point in polygon analysis for moving objects farid karimipour mahmoud r. delavar andrew u. frank...

Point in Polygon Point in Polygon AnalysisAnalysis

for Moving Objectsfor Moving ObjectsFarid KarimipourFarid KarimipourMahmoud R. Mahmoud R. DelavarDelavarAndrew U. FrankAndrew U. FrankHani RezayanHani Rezayan

University of Tehran, IranUniversity of Tehran, IranUniversity of Tehran, IranUniversity of Tehran, Iran

Technical University Vienna, Technical University Vienna, AustriaAustria

University of Tehran, IranUniversity of Tehran, Iran

Pontypridd, Wales, UKPontypridd, Wales, UKSeptember 5 – 8, 2005September 5 – 8, 2005

DMGIS’ 05DMGIS’ 05

Presenter:Presenter:Eva GrumEva Grum

Technical University Vienna, Technical University Vienna, AustriaAustria

Farid KarimipourFarid KarimipourMahmoud R. DelavarMahmoud R. DelavarAndrew U. Frank Andrew U. Frank Hani RezayanHani Rezayan

Point in Polygon Point in Polygon Analysis for Moving Analysis for Moving

ObjectsObjects

DMGIS’ 05DMGIS’ 05Pontypridd, Wales, UKPontypridd, Wales, UKSeptember 5September 5 – – 8, 20058, 2005

OverviewOverview

Time in GIScience Moving objects Computational model formalization Case Study: Point in polygon analyses



ObjectsObjects


GoalsGoals

Demonstrate a uniform approach to analysis of static and dynamic situations using time lifting.

Show how it applies to operations used for the analysis of spatio-temporal data.



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Everything changes: Time is a dimension of reality

Interest in change, seldom in static situation

Deficiencies in current GIS: Lack of comprehensive ontology Discrete or partial continuous treatments Dominance of analytical approaches Context-based viewpoints

GI Science and Theory : GI Science and Theory : TimeTime



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GI Theory Development:GI Theory Development:Category TheoryCategory Theory Fundamental concepts

Category A collection of primitive element types

(domains), a set of operations on those types

(morphisms) Composition of morphism: ·

Identity morphism id (do nothing op!).

A

C

B

D

f1

f2 f4f3

f5



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Functors give a principled Functors give a principled way of extending an way of extending an algebraalgebra“Many constructions of a new algebraic system

from a given one also construct suitable morphism of the new algebraic system from morphism between the given ones. These constructions will be called 'functors' when they preserve identity morphism and composites of morphisms."

(Mac Lane and Birkhoff 1967 p.131)



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Examples for well known Examples for well known Functors:Functors: In school you were faced with the

problem of dividing 2 apples among 4 people. No solution with integers! We added

rational numbers. To find square roots to all real numbers,

we went to complex numbers. To solve geometric operations in 3d all

the time, use homogenous coordinates!



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GI Theory Development:GI Theory Development:FunctorFunctorA mapping between categories: Let C and D be categories. A functor F from C to

D is a mapping that: Maps each object X in C to an object F(X) in D, Maps each morphism f : R → V in C to a

morphism F(f) : F(R) → F(V) in D Such that:

Identity maps to identity: F(id(X)) = id(F(X)) for every object

Composition is preserved: F(g ·f) = F(g) · F(f) for all morphisms f:XY and g:YZ .



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Functional Formalization of Functional Formalization of TimeTimeChange and movement is formalized

as a function from time to a position or an object property value.

Changing v = Time → v where v = any static type

Time = Data type for timeThese functions are Functors!



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Extending analytical Extending analytical functions to work with functions to work with temporal datatemporal dataGiven the basic operations (e.g. +, -, *)

for static and dynamic data.Construct analytical functions which work

for static data to apply to dynamic data:Automatically change the base operations

they are constructed from.This is called “Lifting” (and is achieved by

overloading)



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Case Study: Point in Polygon Case Study: Point in Polygon Analysis for Moving ObjectsAnalysis for Moving ObjectsGiven a method for “point in polygon” test for

static points,Construct a test for moving points.

Part of a series of tests to lift all low level analytical functions used in GIS to work on moving points: Convex hull Voronoï diagrametc.



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Time Lifting of Primitive Elements

Changing Number Changing Point

Time Lifting of Basic Operators

Time LiftingTime Lifting

type Changing v = time → v

lift0 a = \t → alift1 op a = \t → op (a t)lift2 op a b = \t → op (a t) (b t)

(+) = lift2 (+)(-) = lift2 (-)(*) = lift2 (*)(/) = lift2 (/)



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Lifting the analytical Lifting the analytical functionfunctionpointInPolygon::

Point a -> [Polygon a] -> Id_of_Polygon

Without programming lifted by overloading to:

pointInPolygon:: Changing (Point a) -> [Polygon a] -> Changing Id_of_Polygon



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Case Study: Case Study:

Point ID t=0 t=10 t=20

1 4 5 3

2 9 7 4

3 10 6 1

4 2 3 8

5 6 7 5

t=0

Determine for a moving point the polygon it is in for different time points



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ConclusionsConclusions

Category theory is the high level abstraction that provides the environment in which a theory of space-time fields and objects is possible.

Models for static analysis can be lifted to apply to dynamic situations without reprogramming.

point in polygon analysis for moving objects farid karimipour mahmoud r. delavar andrew u. frank...

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