  Cohomology and cohomology ring of three-dimensional (3D) objects are
topological invariants that characterize holes and their relations. Cohomology
ring has been traditionally computed on simplicial complexes. Nevertheless,
cubical complexes deal directly with the voxels in 3D images, no additional
triangulation is necessary, facilitating efficient algorithms for the
computation of topological invariants in the image context. In this paper, we
present formulas to directly compute the cohomology ring of 3D cubical
complexes without making use of any additional triangulation. Starting from a
cubical complex $Q$ that represents a 3D binary-valued digital picture whose
foreground has one connected component, we compute first the cohomological
information on the boundary of the object, $\partial Q$ by an incremental
technique; then, using a face reduction algorithm, we compute it on the whole
object; finally, applying the mentioned formulas, the cohomology ring is
computed from such information.
