The Euler-Poincará formula through contact surfaces of voxelized objects

Two new versions of the Euler-Poincaré formula are proposed considering two new defined cuboids: the tetra-voxel and the octo-voxel, without losing information on the number of vertices and edges. The well-known relationship between contact and enclosing surface concepts, as well as the relationships between vertices, edges and enclosing surfaces, allowed us to compute an innovative algorithm for obtaining alternative versions of the Euler-Poincare formula. This is a very important topological descriptor of 3D binary images. We considered not only topological but geometric aspects. Our method was compared to other proposals, obtaining that our proposed contact surface-based method offers more advantages.