153 research outputs found
Typical Geometry, Second-Order Properties and Central Limit Theory for Iteration Stable Tessellations
Since the seminal work of Mecke, Nagel and Weiss, the iteration stable (STIT)
tessellations have attracted considerable interest in stochastic geometry as a
natural and flexible yet analytically tractable model for hierarchical spatial
cell-splitting and crack-formation processes. The purpose of this paper is to
describe large scale asymptotic geometry of STIT tessellations in
and more generally that of non-stationary iteration infinitely
divisible tessellations. We study several aspects of the typical first-order
geometry of such tessellations resorting to martingale techniques as providing
a direct link between the typical characteristics of STIT tessellations and
those of suitable mixtures of Poisson hyperplane tessellations. Further, we
also consider second-order properties of STIT and iteration infinitely
divisible tessellations, such as the variance of the total surface area of cell
boundaries inside a convex observation window. Our techniques, relying on
martingale theory and tools from integral geometry, allow us to give explicit
and asymptotic formulae. Based on these results, we establish a functional
central limit theorem for the length/surface increment processes induced by
STIT tessellations. We conclude a central limit theorem for total edge
length/facet surface, with normal limit distribution in the planar case and
non-normal ones in all higher dimensions.Comment: 51 page
Cell shape analysis of random tessellations based on Minkowski tensors
To which degree are shape indices of individual cells of a tessellation
characteristic for the stochastic process that generates them? Within the
context of stochastic geometry and the physics of disordered materials, this
corresponds to the question of relationships between different stochastic
models. In the context of image analysis of synthetic and biological materials,
this question is central to the problem of inferring information about
formation processes from spatial measurements of resulting random structures.
We address this question by a theory-based simulation study of shape indices
derived from Minkowski tensors for a variety of tessellation models. We focus
on the relationship between two indices: an isoperimetric ratio of the
empirical averages of cell volume and area and the cell elongation quantified
by eigenvalue ratios of interfacial Minkowski tensors. Simulation data for
these quantities, as well as for distributions thereof and for correlations of
cell shape and volume, are presented for Voronoi mosaics of the Poisson point
process, determinantal and permanental point processes, and Gibbs hard-core and
random sequential absorption processes as well as for Laguerre tessellations of
polydisperse spheres and STIT- and Poisson hyperplane tessellations. These data
are complemented by mechanically stable crystalline sphere and disordered
ellipsoid packings and area-minimising foam models. We find that shape indices
of individual cells are not sufficient to unambiguously identify the generating
process even amongst this limited set of processes. However, we identify
significant differences of the shape indices between many of these tessellation
models. Given a realization of a tessellation, these shape indices can narrow
the choice of possible generating processes, providing a powerful tool which
can be further strengthened by density-resolved volume-shape correlations.Comment: Chapter of the forthcoming book "Tensor Valuations and their
Applications in Stochastic Geometry and Imaging" in Lecture Notes in
Mathematics edited by Markus Kiderlen and Eva B. Vedel Jense
Geometry of iteration stable tessellations: Connection with Poisson hyperplanes
Since the seminal work by Nagel and Weiss, the iteration stable (STIT)
tessellations have attracted considerable interest in stochastic geometry as a
natural and flexible, yet analytically tractable model for hierarchical spatial
cell-splitting and crack-formation processes. We provide in this paper a
fundamental link between typical characteristics of STIT tessellations and
those of suitable mixtures of Poisson hyperplane tessellations using martingale
techniques and general theory of piecewise deterministic Markov processes
(PDMPs). As applications, new mean values and new distributional results for
the STIT model are obtained.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ424 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm). arXiv admin
note: text overlap with arXiv:1001.099
Shape-Driven Nested Markov Tessellations
A new and rather broad class of stationary (i.e. stochastically translation
invariant) random tessellations of the -dimensional Euclidean space is
introduced, which are called shape-driven nested Markov tessellations. Locally,
these tessellations are constructed by means of a spatio-temporal random
recursive split dynamics governed by a family of Markovian split kernel,
generalizing thereby the -- by now classical -- construction of iteration
stable random tessellations. By providing an explicit global construction of
the tessellations, it is shown that under suitable assumptions on the split
kernels (shape-driven), there exists a unique time-consistent whole-space
tessellation-valued Markov process of stationary random tessellations
compatible with the given split kernels. Beside the existence and uniqueness
result, the typical cell and some aspects of the first-order geometry of these
tessellations are in the focus of our discussion
Intrinsic Volumes of the Maximal Polytope Process in Higher Dimensional STIT Tessellations
Stationary and isotropic iteration stable random tessellations are
considered, which can be constructed by a random process of cell division. The
collection of maximal polytopes at a fixed time within a convex window
is regarded and formulas for mean values, variances, as
well as a characterization of certain covariance measures are proved. The focus
is on the case , which is different from the planar one, treated
separately in \cite{ST2}. Moreover, a multivariate limit theorem for the vector
of suitably rescaled intrinsic volumes is established, leading in each
component -- in sharp contrast to the situation in the plane -- to a
non-Gaussian limit.Comment: 27 page
- …