Topological Relations from Metric Refinements
Max J. Egenhofer &Matthew P. Dube
ACM SIGSPATIAL GIS 2009 – Seattle, WA
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
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
d
m
o
cB cv
cti e
Inner Traversal Splitting Inverse
B
A
Inner Traversal Splitting Inverse
€
ITS-1=length(A°∩∂B)length(∂A)
Outer Traversal Splitting Inverse
B
A
Outer Traversal Splitting Inverse
€
OTS−1 =length(A− ∩ ∂B)
length(∂A)
Splitting Metrics
B
A
Inner Area Splitting
Inner Traversal Splitting
Outer Area Splitting
Alongness Splitting
Outer Area Splitting Inverse
Inner Traversal Splitting Inverse
Outer Traversal Splitting
Outer Traversal Splitting Inverse
Exterior Splitting
Refinement Opportunity
B InteriorB
BoundaryB 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
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
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 ES = 0 Dependency
Sample method for inside = Possible = Not Possible
Redundancies
Are there any redundancies that can be exploited?
Utilize the process of subsumption
Construct Hasse Diagrams
meet Hasse DiagramSpeci
fici
ty o
f re
finem
en
t:
Low
at
top;
hig
h a
t bott
om
Redundant Information
Explicit Definition
Fewest Refinements
Minimal set of refinements for the eight simple region-region relations:
IAS = 0
0 < IAS < 1
IAS = 1
OTS-1 = 0
0 < OTS-1
EC = 0
0 < EC < 1
CC = 0
0 < CC < 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
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
metric compositio
n
sensor informati
cs
3D worlds
sketch to
speech