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How robust is your spatial queryHow robust is your spatial query??A formal taxonomy to express A formal taxonomy to express
spatial intersectionsspatial intersections
Yao Cui and Michael RossIntegrated Land Management Bureau
Victoria, BC
Martin DavisRefractions Research Inc.
Victoria, BC
FOSS4G 2007, Victoria, BC, Canada
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1.1. What is an intersection taxonomy?What is an intersection taxonomy?
2.2. Why do we need this taxonomy?Why do we need this taxonomy?
3.3. How was the taxonomy created?How was the taxonomy created?
4.4. How was the taxonomy validatedHow was the taxonomy validated
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Why do we need a taxonomy?Why do we need a taxonomy?
Started when the first version of Java Started when the first version of Java Topology Suite (JTS) was developed by Topology Suite (JTS) was developed by Martin Davis right here in Victoria, BC, Martin Davis right here in Victoria, BC, back in 2001…back in 2001…
One of the objectives: to test JTS binary One of the objectives: to test JTS binary predicatespredicates
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A list of binary predicatesA list of binary predicates
EqualEqual DisjointDisjoint IntersectsIntersects TouchesTouches CrossesCrosses WithinWithin ContainContain OverlapOverlap
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Java Topology Suite Test BuilderJava Topology Suite Test Builder
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Adding another test caseAdding another test case
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Adding another test caseAdding another test case
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Adding more cases …Adding more cases …
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Now the questions:Now the questions:
How many more test cases?How many more test cases? How do we know if we have enough test How do we know if we have enough test
cases?cases? How do we know if important How do we know if important
representative test cases are not missing?representative test cases are not missing?
How are we going to define the different How are we going to define the different categories of spatial intersects?categories of spatial intersects?
How to organize these test cases?How to organize these test cases?
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We need four thingsWe need four things
Taxonomy to define and express Taxonomy to define and express intersection componentsintersection components
Taxonomy to express intersectionsTaxonomy to express intersections Taxonomy to help organizing the test Taxonomy to help organizing the test
casescases A grammar to define the taxonomyA grammar to define the taxonomy
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How was the taxonomy created?How was the taxonomy created?
To search and define a path To search and define a path
to decompose a geometryto decompose a geometry
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Geometry Object ModelGeometry Object Model(as defined by OGC SFS)(as defined by OGC SFS)
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Combinations of Geometric Subtypes between Geometry A and B
Geometry B
P mP L LR nsL mL A mA C
Ge
om
etry
A
P P/P P/mP P/L P/LR P/nsL P/mL P/A P/mA P/C
mP mP/mP mP/L mP/LR mP/nsL mP/mL mP/A mP/mA mP/C
L L/L L/LR L/nsL L/mL L/A L/mA L/C
LR LR/LR LR/nsL LR/mL LR/A LR/mA LR/C
nsL nsL/nsL nsL/mL nsL/A nsL/mA nsL/C
mL mL/mL mL/A mL/mA mL/C
A A/A A/mA A/C
mA mA/mA mA/C
C C/C
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Dimensionally Extended Dimensionally Extended Nine-Intersection Model (DE-9IM)Nine-Intersection Model (DE-9IM)
Interior Boundary Exterior
Interior dim(I(a)∩I(b)) dim(I(a)∩B(b)) dim(I(a)∩E(b))
Boundary dim(B(a)∩I(b)) dim(B(a)∩B(b)) dim(B(a)∩E(b))
Exterior dim(E(a)∩I(b)) dim(E(a)∩B(b)) dim(E(a)∩E(b))
Source: Egenhofer and Clementini
B
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Interior Boundary Exterior
Interior 1 0 2
Boundary 0 -1 1
Exterior 1 0 22
ABAB BABA
EqualEqual FF FF
DisjointDisjoint FF FF
IntersectsIntersects TT TT
TouchesTouches FF FF
CrossesCrosses TT TT
WithinWithin FF FF
ContainsContains FF FF
OverlapsOverlaps FF FF
ABB
A BB
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Geometry
Geometry DecomposingGeometry Decomposing
Dimensions
Instantiable Geometry Types
BoundaryInterior Exterior
Intersection Component
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Intersection components in BNFIntersection components in BNF
<intersection_component> ::= <interior_component>| <boundary_component>| <exterior_component>
<geometry> ::= <Point> | <LineString> | <linearRing> | <non_simple_LineString> | <Polygon> | <MultiPoint>| <MultiLineString>| <MultiPolygon>
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Examples of Interior Intersection Components in BNFExamples of Interior Intersection Components in BNF
<interior_component> ::= <interior_component_0d> | <interior_component_1d>| <interior_component_1d> <separator> <reverse>
<interior_component_0d> ::= <interior> <separator> <close_point>| <interior> <separator> <vertex_choice>| <interior> <separator> <close_point> <multipass>| <interior> <separator> <vertex_choice> <multipass>
<interior_component_1d> ::= <interior> <separator> <start_point> <to> <end_point>| <interior> <separator> <end_point> <to> <vertex_choice> | <interior> <separator> <vertex> <to> <nonVertex>| <interior> <separator> <end_point> <to> <vertex_choice>
<multipass>| <interior> <separator> <vertex> <to> <nonVertex>
<multipass>| <interior> <separator> <end_point> <multipass> <to>
<vertex_choice>| <interior> <separator> <vertex> <multipass> <to>
<nonVertex>| <interior> <separator> <end_point> <multipass> <to>
<vertex_choice> <multipass>| <interior> <separator> <vertex> <multipass> <to>
<nonVertex> <multipass>
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Examples of boundary intersection components in BNFExamples of boundary intersection components in BNF
<boundary_component> ::= <line_boundary> | <polygon_boundary>
<line_boundary> ::= <boundary> <separator> <end_point> | <boundary> <separator> <end_point> <multipass>
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Examples of Examples of polygonpolygon boundary intersection component in BNF boundary intersection component in BNF
<polygon_boundary> ::= <polygon_boundary_0d> | <polygon_boundary_1d>| <polygon_boundary_1d> <separator> <reverse>
<polygon_boundary_0d> ::= <polygon_boundary_types> <separator> <close_point><polygon_boundary_types> <separator> <vertex_choice><polygon_boundary_types> <separator> <touch_point><boundary> <separator> <close_point><boundary> <separator> <vertex_choice><boundary> <separator> <touch_point> ; touch_point: at the connecting point between polygon boundaries
<polygon_boundary_1d> ::= <polygon_boundary_types> <separator> <start_point> <to> <end_point>| <polygon_boundary_types> <separator> <end_point> <to> <vertex_choice>| <polygon_boundary_types> <separator> <end_point> <to> <touch_point>| <polygon_boundary_types> <separator> <vertex> <to> <nonVertex> | <polygon_boundary_types> <separator> <vertex_choice> <to> touch_point>| <boundary> <separator> <start_point> <to> <end_point>| <boundary> <separator> <end_point> <to> <vertex_choice>| <boundary> <separator> <end_point> <to> <touch_point>| <boundary> <separator> <vertex> <to> <nonVertex> | <boundary> <separator> <vertex_choice> <to> touch_point>……
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Terminal DefinitionsTerminal Definitions
Terminal Definitions
<boundary> ::= Bdy
<both> ::= “b”
<close_brace> ::= “}”
<close_paren> ::= “)”
<close_point> ::= CP
<crossing> ::= “x”
<end_point> ::= EP
<exterior> ::= Ext
<hole> ::= “h”
<inner_boundary> ::= iBdy
<interior> ::= Int
<intersection_operator> ::= “=”
<linearRing> ::= LR
<LineString> ::= “L”
<MultiPoint> ::= mP
<MultiLineString> ::= mL
<MultiPolygon> ::= mA
<non_simple_LineString> ::= nsL
<nonVertex> ::= NV
<open_brace> ::= “{”
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Examples of Intersection ComponentsExamples of Intersection ComponentsDimensions of Intersection Components
0 1 2
Point Line Polygon Line Polygon Point Line Polygon
Interior Int
Int.CPInt.VInt.NVInt.CPxInt.Vx……..
Int.EP-SPInt.EP-VInt.EP-NVInt.V-NVInt.CP-V……..
Int
Boundary
Bdy.EPBdy.EPxBdy.EPoBdy.EPb
Bdy.CPBdy.VBdy.NVBdy.TPoBdy.CPoBdy.VoBdy.NViBdy.CPiBdy.V………….
Bdy.EP-SPBdy.EP-VBdy.EP-NVBdy.EP-TPBdy.V-NVBdy.TP-EPBdy.TP-VoBdy.EP-SPoBdy.EP-V………..
Exterior Ext ExtExtExt.h
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Hierarchy of taxonomy intersection componentsHierarchy of taxonomy intersection components
Geometry
Intersection Components
Decom
posing
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Examples of Intersection Components Examples of Intersection Components
Geometry Expression Description
Interior intersection components in zero dimension
A.P.Int The interior a Point
A.L.Int.V The interior of a LineString at a vertex
A.L.Int.NV The interior of a LineString at a non-vertex
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Examples of Intersection Components in Examples of Intersection Components in one dimensionone dimension
Interior intersection components in one dimension
A.L.Int.SP-EP A Line: from the start point to the end point
A.L.Int.EP-V A LineString: from the end point to a vertex
A.L.Int.EP-NV A LineString: from the end point to a non-vertex
A.LR.Int.EP-NVA closed LineString: from the closing point to a non-vertex
A.L.Int.V-NV A LineString: from a vertex to a non-vertex
A.L.Int.NV-NV A LineString: from a non-vertex to a non-vertex
Interior intersection component in two dimension
A.A.Int A Polygon: in the interior
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Taxonomy examples on Intersections Taxonomy examples on Intersections
GeometryIntersection Expression
A B
dim(0){A.P.Int ∩ B.P.Int}
dim(0){A.P.Int ∩ B.L.Bdy.EP}
dim(0){A.L.Int.NV ∩ B.L.Int.VN}
dim(0){A.P.Int ∩ B.A.Ext.h}
dim(1){A.L.Int.EP-SP ∩ B.L.Int.EP-NV}
dim(1){A.L.Int.SP-EP ∩ B.A.Bdy.NV-NV}
dim(2){A.A.Int ∩ B.A.Int}
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In summaryIn summary
Defined an intersection taxonomy and Defined an intersection taxonomy and equivalent formal language with a validatorequivalent formal language with a validator
Generated a list of over 60,000 potential Generated a list of over 60,000 potential test cases by enumerating through test cases by enumerating through intersection componentsintersection components
Compiled 400 representative test cases Compiled 400 representative test cases and organized them by following the and organized them by following the taxonomytaxonomy
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IntersectionTalk ValidatorIntersectionTalk Validator
Built a parser to validate the grammar of Built a parser to validate the grammar of the predicate taxonomythe predicate taxonomy
Written in GNU SmallEiffel usingGobo Written in GNU SmallEiffel usingGobo Eiffel Yacc and Gobo Eiffel LexEiffel Yacc and Gobo Eiffel Lex Yacc takes a BNF grammar as input and Yacc takes a BNF grammar as input and
generates a parsergenerates a parser Took only two hours to buildTook only two hours to build
Most of that time was spent typing in the Most of that time was spent typing in the grammargrammar
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Yacc Input Grammar FormatYacc Input Grammar FormatVery Similar to EBNFVery Similar to EBNF
Intersection_Component: Interior_Component | Boundary_Component | Exterior_Component ;
Interior_Component: Interior_Component_0D | Interior_Component_1D | Interior_Component_1D SEPARATOR REVERSE | INTERIOR ;
Interior_Component_0D: INTERIOR SEPARATOR CLOSE_POINT | INTERIOR SEPARATOR Vertex_Choice | INTERIOR SEPARATOR CLOSE_POINT Multipass | INTERIOR SEPARATOR Vertex_Choice Multipass ;
Interior_Component_1D: INTERIOR SEPARATOR START_POINT TO END_POINT | INTERIOR SEPARATOR END_POINT TO Vertex_Choice | INTERIOR SEPARATOR CLOSE_POINT TO Vertex_Choice | INTERIOR SEPARATOR Vertex_Choice TO Vertex_Choice | INTERIOR SEPARATOR END_POINT TO Vertex_Choice Multipass | INTERIOR SEPARATOR CLOSE_POINT TO Vertex_Choice Multipass | INTERIOR SEPARATOR Vertex_Choice TO Vertex_Choice Multipass | INTERIOR SEPARATOR END_POINT Multipass TO Vertex_Choice | INTERIOR SEPARATOR CLOSE_POINT Multipass TO Vertex_Choice | INTERIOR SEPARATOR Vertex_Choice Multipass TO Vertex_Choice | INTERIOR SEPARATOR END_POINT Multipass TO Vertex_Choice Multipass | INTERIOR SEPARATOR CLOSE_POINT Multipass TO Vertex_Choice Multipass | INTERIOR SEPARATOR Vertex_Choice Multipass TO Vertex_Choice Multipass ;……
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Future work and the challengeFuture work and the challenge
More rigorous decomposing of intersection More rigorous decomposing of intersection components?components?
Adding taxonomy for spatial operations Adding taxonomy for spatial operations and functions?and functions?
Automatically “grow” specific geometries: Automatically “grow” specific geometries: taxonomy grammar, rewriting rules (Koch taxonomy grammar, rewriting rules (Koch and Mandelbrot), Chomsky grammars, L-and Mandelbrot), Chomsky grammars, L-systems (Lindenmayer), …?systems (Lindenmayer), …?
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Thank you!Thank you!
Fore more information, contact Yao Cui E-mail: [email protected]: (250) 387-9310
Link to JTS and JTS Validation Suite: http://www.vividsolutions.com/jts/jtshome.htm
