27 research outputs found
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
Grupo genético e heterose na produção de leite de vacas de corte submetidas a diferentes sistemas de alimentação
Produção e composição do leite de vacas de quatro grupos genéticos submetidas a dois manejos alimentares no perÃodo de lactação
Stein's method and point process approximation
The Stein-Chen method for Poisson approximation is adapted into a form suitable for obtaining error estimates for the approximation of the whole distribution of a point process on a suitable topological space by that of a Poisson process. The adaptation involves consideration of an immigration-death process on the topological space, whose equilibrium distribution is that of the approximating Poisson process; the Stein equation has a simple interpretation in terms of the generator of the immigration-death process. The error estimates for process approximation in total variation do not have the ‘magic’ Stein-Chein multiplying constants, which for univariate approximation tend to zero as the mean gets larger, but examples, including Bernoulli trials and the hard-core model on the torus, show that this is not possible. By choosing weaker metrics on the space of distributions of point processes, it is possible to reintroduce these constants. The proofs actually yield an improved estimate for one of the constants in the univariate case