Topological Relations from Metric Refinements

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# Topological Relations from Metric Refinements - PowerPoint PPT Presentation

Topological Relations from Metric Refinements. Max J. Egenhofer & Matthew P. Dube ACM SIGSPATIAL GIS 2009 – Seattle, WA. The Metric World…. How many? How much?. The Not-So-Metric World…. When geometry came up short, math adapted Distance became connectivity

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## Topological Relations from Metric Refinements

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Presentation Transcript
Topological Relations from Metric Refinements

Max J. Egenhofer &

Matthew P. Dube

ACM SIGSPATIAL GIS 2009 – Seattle, WA

The Metric World…
• How many?
• How much?
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!

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
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?
The \$64,000 Question:
• The reverse has not been investigated:
• Can metric properties tell us anything about the spatial configuration of objects?
Importance?
• Why is this an important concern?
• Instrumentation
• Sensor Systems
• Databases
• Programming

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

Inner Area Splitting

Inner Area Splitting

A

B

Outer Area Splitting

Outer Area Splitting

A

B

Outer Area Splitting Inverse

A

B

Outer Area Splitting Inverse

Exterior Splitting

Exterior Splitting

A

B

Inner Traversal Splitting

Inner Traversal Splitting

A

B

Outer Traversal Splitting

Outer Traversal Splitting

A

B

Alongness Splitting

A

B

Alongness Splitting

Inner Traversal Splitting Inverse

Inner Traversal Splitting Inverse

A

B

Outer Traversal Splitting Inverse

Outer Traversal Splitting Inverse

A

B

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

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
Closeness Metrics

Expansion Closeness

Contraction Closeness

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
disjoint

ITS = 0

ITS-1 = 0

OAS-1, OTS-1 = 1

OAS, OTS = 1

IAS = 0

AS = 0

ES = 0

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?
Combined Approach
• Find values of metrics relevant for a topological relation
• Find which relations satisfy that particular value for that particular metric
• Combine information
Redundancies
• Are there any redundancies that can be exploited?
• Utilize the process of subsumption
• Construct Hasse Diagrams
meet Hasse Diagram

Redundant Information

Specificity of refinement: Low at top; high at bottom

Explicit Definition

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

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
• 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.
How can this be used?

spherical relations

metric composition

sensor informatics

3D worlds

sketch to speech

Questions?

I will now attempt to provide some metrics or topologies to your queries!

National Geospatial Intelligence Agency

National Science Foundation