Cubical Cohomology Ring of 3D Photographs

Cohomology and cohomology ring of three-dimensional (3D) objects are topological invariants that characterize holes and their relations. Cohomology ring has been traditionally computed on simplicial complexes. Nevertheless, cubical complexes deal directly with the voxels in 3D images, no additional triangulation is necessary, facilitating efficient algorithms for the computation of topological invariants in the image context. In this paper, we present formulas to directly compute the cohomology ring of 3D cubical complexes without making use of any additional triangulation. Starting from a cubical complex $Q$ that represents a 3D binary-valued digital picture whose foreground has one connected component, we compute first the cohomological information on the boundary of the object, $\partial Q$ by an incremental technique; then, using a face reduction algorithm, we compute it on the whole object; finally, applying the mentioned formulas, the cohomology ring is computed from such information

Cups products in Z2-cohomology of 3D polyhedral complexes

Let I=(Z3,26,6,B) be a 3D digital image, let Q(I) be the associated cubical complex and let ∂Q(I) be the subcomplex of Q(I) whose maximal cells are the quadrangles of Q(I) shared by a voxel of B in the foreground -- the object under study -- and by a voxel of Z3∖B in the background -- the ambient space. We show how to simplify the combinatorial structure of ∂Q(I) and obtain a 3D polyhedral complex P(I) homeomorphic to ∂Q(I) but with fewer cells. We introduce an algorithm that computes cup products on H∗(P(I);Z2) directly from the combinatorics. The computational method introduced here can be effectively applied to any polyhedral complex embedded in R3

Encoding Specific 3D Polyhedral Complexes Using 3D Binary Images

We build upon the work developed in [4] in which we presented a method to “locally repair” the cubical complex Q(I) associated to a 3D binary image I, to obtain a “well-composed” polyhedral complex P(I), homotopy equivalent to Q(I). There, we developed a new codification system for P(I), called ExtendedCubeMap (ECM) representation, that encodes: (1) the (geometric) information of the cells of P(I) (i.e., which cells are presented and where), under the form of a 3D grayscale image gP ; (2) the boundary face relations between the cells of P(I), under the form of a set BP of structuring elements. In this paper, we simplify ECM representations, proving that geometric and topological information of cells can be encoded using just a 3D binary image, without the need of using colors or sets of structuring elements. We also outline a possible application in which well-composed polyhedral complexes can be useful.Junta de Andalucía FQM-369Ministerio de Economía y Competitividad MTM2012-32706Ministerio de Economía y Competitividad MTM2015-67072-

Persistent homology for 3D reconstruction evaluation

Space or voxel carving is a non-invasive technique that is used to produce a 3D volume and can be used in particular for the reconstruction of a 3D human model from images captured from a set of cameras placed around the subject. In [1], the authors present a technique to quantitatively evaluate spatially carved volumetric representations of humans using a synthetic dataset of typical sports motion in a tennis court scenario, with regard to the number of cameras used. In this paper, we compute persistent homology over the sequence of chain complexes obtained from the 3D outcomes with increasing number of cameras. This allows us to analyze the topological evolution of the reconstruction process, something which as far as we are aware has not been investigated to date

Cup products on polyhedral approximations of 3D digital images

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

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

Well-Composed Cell Complexes

Well-composed 3D digital images, which are 3D binary digital images whose boundary surface is made up by 2D manifolds, enjoy important topological and geometric properties that turn out to be advantageous for some applications. In this paper, we present a method to transform the cubical complex associated to a 3D binary digital image (which is not generally a well-composed image) into a cell complex that is homotopy equivalent to the first one and whose boundary surface is composed by 2D manifolds. This way, the new representation of the digital image can benefit from the application of algorithms that are developed over surfaces embedded in ℝ3

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|