Topological Relations from Metric Refinements
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Transcript 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…
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
9-Intersection Matrix
B Interior
A Interior
A Boundary
A Exterior
B Boundary
B Exterior
Neighborhood Graphs
d
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
m
o
cB
i
meet
cv
e
ct
overlap
Relevant Metrics
A
B
Inner Area Splitting
Inner Area
Splitting
A
IAS
B
area (A B)
area (A )
Outer Area Splitting
Outer Area
Splitting
A
OAS
area(A B )
area(A)
B
Outer Area Splitting
Inverse
A
B
Outer Area
Splitting
Inverse
-1
OAS
=
area(A B)
area(A )
Exterior Splitting
Exterior
Splitting
area(bounded(A B ))
A
ES =
area(A)
B
Inner Traversal Splitting
Inner Traversal
Splitting
A
IT S =
B
length(A B)
length(A )
Outer Traversal Splitting
Outer
Traversal
Splitting
A
OTS =
length(A B )
length(A)
B
Alongness Splitting
A
B
Alongness
Splitting
AS =
length(A B)
length(A)
Inner Traversal Splitting
Inverse
Inner Traversal
Splitting
Inverse
A
ITS
B
-1
=
length(A B )
length( A )
Outer Traversal Splitting
Inverse
Outer
Traversal
Splitting
Inverse
A
length(A
B)
1
OTS
length(A)
B
Splitting Metrics
Exterior
Splitting
Outer Area
Splitting
Outer
Traversal
Splitting
Alongness
Splitting
Inner Traversal
Splitting
Inverse
Inner Traversal
Splitting
A
B
Outer Area
Splitting
Inverse
Outer
Traversal
Splitting
Inverse
Inner Area
Splitting
Refinement Opportunity
B Interior
B Boundary
B Exterior
A Interior
IAS
ITS-1
OAS
A Boundary
ITS
AS
OTS
A Exterior
OAS-1
OTS-1
ES
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
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
OAS, OTS = 1
ITS-1 = 0
OAS-1, OTS-1 = 1
IAS = 0
AS = 0
ES = 0
Inner Traversal Splitting
0
0
(0,1)
(0,1]
1
0
0
0
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?
Combined Approach
Find values of metrics relevant for a
topological relation
Find which relations satisfy that particular
value for that particular metric
Combine information
IAS = 1
ITS-1 = 0
OAS = 0
0 < EC < 1
ITS = 1
AS = 0
OTS = 0
CC = 0
0 < EC < 1
&
OTS = 0
0 < OAS-1
0 < OTS-1
Sample method for inside
ES = 0
= Possible
Dependency
= Not Possible
Redundancies
Are there any redundancies that can be
exploited?
Utilize the process of subsumption
Construct Hasse Diagrams
meet Hasse Diagram
Specificity of refinement: Low at
top; high at bottom
Redundant
Information
Explicit
Definition
Hasse Diagrams
disjoint
meet
overlap
equal
coveredBy
inside
covers
contains
Fewest Refinements
Minimal set of refinements for the eight
simple region-region relations:
OTS-1 = 0
0 < OTS-1
IAS = 0
0 < IAS < 1
IAS = 1
EC = 0
ITS = 0
0 < EC < 1
AS < 1
CC = 0
0 < CC < 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
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?
spherical
relations
3D
worlds
metric
composition
sensor
informatics
sketch to
speech
Questions?
I will now attempt to provide some
metrics or topologies to your queries!
National Geospatial Intelligence Agency
National Science Foundation