Nonparametric Morphological Estimators in Statistical Image Analysis
Volker Schmidt
Faculty of Mathematics and Economics
University of Ulm
Wednesday, April 13, 2005
4:30 - 5:45 PM
Terman Engineering Center, Room 453
Abstract:
The classification of binary images with respect to their
morphological properties is an actual topic in statistical image
analysis, both in macroscopic-cartographic and microscopic
scales. Specific Minkowski functionals or, equivalently, specific
intrinsic volumes are important characteristics in order to describe
the morphological structure of binary images. In particular, these
functionals characterize such parameters of the foreground (or
background) phase of binary images like the area fraction, the
specific boundary length (per unit area), and the
connectivity/porosity of binary images. Examples of application
include cellular structures in telecommunication networks as well as
in biological and technical materials.
A family of nonparametric estimators is investigated for the
simultaneous estimation of the vector of all specific intrinsic
volumes. Using techniques from stochastic geometry like Steiner's
polynomial series expansion for the area of the epsilon-neighborhood
of polyconvex sets, or the principal kinematic formula of convex
geometry, these estimators can be represented by integrals of
stationary random fields. This implies in particular that the
estimators are unbiased.
Moreover, asymptotic properties of the estimators are derived for
unboundedly increasing sampling windows, like mean-square consistency
and asymptotic normality. These properties can be used for
classification of binary images on the basis of asymptotic
significance tests. Special emphasis is put on so-called germ-grain
models, in particular the Boolean model, which is generated by a
Poisson point process (of germs) and an independent sequence of iid
random sets (of grains associated with the germs).
References:
S. Klenk, V. Schmidt, E. Spodarev (2005) A new algorithmic approach to the computation
of Minkowski functionals of polyconvex sets Computational Geometry: Theory and
Applications (submitted)
U. Pantle, V. Schmidt, E. Spodarev (2005) Central limit theorems for
functionals of stationary germ-grain models. Advances in Applied
Probability (submitted)
V. Schmidt, E. Spodarev (2005) Joint estimators for the specific intrinsic volumes of
stationary random sets. Stochastic Processes and their Applications
(to appear)
E. Spodarev, V. Schmidt (2005) On the local connectivity number of stationary random
closed sets. In: C. Ronse, L. Najman, E. Decenciere Fernandiere (eds.) Proceedings of the
7th International Symposium on Mathematical Morphology. Kluwer, Dordrecht, 343-556.
Operations Research Colloquia: http://or.stanford.edu/oras_seminars.html