An iterative algorithm for homology computation on simplicial shapes

Abstract

We propose a new iterative algorithm for computing the homology of arbitrary shapes discretized through simplicial complexes, We demonstrate how the simplicial homology of a shape can be effectively expressed in terms of the homology of its sub-components. The proposed algorithm retrieves the complete homological information of an input shape including the Betti numbers, the torsion coefficients and the representative homology generators.To the best of our knowledge, this is the first algorithm based on the constructive Mayer-Vietoris sequence, which relates the homology of a topological space to the homologies of its sub-spaces, i.e. the sub-components of the input shape and their intersections. We demonstrate the validity of our approach through a specific shape decomposition, based only on topological properties, which minimizes the size of the intersections between the sub-components and increases the efficiency of the algorithm.

Homology Computation During an Incremental Construction Process

“…In particular, note that the SES theorem applies for the inverse operation of the identification. At last, note that these results generalize in some sense the results described in [5,6], since the identification operation is a more basic operation than the gluing of connected components. In other words, gluing connected components can be achieved by identifying parts of their boundaries; but the identification operation can also be applied to subsets of simplices belonging to the same connected component.…”

“…This algorithm has been applied for the Manifold-Connected decomposition of abstract simplicial complexes [6]. The basic idea is the following: let B and C be sub-complexes of A, such that A = B∪C, and Υ B∩C , Υ B and Υ C are homological equivalences associated with B ∩ C, B and C; a short exact sequence ((B ∩ C), (B ⊕ C), A = (B ∪ C), i, j, r, s) can be defined 5 , and a homological equivalence associated with A can be computed by applying the SES theorem.…”

“…They can also be applied to cellular combinatorial structures as incidence graphs 6 [10] and combinatorial maps [7]. An incidence graph is represented in figure 2; results in [3] are obtained for combinatorial maps, which generalize incidence graphs in order to take multi-incidence into account 7 , and these results directly apply to incidence graphs.…”

“…Although some aspects are close to the work described in [6], the approach here focuses on the application of construction operations and the related complexities. Since only the structure of an object has to be taken into account in order to compute its homology, it will be assumed that it is represented by a semi-simplicial set or a cellular combinatorial structure (incidence graph or combinatorial map).…”

Choking Loops on Surfaces

“…In practice, such noncontractible loops have a multitude of potential applications in segmentation, parameterization, topological simplification and repair, path planning, detection of geometrical and topological features, biomedical imaging, and determining integrability of partial differential equations; see, e.g., [1], [2], [3], [4], [5], [6], [7], [8]. A number of algorithms based on surface homology (equivalence classes of such loops, equivalent when they form the boundary of a patch) and homotopy (equivalence classes of such loops, equivalent when they can deform continuously from one to the other) have been proposed.…”