The Euler Number of Discretized Sets — On the Choice of Adjacency in Homogeneous Lattices

  • Joachim Ohser
  • Werner Nagel
  • Katja Schladitz
Part of the Lecture Notes in Physics book series (LNP, volume 600)


Two approaches for determining the Euler-Poincaré characteristic of a set observed on lattice points are considered in the context of image analysis — the integral geometric and the polyhedral approach. Information about the set is assumed to be available on lattice points only. In order to retain properties of the Euler number and to provide a good approximation of the true Euler number of the original set in the Euclidean space, the appropriate choice of adjacency in the lattice for the set and its background is crucial. Adjacencies are defined using tessellations of the whole space into polyhedrons. In ℝ3, two new 14 adjacencies are introduced additionally to the well known 6 and 26 adjacencies. For the Euler number of a set and its complement, a consistency relation holds. Each of the pairs of adjacencies (14.1, 14.1), (14.2, 14.2), (6, 26), and (26, 6) is shown to be a pair of complementary adjacencies with respect to this relation. That is, the approximations of the Euler numbers are consistent if the set and its background (complement) are equipped with this pair of adjacencies. Furthermore, sufficient conditions for the correctness of the approximations of the Euler number are given. The analysis of selected microstructures and a simulation study illustrate how the estimated Euler number depends on the chosen adjacency. It also shows that there is not a uniquely best pair of adjacencies with respect to the estimation of the Euler number of a set in Euclidean space.


Lattice Point Convex Body Lattice Cell Euler Number Consistency Relation 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Joachim Ohser
    • 1
  • Werner Nagel
    • 2
  • Katja Schladitz
    • 1
  1. 1.Fraunhofer Institut für Techno- und WirtschaftsmathematikKaiserslauternGermany
  2. 2.Fakultät für Mathematik und InformatikFriedrich-Schiller-Universität JenaJenaGermany

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