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.

对于3D立方网格中组成良好（流形）的对象，欧拉特性等于对象边界的欧拉特性的一半，而欧拉特性又等于边界顶点的数量减去边界面的数量。通过调整顶点和面邻接的边界单元的数量，我们将该公式扩展到任意对象，不一定要很好地组成对象。我们通过构造两个等效的给定对象等效的同构的多面体复合体，证明了我们方法的正确性。所提出的用于计算欧拉特性的公式简单，易于实现且高效。实验表明，与基于体积的实际输入公式相比，我们的公式可以更快地进行评估，并且比经典的基于表面的公式更快。