Cup Products on Polyhedral Approximations of 3D Digital Images

Abstract

Abstract. 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.

Human Gait Identification Using Persistent Homology

“…A new approach called reduced persistence barcode is used to improve the discriminative capacity of the representation. This approach adds the possibility to use more discriminative topological invariants such as cup product [7]. Furthermore, we plan to use the bottleneck distance as similarities measures between barcodes, which guarantees stability in Persistent Homology.…”

Distributed computation of low-dimensional cup products

“…To implement (2) one can use the Alexander-Whitney diagonal approximation formula in the case of simplicial spaces, and Serre's analogue of this for cubical spaces. Details of the cubical analogue can be found in [26], and details on practical implementations of these two formulae can be found in [16,18,14,15,28,17,24]. In this section we assume that X is an arbitrary connected regular finite CW-space and observe that for k = 2 the homomorphism (2), and hence the cup product (1), can be read directly from a group presentation P = x | r for the fundamental group π 1 X.…”

“…In Section 3 we illustrate the method on the integral cohomology ring of a 3-dimensional digital image. Previous papers [16,18,14,15] have described different approaches to computing the cohomology ring, over Z/2Z, of cubical and simplicial spaces arising from 3-dimensional digital images; these papers are based on techniques in [28,17,24]. The fundamental group algorithm in [1] involves the construction of an admissible discrete vector field on X, and this construction can consume significant memory and time for large CW-spaces X.…”

A Tutorial on Well-Composedness

“…Gonzalez-Diaz et al [67] introduced in 2011 a method able to topologically repair a cubical complex associated with a 3D binary digital image into a polyhedral complex which is homotopy equivalent and wellcomposed in the sense that the boundary of its underlying polyhedron is a 2-manifold; cohomological information [71,72,65,66,70] is then computable directly on this manifold. The proposed (local) method is homotopy preserving, which implies that the resulting cohomological informations can be used to recognition or characterization tasks.…”