A Tool for Integer Homology Computation: Lambda-At Model

In this paper, we formalize the notion of lambda-AT-model (where $\lambda$ is a non-null integer) for a given chain complex, which allows the computation of homological information in the integer domain avoiding using the Smith Normal Form of the boundary matrices. We present an algorithm for computing such a model, obtaining Betti numbers, the prime numbers p involved in the invariant factors of the torsion subgroup of homology, the amount of invariant factors that are a power of p and a set of representative cycles of generators of homology mod p, for each p. Moreover, we establish the minimum valid lambda for such a construction, what cuts down the computational costs related to the torsion subgroup. The tools described here are useful to determine topological information of nD structured objects such as simplicial, cubical or simploidal complexes and are applicable to extract such an information from digital pictures.Comment: Journal Image and Vision Computing, Volume 27 Issue 7, June, 200

A Graph-with-Loop Structure for a Topological Representation of 3D Objects

Given a cell complex K whose geometric realization |K| is embedded in R 3 and a continuous function h: |K|→R (called the height function), we construct a graph G h (K) which is an extension of the Reeb graph R h (|K|). More concretely, the graph G h (K) without loops is a subdivision of R h (|K|). The most important difference between the graphs G h (K) and R h (|K|) is that G h (K) preserves not only the number of connected components but also the number of “tunnels” (the homology generators of dimension 1) of K. The latter is not true in general for R h (|K|). Moreover, we construct a map ψ: G h (K)→K identifying representative cycles of the tunnels in K with the ones in G h (K) in the way that if e is a loop in G h (K), then ψ(e) is a cycle in K such that all the points in |ψ(e)| belong to the same level set in |K|

Extending the notion of AT-Model for integer homology computation

When the ground ring is a field, the notion of algebraic topological model (AT-model) is a useful tool for computing (co)homology, representative (co)cycles of (co)homology generators and the cup product on cohomology of nD digital images as well as for controlling topological information when the image suffers local changes [6,7,9]. In this paper, we formalize the notion of λ-AT-model (λ being an integer) which extends the one of AT-model and allows the computation of homological information in the integer domain without computing the Smith Normal Form of the boundary matrices. We present an algorithm for computing such a model, obtaining Betti numbers, the prime numbers p involved in the invariant factors (corresponding to the torsion subgroup of the homology), the amount of invariant factors that are a power of p and a set of representative cycles of the generators of homology mod p, for such p

Combinatorial models for topology-based geometric modeling

Many combinatorial (topological) models have been proposed in geometric modeling, computational geometry, image processing or analysis, for representing subdivided geometric objects, i.e. partitionned into cells of different dimensions: vertices, edges, faces, volumes, etc. We can distinguish among models according to the type of cells (regular or not regular ones), the type of assembly ("manifold" or "non manifold"), the type of representation (incidence graphs or ordered models), etc

Avances en Matemática Discreta en Andalucía. V Encuentro andaluz de Matemática Discreta. La Línea de la Concepción (Cádiz), 4-5 de julio de 2007

V Encuentro andaluz de Matemática Discreta. La Línea de la Concepción (Cádiz), 4-5 de julio de 200

Homology of Cellular Structures Allowing Multi-incidence

International audienceThis paper focuses on homology computation over ‘cellular’ structures that allow multi-incidence between cells. We deal here with combinatorial maps, more precisely chains of maps and subclasses such as maps and generalized maps. Homology computation on such structures is usually achieved by computing simplicial homology on a simplicial analog. But such an approach is computationally expensive because it requires computing this simplicial analog and performing the homology computation on a structure containing many more cells (simplices) than the initial one. Our work aims at providing a way to compute homologies directly on a cellular structure. This is done through the computation of incidence numbers. Roughly speaking, if two cells are incident, then their incidence number characterizes how they are attached. Having these numbers naturally leads to the definition of a boundary operator, which induces a homology. Hence, we propose a boundary operator for chains of maps and provide optimization for maps and generalized maps. It is proved that, under specific conditions, the homology of a combinatorial map as defined in the paper is equivalent to the homology of its simplicial analogue