Mean curvatures of stationary random sets and associated random fields

This is joint work with Simone Klenk, Ursa Pantle and Volker Schmidt.
In applications, one often has to distinguish automatically between two spatially homogeneous images with similar structure. Examples range from the automatic cancer diagnosing in medicine to testing the physical properties of materials in materials science. In this talk, a method for such analysis is proposed using the tools of stochastic geometry and spatial statistics. The core of the method is the estimation of morphological image characteristics combined with asymptotic Gauss tests of their distribution.
We introduce estimators for the specific intrinsic volumes (comprising the volume fraction, the specific surface area and the Euler–Poincaré characteristic (porosity)) of stationary random closed sets by estimating the mean of associated random fields. It is shown that these estimators are unbiased, $L^2$-consistent and normally distributed. The corresponding central limit theorem is proved using the techniqiues for $m$-dependent and $\beta$-mixing random fields. These results are applied to construct the corresponding asymptotic Gauss tests.
Finally, efficient computer algorithms implementing the above methods are touched upon. In two dimensions, numerical results are presented.