Homological Computation Using Spanning Trees

Abstract

Abstract. We introduce here a new F2 homology computation algorithm based on a generalization of the spanning tree technique on a finite 3-dimensional cell complex K embedded in R 3 . We demonstrate that the complexity of this algorithm is linear in the number of cells. In fact, this process computes an algebraic map φ over K, called homology gradient vector field (HGVF), from which it is possible to infer in a straightforward manner homological information like Euler characteristic, relative homology groups, representative cycles for homology generators, topological skeletons, Reeb graphs, cohomology algebra, higher (co)homology operations, etc. This process can be generalized to others coefficients, including the integers, and to higher dimension.

Homological spanning forest framework for 2D image analysis

“…The AT-model is based in the description of homology in terms of chain homotopies. This idea is not new (comes back to the Eilenberg-MacLane work on Algebraic Topology [10]) and has been developed in algebraic-topological methods like Effective Homology [37,40] and Homological Perturbation Theory [20] and in the discrete settings in Discrete Morse [13] and AT-model [15,33] theories. For example, in Discrete Morse Theory, the HSF structure can be specified as an appropriate graph description of an optimal discrete gradient vector field.…”

“…In the case of coefficients on a field, it has been proved (see [15,22,33]) that the whole homology computation process can be exclusively specified by the exhaustive use of these operations. In that case, they are a chain homotopy equivalence version of the classical cell collapsing operation (see, for example, [4]).…”

Homological optimality in Discrete Morse Theory through chain homotopies

“…The underlying idea is to classify chain homotopies as gradient vector fields on a finite cell complex and vice versa. In order to do this, we use a chain homotopy operator (see González-Dí az and Real (2005), González-Díaz et al (2009)) at algebraic level, and its associated graph-based representation (homological spanning forest or, HSF for short Molina-Abril and Real (2009)) at combinatorial level.…”

A Homological–Based Description of Subdivided nD Objects

“…We can progress in this way in the task to "combinatorialize" homological information at two level: local (DMT) and global (in terms of coordinated-based hierarchical forests based on chain homotopies [10]). The algebraic nature of the global approach can be combinatorialized if we place these chain homotopies and critical cells (homology generators) in a graph-based ambiance.…”