Topological Relations from Metric Refinements

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 Rn or pixel space Zn

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

9. cB cv m ct d o e i Neighborhood Graphs • Moving from one configuration directly to another without a different one in between • Continue the process and we end up with this: disjoint meet disjoint meet overlap

10. Relevant Metrics A B

11. Inner Area Splitting Inner Area Splitting A B

12. Outer Area Splitting Outer Area Splitting A B

13. Outer Area Splitting Inverse A B Outer Area Splitting Inverse

14. Exterior Splitting Exterior Splitting A B

15. Inner Traversal Splitting Inner Traversal Splitting A B

16. Outer Traversal Splitting Outer Traversal Splitting A B

17. Alongness Splitting A B Alongness Splitting

18. Inner Traversal Splitting Inverse Inner Traversal Splitting Inverse A B

19. Outer Traversal Splitting Inverse Outer Traversal Splitting Inverse A B

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

21. Refinement Opportunity

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, yetAlongness 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-1, OTS-1 = 1 OAS, OTS = 1 IAS = 0 AS = 0 ES = 0

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

27. Key Questions: • Can all eight topological relations be uniquely determined from refinement specifications? • Can all eight topological relations be uniquely determined by apair 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

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

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

32. Hasse Diagrams

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