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

OAS
-1
=
area(A  B)
area(A )
Exterior Splitting
Exterior
Splitting
a rea(b o u n d ed
(A   B ) )
A
ES =
a rea(A)
B

Inner Traversal Splitting
Inner Traversal
Splitting
A
ITS =
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
-1
ITS =
B

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