Integral Operators for Computing Homology Generators at Any Dimension

Abstract

Starting from an nD geometrical object, a cellular subdivision of such an object provides an algebraic counterpart from which homology information can be computed. In this paper, we develop a process to drastically reduce the amount of data that represent the original object, with the purpose of a subsequent homology computation. The technique applied is based on the construction of a sequence of elementary chain homotopies (integral operators) which algebraically connect the initial object with a simplified one with the same homological information than the former.

Associating Cell Complexes to Four Dimensional Digital Objects

“…Other tools used for obtaining the cell complex associated to a digital object are the integral operators (see [3]). …”

“…Now, we only need to join these pieces to obtain the searched cell complex. Let us note that such that complex is homologically equivalent to the given 4-dimensional digital object according to Proposition 1 in [3].…”

Homological Computation Using Spanning Trees

“…An algebraic GVF satisfying the conditions φ∂φ = φ and ∂φd = ∂ will be called a homology GVF [6]. If φ is a combinatorial GVF which is only non-null for a unique cell a ∈ K q and satisfying the extra-condition φ∂φ = φ, then it is called a (combinatorial) integral operator [3]. An algebraic GVF φ is called strongly nilpotent if it satisfies the following property:…”

Getting Topological Information for a 80-Adjacency Doxel-Based 4D Volume through a Polytopal Cell Complex

“…The idea is to deform Q 4 using integral operators (elementary chain homotopy operators increasing the dimension by 1, which are non null only acting on one element [5]) to get the convex hull of this configuration. Now, we give an orientation to each cell c which preserves the global coherence on the cell complex (see Algorithm 2).…”

“…A three dimensional cell complex consists of vertices (0-cells), edges (1-cells), faces (2-cells) and polyhedra (3-cells). In particular, each edge connects two vertices, each face is enclosed by a loop of edges, and each 3-cell is enclosed by an envelope of faces; (b) (homology analysis level) Homology information about K(V ) is codified in homological algebra terms [5,6]. This method has recently evolving to a technique which for generating a Z/2Z-coefficient AT-model for a 26-adjacency voxel-based digital binary volume V uses a polyhedral cell complex at geometric modeling level [11,12,17,19] and a chain homotopy map (described by a vector fields or by a discrete differential form) at homology analysis level [20,24].…”