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 InteriorB

BoundaryB Exterior

A Interior

A Boundary

A Exterior

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

Relevant Metrics

B

A

Inner Area Splitting

B

A

Inner Area Splitting

€

IAS=area(A°∩B°)area(A)

Outer Area Splitting

B

A

Outer Area Splitting

€

OAS=area(A°∩B−)

area(A)

Outer Area Splitting Inverse

B

A

Outer Area Splitting Inverse

€

OAS-1=area(A−∩B°)area(A)

Exterior Splitting

B

A

Exterior Splitting

€

ES=area(bounded(A−∩B−))area(A)

Inner Traversal Splitting

B

A

Inner Traversal Splitting

€

ITS=length(∂A∩B°)length(∂A)

Outer Traversal Splitting

B

A

Outer Traversal Splitting

€

OTS=length(∂A∩B−)

length(∂A)

Alongness Splitting

B

A

Alongness Splitting

€

AS=length(∂A∩∂B)length(∂A)

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

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 = 0OAS, OTS = 1 OAS-1, OTS-1 = 1

IAS = 0

AS = 0

ES = 0

Inner Traversal Splitting

0

0

(0,1)

(0,1] 0

01 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 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

Hasse Diagrams

disjoint meet overlap equal

coveredBy inside covers contains

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

Questions?

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

queries!

National Geospatial Intelligence Agency

National Science Foundation