1. Topological relations from metric refinements Egenhofer, M. J. and Dube, M. P ACM SIGSPATIAL GIS '09 Presenter : Murat Demiray Narrative : Mustafa Karamavus

2. Outline Motivation Problem Statement Challenges Contributions Key Concepts Conclusion

3. Motivation In GIS, understanding the spatial relationship (topology) between spatial objects is very important for spatial reasoning. The spatial properties of a topology doesn’t change under continuous deformations of objects, such as twisting, stretching or shrinking.) e.g. the relation, neighborhood (meet) of the countries. linkage between topology and metric has not yet been established sufficiently Topology matters. Why ? - People easily understand - Mostly queried in SDBMS Metric matters. Why ? - Topological properties can be induced by calculation from metric space

4. Problem Statement: Can eight topological relations between two spatial objects be derived uniquely from a combination of metric properties? Input: 11 metric refinements that apply to two spatial objects and eight topological relations between two spatial objects Output: minimum set of refinements specifications from which eight topological relations between two spatial objects can be uniquely derived Objective: find one or more refinement specifications from which eight topological relations can be uniquely derived. Constraints: spatial regions (a continuous boundary, no holes, no spikes, no cuts) in R 2 topological relations : disjoint, meet, overlap, equal, coveredby, inside, covers, contains

5. Challenges 11 metrics and many refinement specifications of these metrics. metric space based on the notion of distance and the distance changes under continuous deformations of objects it is needed to determine metric refinement specifications which don’t change changes under continuous deformations of objects. one or more refinement specifications can apply one or more topological relations. objective is to find the smallest set of refinement specifications from which eight topological relations can be uniquely derived.

6. Contributions A comprehensive set of eleven metrics (9 splitting measure and 2 closeness measures) Metric refinement specifications that apply to eight topological relations Pruning to the set of metric refinement specifications from which eight topological relations can be derived uniquely Eliminating the redundancy in the pruned set and pruning for the smallest set of metric refinement specifications from which eight topological relations can be derived uniquely

7. Key Concepts (reminder) Assumptions : - two spatial regions in R 2 -no spikes, no holes, no cuts The conceptual neighborhood

8. Key Concepts (Metric Refinements) Each metric is standardized

9. Key Concepts (mapping to metrics) specification of a metric. - e.g. IAS value for disjoint 0 since A 0 ∩ B 0 = Ø

10. Key Concepts (Refinement Specifications) Can topological relation can be uniquely derived from specifications? Exercise : Find another intersection that uniquely determines disjoint.

11. Key Concepts (Refinement Specifications) Can topological relation can be uniquely derived from specifications?

12. Eliminating Redundancies Two refinement specifications are redundant if they apply same topological relation The smallest set of refinement specifications necessary to infer all 8 topological relations Equal and meet from 1 ref spec coveredBy and covers from 3 ref specs Other four from 2 ref specs

13. Conclusion Pros A systematic metric model From metric to topology Pruning to find minimum required refinement specifications Cons Motivation needs to be extended for clarification Limited background information

14. Questions