1. Topological Relations from Metric Refinements Max J. Egenhofer & Matthew P. Dube ACM SIGSPATIAL GIS 2009 – Seattle, WA

2. The Metric World… How many? How much?

3. The Not-So-Metric World… When geometry came up short, math adapted Distance became connectivity Area and volume became containment Thus topology was born Metrics still here!

4. Interconnection Topology is an indicator of “nearness” –Open sets represent locality Metrics are measurements of “nearness” –Shorter distance implies closer objects Euclidean distance imposes a topology upon any real space R n or pixel space Z n

5. The $32,000 Question: Metrics have been used in spatial information theory to refine topological relations No different; different only in your mind! - The Empire Strikes Back Is the degree of the overlap of these objects different?

6. The $64,000 Question: The reverse has not been investigated: Can metric properties tell us anything about the spatial configuration of objects?

7. Importance? Why is this an important concern? –Instrumentation –Sensor Systems –Databases –Programming

8. 9-Intersection Matrix B InteriorB BoundaryB Exterior A Interior A Boundary A Exterior

9. Neighborhood Graphs Moving from one configuration directly to another without a different one in between Continue the process and we end up with this: disjointmeetdisjointmeetoverlap dmocBcvctie

10. Relevant Metrics B A

11. Inner Area Splitting B A

12. Outer Area Splitting B A

13. Outer Area Splitting Inverse B A

14. Exterior Splitting B A

15. Inner Traversal Splitting B A

16. Outer Traversal Splitting B A

17. Alongness Splitting B A

18. Inner Traversal Splitting Inverse B A

19. Outer Traversal Splitting Inverse B A

20. Splitting Metrics B A Inner Area Splitting Inner Traversal Splitting Outer Area Splitting Alongness Splitting Outer Area Splitting Inverse Inner Traversal Splitting Inverse Outer Traversal Splitting Outer Traversal Splitting Inverse Exterior Splitting

21. Refinement Opportunity B InteriorB BoundaryB Exterior A InteriorIASITS -1 OAS A BoundaryITSASOTS A ExteriorOAS -1 OTS -1 ES

22. Refinement Opportunity How does the refinement work in the case of a boundary? Refinement is not done by presence ; it is done by absence Consider two objects that meet at a point. Boundary/Boundary intersection is valid, yet Alongness Splitting = 0

23. Closeness Metrics Expansion Closeness Contraction Closeness

24. Dependencies Are there dependencies to be found between a well-defined topological spatial relation and its metric properties? To answer, we must look in two directions: –Topology gives off metric properties –Metric values induce topological constraints

25. disjoint ITS = 0 ITS -1 = 0 OAS, OTS = 1 OAS -1, OTS -1 = 1 IAS = 0 AS = 0 ES = 0

26. Inner Traversal Splitting 00 (0,1)(0,1] 0010

27. Key Questions: Can all eight topological relations be uniquely determined from refinement specifications? Can all eight topological relations be uniquely determined by a pair of refinement specifications, or does unique inference require more specifications? Do all eleven metric refinements contribute to uniquely determining topological relations?

28. Combined Approach Find values of metrics relevant for a topological relation Find which relations satisfy that particular value for that particular metric Combine information

29. IAS = 1ITS -1 = 0OAS = 00 < EC < 1 ITS = 1AS = 0OTS = 0CC = 0 0 < EC < 1 & OTS = 0 0 < OAS -1 0 < OTS -1 ES = 0Dependency Sample method for inside = Possible = Not Possible

30. Redundancies Are there any redundancies that can be exploited? Utilize the process of subsumption Construct Hasse Diagrams

31. meet Hasse Diagram Specificity of refinement: Low at top; high at bottom Redundant Information Explicit Definition

32. Hasse Diagrams disjointmeetoverlapequal coveredByinsidecoverscontains

33. Fewest Refinements Minimal set of refinements for the eight simple region-region relations: IAS = 0 0 < IAS < 1 IAS = 1 OTS -1 = 0 0 < OTS -1 EC = 0 0 < EC < 1 CC = 0 0 < CC < 1 ITS = 0 AS < 1

34. coveredBy Intersection of all graphs of values produces relation Can we get smaller? –Coupled with inside –Coupled with equality What metrics can strip each coupling? –EC can strip inside –ITS/AS can strip equality

35. Key Questions Answered: All eight topological relations are determined by metric refinements. covers and coveredBy require a third refinement to be uniquely identified. Some metric information is redundant and thus not necessary.

36. How can this be used? spherical relations metric composition sensor informatics 3D worlds sketch to speech

37. Questions? I will now attempt to provide some metrics or topologies to your queries! National Geospatial Intelligence Agency National Science Foundation