The Euler characteristic on the face-centred cubic lattice

The Euler characteristic is an important topological invariant of an object, and it has been used in two and three dimensions to provide information about objects in binary digital pictures. Generally it is calculated by counting certain local patterns appearing in an object, and as such it has been defined for the cartesian lattices Z(2) and Z(3), and the hexagonal lattice. In this paper we extend these results to the face-centred cubic lattice; this lattice being the natural extension of the hexagonal lattice to three dimensions, We show how the Euler characteristic may be defined, and that the definition is consistent with classical results. We also investigate means of implementing our definition. (C) 1997 Elsevier Science B.V.