø) o{B,C}(0,{B},ø) o{B,C}(1,{C},ø) hull does not account for dimensionof an intersection renders these scenes identical

36. Europe Revisited ` Switzerland inside the hull of its neighbors Switzerland meets the hull of its neighbors

37. DE LI FR AU IT CH Europe Revisited One 1-dimension outside touches between Switzerland and Liechtenstein Two 1-dimension outside touches between Switzerland and Liechtenstein

38. Summary • Hull and o-notation developed for Spatial Scenes • Capable of capturing topology of Spatial Scenes • Capable of reverse engineering spatial scenes • Some cases need hull and o, some require only one • hull+o as a unifying measure for the topology of scenes and binary relations

39. Future Work • Normalizeo-notation • Compare with MapTree’s expressive power • Adapt hull+o to nonareal objects, such as lines and points • Create an algorithmically definable surrounds relation • Implement hull+o as a Scene Topology evaluator • Employhull+o as a scene similarity calculator

40. Topologically Truthful Sketch manually hull+o notation automatically hull+o notation Vasardani

42. The Topology of Spatial Scenes in R2