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Spatio-Temporal Predicates

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Spatio-Temporal Predicates Martin Erwig and Markus Schneider IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING Presented by Mamadou Hassimiou Diallo * – PowerPoint PPT presentation

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Title: Spatio-Temporal Predicates


1
Spatio-Temporal Predicates
Martin Erwig and Markus Schneider IEEE
TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING
Presented by Mamadou Hassimiou Diallo
2
Overview
  • Challenges
  • Dealing with large collections of relatively
    simple geometric objects
  • Modeling, storing, querying spatio-temporal
    objects
  • ExampleDatabase information about the flights
    of airplanes and about weather conditionsQuery
    whether an airplane crossed a certain storm
  • Focus
  • Nature and formal definition of spatio-temporal
    relationships gt Spatio-temporal predicates
  • Spatio-temporal predicates describe developments
    of spatial topological relationships
  • Approach
  • Integrates the two-dimensional topological
    relationships and the change of spatial
    information over time
  • Framework Spatio-temporal predicates obtained by
    temporal aggregation of elementary spatial
    predicates and sequential composition

3
Foundations Basic and Spatial Data Types
  • Formal definition
  • Based on point set
  • Used for storing, retrieving, manipulating, and
    querying spatial objects
  • Data types
  • Undefined
  • Boolean
  • Points elements of the Euclidean plane
  • Lines two-dimensional curves
  • Regions point sets with a two-dimensional extent
    and are bounded by lines

4
Foundations Basic and Spatial Data Types
  • Problem
  • Modeling regions as arbitrary point sets
  • Can result in undesired geometric anomalies
  • Solution
  • Point set topology regularization process
  • Point set topology point set different parts
  • Point set A
  • Parts
  • Boundary interior closure Not(A)
  • interior eliminate dangling points, dangling
    lines, and boundary parts
  • closure eliminate cuts and punctures
  • Regularity a point set is regular closed if
  • Regularization function
  • Regions type

5
Foundations Topological Predicates
  • Spatial data modeling and reasoning
  • Topological predicates between spatial objects in
    the two-dimensional space
  • 9 possible intersections of boundary interior,
    and exterior
  • Matrix for evaluating two dimensional spaces A
    and B
  • 29 512 different configurations
  • 8 meaningful configurations

6
Foundations Topological Predicates
7
Foundations Topological Predicates
8
Foundations Spatio-Temporal Data Types
  • Definition
  • continuous model of time time R
  • temporal function t(a) time ---gt a (all total
    functions from time a)
  • Algebraic model
  • Moving points t(point), evolving lines t(lines),
    evolving regions t(region)
  • Focus moving points and evolving regions
  • Temporal lifting
  • Flat functions -----gt temporal functions
  • Flat function f a1 xx an ------gt ß
  • Lifted function ?f t(a1) x x t(an) ------gt
    t(ß)
  • Example
  • Distance ?distance
  • distance (point) x (region) -----gt real
  • Distance t(point) x t(region) -----gt t(real)

9
Spatio-Temporal Predicates Nature
  • Definition
  • Predicates can be used to express facts (true or
    false)
  • spatial predicate function spatial objects
    -----gt boolean
  • temporally lifted spatial predicate function
    spatio-temporal objects -----gt temporal Booleans
  • A spatio-temporal predicate is a function of type
    t(a) x t(ß) ---gt ? for a, ß in point, region
  • Examples
  • inside point x region ---gt bool (Yields true,
    undefined, false)
  • ? inside Point x Region ---gt Bool (Yields true,
    undefined, false)
  • Always-inside true iff î inside true for all
    times
  • this definition is a bit problematic

10
Spatio-Temporal Predicates Temporal Aggregation
  • Universal and existential aggregation
  • Operator spatial predicate -----gt
    spatio-temporal predicate
  • (a x ß ---gt bool) -----gt (t(a) x t(ß) ---gt B)
  • Existential quantification semantic is true iff
    p is true for the values of S1 and S2 at some
    time
  • lambda-notation
  • Universal quantification semanticdepends on
    time, t1 U t2, t1, t2, t1 ? t2
  • Creation Operators
  • spatio-temporal predicates from spatial
    predicates
  • arrowhead indicates which object's lifetime

11
Spatio-Temporal Predicates Basic Spatio-Temporal
Predicates
  • Can be defined by temporal lifting and
    aggregation
  • Default expected aggregation behavior (universal
    quantifier)
  • Relaxing symmetric definitions for Meet and
    Overlap
  • Predicate that yields true for two arbitrary
    spatio-temporal objects

12
Spatio-Temporal Predicates Developments
  • Developments
  • Sequences of spatio-temporal predicates
  • Example A moving point P is located at time t1
    outside of an evolving region R and changes
    (continuously) its location
  • If P, at time t3, is inside of R ----gt P enters R
  • If P, at some time t5, is again outside of R
    ----gt P crosses R
  • P is located on the border of R at some time t2
    and at some time t4
  • Table Development of P
  • Observations

13
Spatio-Temporal Predicates Development
  • Observation
  • developments of objects a need to restrict the
    validity of spatio-temporal predicates to
    intervals
  • Definition Predicate Constriction
  • Let P be a spatio-temporal predicate, and let I
    be a (half-) open or closed interval. Then,
  • Example Inside (P, R) is false, Inside (P, R) is
    false
  • Two classes of predicates

14
Spatio-Temporal Predicates Development
  • Combination operations
  • from defines a spatio-temporal predicate that
    for some time t0 checks p and then enforces P for
    all t gt t0
  • until P must hold until p is true at some time
    t0
  • then is true if there is some time point t0 when
    p is true so that P holds before and Q holds
    after t0
  • Definition Temporal Composition
  • Let p be a spatial predicate, and let P and Q be
    spatio-temporal predicates. Then,
  • Example

15
Questions?
16
Spatio-Temporal Data Modeling
  • Spatio-Temporal Objects
  • Examples
  • flight of an airplane, the migration of whales,
    the raging of a storm, or the spreading of a fire
    region
  • Characteristic features spatial entities
    changing over time continuously
  • Changes motion, shrinking, growing, shape
    transformation, splitting, merging, disappearing,
    or reappearing of spatio-temporal objects
  • Temporal changes modifications of mutual
    topological relationships over time
  • disjoint, intersect
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