The euler number of discretised sets-surprising results in three dimensions

Abstract

The problem of estimating the Euler-Poincare characteristic (Euler number for short) of a set in the 3d Euclidean space is considered, given that this set is observed in the points of a lattice. In this situation, which is typical in image analysis, the choice of an appropriate data-based discretisation of the set is crucial. Four versions of a discretisation method which is based on the notion of adjacency systems are presented; these versions are referred to as (14.1,14.1), (14.2,14.2), (6,26), and (26,6). A comparative assessment of the four approaches is performed with respect to the systematic error occuring in application to Boolean models. It is a surprising result that, except for (26,6), the estimators yield infinitely large systematic errors when the lattice spacing goes to zero. Furthermore, the measurements of the Euler number from 3d data of autoclaved aerated concrete illustrate the influence of the choice of adjacency and the behaviour of the estimators.