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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
Topological Relations from Metric Refinements

Max J. Egenhofer &

Matthew P. Dube

ACM SIGSPATIAL GIS 2009 – Seattle, WA

the metric world
The Metric World…
  • How many?
  • How much?
the not so metric world
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
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
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 $64,000 Question:
  • The reverse has not been investigated:
  • Can metric properties tell us anything about the spatial configuration of objects?
importance
Importance?
  • Why is this an important concern?
    • Instrumentation
    • Sensor Systems
    • Databases
    • Programming
neighborhood graphs

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

Inner Area Splitting

A

B

outer area splitting
Outer Area Splitting

Outer Area Splitting

A

B

outer area splitting inverse
Outer Area Splitting Inverse

A

B

Outer Area Splitting Inverse

exterior splitting
Exterior Splitting

Exterior Splitting

A

B

inner traversal splitting
Inner Traversal Splitting

Inner Traversal Splitting

A

B

outer traversal splitting
Outer Traversal Splitting

Outer Traversal Splitting

A

B

alongness splitting
Alongness Splitting

A

B

Alongness Splitting

inner traversal splitting inverse
Inner Traversal Splitting Inverse

Inner Traversal Splitting Inverse

A

B

outer traversal splitting inverse
Outer Traversal Splitting Inverse

Outer Traversal Splitting Inverse

A

B

splitting metrics
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 opportunity1
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
Closeness Metrics

Expansion Closeness

Contraction Closeness

dependencies
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
disjoint

ITS = 0

ITS-1 = 0

OAS-1, OTS-1 = 1

OAS, OTS = 1

IAS = 0

AS = 0

ES = 0

key questions
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
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
Redundancies
  • Are there any redundancies that can be exploited?
  • Utilize the process of subsumption
  • Construct Hasse Diagrams
meet hasse diagram
meet Hasse Diagram

Redundant Information

Specificity of refinement: Low at top; high at bottom

Explicit Definition

fewest refinements
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
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
key questions answered
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.
how can this be used
How can this be used?

spherical relations

metric composition

sensor informatics

3D worlds

sketch to speech

questions
Questions?

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

National Geospatial Intelligence Agency

National Science Foundation