Surface-based computation of the Euler characteristic in the cubical grid

Abstract

For well-composed (manifold) objects in the 3D cubical grid, the Euler characteristic is equal to half of the Euler characteristic of the object boundary, which in turn is equal to the number of boundary vertices minus the number of boundary faces. We extend this formula to arbitrary objects, not necessarily well-composed, by adjusting the count of boundary cells both for vertex- and for face-adjacency. We prove the correctness of our approach by constructing two well-composed polyhedral complexes homotopy equivalent to the given object with the two adjacencies. The proposed formulas for the computation of the Euler characteristic are simple, easy to implement and efficient. Experiments show that our formulas are faster to evaluate than the volume-based ones on realistic inputs, and are faster than the classical surface-based formulas.