Modelisation of scattered objects as random closed sets

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Modelisation of scattered objects as random
closed sets

Stefan Rolfes
21.06.2000
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Natural environments
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Observation Objects that occur in natural
scenes tend to form patches (alga, stone fields,
)
We consider natural, unstructured scenes as
collection of marked closed sets
The family of all closed sets
The mark space, covering the possible types of
objects
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Application Estimating amount of living and
dead alga
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Estimating the surface of dead and living alga in
a large area
The mark space is defined as
Complete observation of
Question Can we estimate and
based on partial observations?
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Estimation based on partial observations
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Complete observations
Partial observations
Covering trajectory time and energy consuming,
complete mapping in of the alga
Sample trajectory short path, but just partial
mapping in of the alga
Infer biomass in from partial observations
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Capturing global characteristics of an
environment by statistical models
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Consider a natural scene as a realisation
of a random closed set ,defined by a
random closed set model
describing just global characteristics (no need
of detailed description)

• Distribution of the closed sets (number per unit
  area)
• Spatial and semantic relationship between them
• Distribution of basic morphological
  characteristics (size, shape, .)

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Examples of random closed sets
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Non isotropic distribution
Uniform distribution
Regular structures
Cluster process
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Random closed sets
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Modelisation of the natural scene by a
random closed set
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Parameter estimation of random closed set models
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Intersecting sets Non
intersecting sets
Objects can be counted, direct estimation of
intensity and morphological characteristics
No direct observation of number and morphological
characteristics
The distribution of the compact sets and the
intensity is determined by the hitting capactiy
(Matheron 75)
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The hitting capacity
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Hitting capacity of
hit
hit
miss
Independent realisations of the RCS
Analytical forms of can be found
for some model types
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Boolean model simple model for random closed
sets
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Frequently used to model random scenes, in that
the objects are uniformly distributed (biological
and physical contexts)

• The sequence of locations (germs)
  of the closed sets is a
  stationary Poisson process of intensity

• The sequence (grains)
  are i.i.d. realisations of
  random closed sets with distribution

Boolean model
Goal Estimate and such that
is a typical realisation of
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Hitting capacity for boolean models
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The hitting capacity for boolean models (Stoyan)
Is the primary (typical) grain, often assumed to
be
characterizable by a parameter vector with
distribution
Examples circles, ellipses, rectangles,
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Hitting capacity for isotropic cluster processes 1
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A cluster process is a union of clusters
at locations
The sequence
is Poisson distributed with intensity
Characterisation of the clusters can be done when
the clusters are assumed to be isotropic
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Hitting capacity for isotropic cluster processes 2
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Intersection of a delimiting area with a
boolean model
The sequence
is Poisson distributed with intensity
Hitting capacity for
Isotropy of the boolean model
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Results for isotropic simulated scenes (Boolean
models) 1
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Two hypothesis for primary grains ,
characterized by
(Circles of random radius)
1. 2.
(Squares of random sidelength)
Two hypothesis for the distribution
Estimate intensity and distribution
parameter
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Results for isotropic simulated scenes (Boolean
models) 2
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Model
Observations
Obtained by LSE criterium
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RCSM for maerl mapping
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1. Phase Independend study of living and dead
mearl
The RCSM depends on environmental factors
Non isotropic model
ocean current, temperature, ...
location
RCSM type and
A priori information
learned from data sent video, diver
information, ...
A posteriori probability
(Bayes a priori information observations
during survey)
Confidence estimation
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Conclusions and Future work
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Conclusions

• Modelisation of natural environments as Random
  Closed Sets
• Estimation of amount of living and dead mearl
  from partial observations
• Characterisation of simple isotropic RCSM
  (boolean, cluster model)

Future work

• Use Bayesian approach to the estimation of
  parameters
• Design and characterization of other appropriate
  RCSMs
• Consideration of interdependence between living
  and dead mearl