the topology of spatial scenes in ℝ 2
DESCRIPTION
The Topology of Spatial Scenes in ℝ 2. Beyond Sets of Binary Relations. School of Computing and Information Science National Center for Geographic Information and Analysis University of Maine. Joshua A. Lewis. Matthew P. Dube. Max J. Egenhofer. The Topology of Spatial Scenes in ℝ 2. - PowerPoint PPT PresentationTRANSCRIPT
Max J. Egenhofer
School of Computing and Information Science
National Center for Geographic Information and Analysis
University of Maine
The Topology of Spatial Scenes in
ℝ2
Beyond Sets of Binary Relations
Joshua A. Lewis Matthew P. Dube
School of Computing and Information Science
National Center for Geographic Information and Analysis
University of Maine
The Topology of Spatial Scenes in
ℝ2
Beyond Sets of Binary Relations
3
Topological Relations Between Two Regions in ℝ2
ABEgenhofer Franzosadisjoint meet overlap equal
coveredBy inside covers contains
Randell CohnCui
DC EC PO EQ
TPP NTPP TPPi NTPPi
4-intersection
RCC-8
Objects of Interest
simple region
separated
region
holed and separated
region
holed region
A
BehrSchneider Li
Collection of Regions
in spatial databases in sketches
in sensor networks
Collection of Regions
… or a united region with a hole?
The Topological Relations surroundedBy vs. outside in
ℝ2
EgenhoferVasardani
DE
CH
ITSM
VA
Part of Europe
Part of Europe
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FRAU
CH
IT
LI
Part of Europe
FR DE CH LI AU IT
FR
DE
CH
LI
AU
IT
DE
FRAU
CH
IT
LI
Border Table
neighbors
FR DE CH LI AU IT
FR
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CH
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AU
IT
FRAU
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IT
LI
Border Table
DE
separated
FR DE CH LI AU IT
FR
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CH
LI
AU
IT
FRAU
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IT
LI
Border Table
DE
equal
FR DE CH LI AU IT
FR
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FRAU
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Border Table
neighbors
separated
equal
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FRAU
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Scene Description with 4-Intersection
meet
disjoint
equal
FR DE CH LI AU IT
FR
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LI
AU
IT
eq
eq
eq
eq
eq
eq
mt
mt
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mt
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mt mt
mt
mt
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mt mt
mt
mt
di di
didi
didi
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di di
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Scene Description
but alsoproduces
produces
produces
CHIT
FR
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AU
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FRAU
CH
IT
LIFR DE CH LI AU IT
FR
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eq
eq
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eq
eq
eq
mt
mt
mt
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mt
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di di
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To Capture a Spatial Scene Completely
• Account for placing objects within or outside of a collection of objects
• Account for the boundary sequence interactions
side ofthe exterior
side ofthe interior
sequence ofboundary parts,
but no interactions
MapTree Model
Side of the boundaryInteractions of boundaries Worboys
Modes of Overlap
Galton
boundaries of a pair of objects segment their common exterior (o > 1), either touch or cross has occurred
• objects have intersected (x > 0)
crossing has occurred (a > 1 and b ≥ 1 or a ≥ 1 and b > 1)
despite being able to show that interactions have happened at some point, the overlap matrix cannot discern the order in which types of boundary interactions occur
Modes of Overlap
Galton
• exterior segmentation
• objects have intersected
• crossings
• no order in which types of boundary interactions occur
To Capture a Spatial Scene Completely
• Account for placing objects within or outside of a collection of objects
• Account for the boundary sequence interactions
Hull+o
[A], the topological hull of A
the smallest closed set, homeomorphic to a 2-disk such that A ⊆ [A]
A:
[A]:
closed set, with path-connected interior in ℝ2
[A], the topological hull of A
the smallest closed set, homeomorphic to a 2-disk such that A ⊆ [A]
A:
[A]:
closed set, with path-connected interior in ℝ2
[A], the topological hull of A
the smallest closed set, homeomorphic to a 2-disk such that A ⊆ [A]
A:
[A]:
closed set, with path-connected interior in ℝ2
[B], the aggregate topological hull of BB: closed set in ℝ2
[B]: ⋃p∈P [p]
P: path-connected subsets of B such that ⋃p∈P = B°and |P| is minimized
Considering Boundary: Dimension
AZ
CO
ND
SD
I-95 vs. ME-15 33
Considering Boundary: Touch vs. Cross
Arizona vs. Colorado
Maine vs. New Hampshire
N/S Highway vs. E/W Highway
cv
Egenhofer Franzosa
cv
Clementini Di Felice34
Considering Boundary: Touch vs. Cross
1-cross1-touch
0-cross0-touch
Considering Boundary: Sequence
Egenhofer Franzosa
cross touch cross cross crosstouch cross cross cross crosscross cross cross cross touchcross cross cross touch cross
cross cross touch cross cross
Spatial Scene
BBCC
AA
BBCC
AA
o-Notationo{B,C}(0,ø,{B})o{C}(1,{C},ø)o{C}(0,{C},{B})
o{B,C}(1,{C},ø)
o{A,C}(0,ø,{C})o{A}(0,ø,{A,C})o{C}(0,ø,{A})
o{A,B}(1,{A},ø)o{A,B}(0,{A},{B})o{A}(1,{A},ø)o{A}(0,ø,{B})
o-Notation for Binary Relations
relation A Bdisjoint o{B} o{A}
meet o{B}(*,B,∅) o{A}(*,A,∅)
overlapo{B}(*,∅,B) o{ø}
(*,∅,B)o{A}(*,∅,A) o{Ø}
(*,∅,A)
equal o{∅}(1,B,∅) o{∅}(1,A,∅)
coveredBy o{∅}(*,B,∅) o{A}(*,A,∅)
inside o{∅} o{A}
covers o{B}(*,B,∅) o{∅}(*,A,∅)
contains o{B} o{∅}
Insufficiency of o-Notation
BB
AA
C C BB
AA
CC
o-notation does not account for diverging boundary on a touch configuration
renders these scenes identical
Insufficiency of o-Notation
BB
AA
C C BB
AA
CC
meet meetcoveredBy
A ∪ B[A ∪ B] [A ∪ B]A ∪ B
[A ∪ B]/(A ∪ B) [A ∪ B]/(A ∪ B)[p1] [p2]
Insufficiency of Hull
hull does not account for dimensionof an intersection
o{B,C}(1,{B},ø)
o{B,C}(0,{B},ø)o{B,C}(1,{C},ø) o{B,C}(0,{C},ø)
renders these scenes identical
Europe Revisited
`
Switzerland meets the hull
of its neighbors
Switzerland inside the hull
of its neighbors
Europe Revisited
One 1-dimension outside touches
between Switzerland and
Liechtenstein
Two 1-dimension outside touches
between Switzerland and
Liechtenstein
CHIT
FR
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Summary• Hull and o-notation developed for Spatial
Scenes
• Capable of capturing topology of Spatial Scenes
• Capable of reverse engineering spatial scenes
• Some cases need hull and o, some require only one
• hull+o as a unifying measure for the topology of scenes and binary relations
Future Work• Normalize o-notation
• Compare with MapTree’s expressive power
• Adapt hull+o to nonareal objects, such as lines and points
• Create an algorithmically definable surrounds relation
• Implement hull+o as a Scene Topology evaluator
• Employ hull+o as a scene similarity calculator
QuickTime™ and aPhoto - JPEG decompressor
are needed to see this picture.hull+o notation
Topologically Truthful Sketch
manually
hull+o notationautomatically
Vasardani
The Topology of Spatial Scenes in
R2