Cubical Cohomology Ring of 3D Photographs

Details Cubical Cohomology Ring of 3D Photographs Cubical Cohomology Ring of 3D Photographs

2011-05-20

arxiv.org/abs/1105.4183v1

Cubical cohomology ring of 3D photographs. International Journal of Imaging Systems and Technology. Volume 21, Issue 1, March

Computer Science

Computer Vision and Pattern Recognition

Scientific paper

10.1002/ima.20271

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.

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