Cup Products on Polyhedral Approximations of 3D Digital Images

Let I be a 3D digital image, and let Q(I) be the associated cubical complex. In this paper we show how to simplify the combinatorial structure of Q(I) and obtain a homeomorphic cellular complex P (I) with fewer cells. We introduce formulas for a diagonal approximation on a general polygon and use it to compute cup products on the cohomology H∗(P (I)). The cup product encodes important geometrical information not captured by the cohomology groups. Consequently, the ring structure of H∗(P (I)) is a finer topological invariant. The algorithm proposed here can be applied to compute cup products on any polyhedral approximation of an object embedded in 3-space.