The Euler-Poincaré Formula Through Contact Surfaces of Voxelized Objects

Abstract

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-Poincaré 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.

A Review for the Euler Number Computing Problem

“…For example, one proposed algorithm, presented by Lee and Poston [52], involves smoothing a 3D image to a differentiable object and applying theorems from differential geometry and algebraic topology. Another alternative method was presented by Sánchez-Cruz et al [53], which calculates the Euler number of a voxelized object with tunnels and/or cavities by analyzing the relationship between contact voxel faces and enclosing surfaces. Moreover, a formula based on surface calculations was proposed by Čomića [54] for computing the Euler number of an arbitrary object in a cubical grid that has either vertex-or face-adjacency.…”