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2 SEMINARI DI MAT. APPLICATA (4 e 5 marzo)
Apologies for cross-posting,
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Giovedì 4 marzo 2004, ore 16.30
Aula 8, Dipartimento di Matematica (Via C. Saldini n.50, Milano)
SEMINARIO DI MATEMATICA APPLICATA
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Joint estimators for the specific intrinsic volumes of stationary random sets
Volker Schmidt
University of Ulm, Germany
An actual question in statistical analysis of image data is the
development of methods by means of which one can automatically
distinguish between two (or more) images of similar structure.
Examples, where this type of decision problems appear, range from
testing the spatial structure of biological cells or tissues in
computer-aided cancer diagnostics, via space-time analysis of
coverage properties in mobile communication systems, to
intelligent management of complex transportation systems. In this
paper, we present a new method for statistical analysis of the
spatial structure of binary image data, which can contribute to
solve the problems mentioned above. The idea behind this method is
a new approach to (indirect) statistical estimation of
morphological image characteristics, using tools of convex and
stochastic geometry. More precisely, we interpret binary images in
the $d$-dimensional Euclidean space as realizations of spatially
homogeneous random closed sets, assuming that they belong to the
extended convex ring. Then, we construct nonparametric joint
estimators for the specific intrinsic volumes (or, equivalently,
the specific Minkowski functionals) of these random closed sets,
including estimators for the specific Euler-Poincare'
characteristic, the specific surface area, and the volume fraction
itself. The estimators are based on an explicit extension of the
classical Steiner formula to the convex ring and they can be
represented by integrals of some stationary random fields. This
implies in particular that the estimators are unbiased. Moreover,
conditions are derived under which they are mean-square
consistent, and a positive-definite and consistent estimator for
their asymptotic covariance matrix is given.
(Work perfomed in collaboration with Evgueni Spodarev (Univ. of Ulm))
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Venerdi 5 marzo 2004, ore 9.30
Sala di Rappresentanza, Dipartimento di Matematica (Via C. Saldini n.50,
Milano)
SEMINARIO DI MATEMATICA APPLICATA
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Limit theorems for stationary tessellations with random inner cell structures
Volker Schmidt
University of Ulm, Germany
Consider a stationary and ergodic tessellation $X$ given in a
convex, compact sampling window $W_{\varrho} \subset R^d$. The
cells of $X$ possess random inner structures (e.g. point
patterns, fibre systems, germ-grain models, tessellations etc.)
generated both independently of each other and independently of
the tessellation $X$ by generic stationary random sets which are
connected with a stationary random vector measure $J_0$ acting on
$R^d\,$. In this paper, we study the asymptotic behavior of some
multivariate random functional which is determined by $X$ and the
individual cell structures contained in the sampling window
$W_{\varrho}\,$, as $W_{\varrho} \uparrow R^d$. It turns out that
this functional provides an unbiased estimator of the intensity
vector associated with $J_0\,$. Furthermore, the asymptotic
covariance matrix of the functional is determined and shown to
depend only on the distribution of the typical cell of $X$ and the
second--order moment and cross covariance measures of the vector
measure $J_0\,$.
Under natural restrictions using the ergodicity of $X$ and
the conditional independence between distinct cell structures we prove
strong laws of large numbers and a multivariate central limit theorem of
the normalized functional.
Finally, some numerical examples are discussed in detail,
for which the inner structure of the
cells of $X$ is induced by iterated Poisson-type tessellations.
The obtained results allow to construct flexible models of random
tessellations and make them tractable for a
statistical analysis including model fitting.
An application to stochastic subscriber line models is briefly
discussed.
(Work perfomed in collaboration with Lothar Heinrich(Univ. of Augsburg) and
Hendrik Schmidt (Univ. of Ulm))
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La home page del seminario e'
<http://users.mat.unimi.it/%7Epavarino/sma/>http://users.mat.unimi.it/~pavarino/sma/
Il seminario e' organizzato dal Dipartimento di Matematica dell'Universita'
di Milano in collaborazione con MIRIAM (MIlan Research group in Industrial
and Applied Mathematics) e il Dottorato di Ricerca MACRO (MAtematica
Computazionale e Ricerca Operativa) e MSSCI (Matematica, Statistica, Scienze
Computazionali e Informatica).
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Dr. Alessandra Micheletti, Research Assistant.
MIRIAM Centre - Dipartimento di Matematica - Universita' degli Studi di Milano
via Saldini 50, I- 20133 Milano, Italy
fax +39-02503-1-6172 ph.:+39-02503-1-6120 (direct) / 6171 (secretariat)
e-mail: alessandra.micheletti@unimi.it
http://www.mat.unimi.it/~miriam/micheletti/alex.html
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