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-

Spatiotemporal Barcodes for Image Sequence Analysis

Taking as input a time-varying sequence of two-dimensional (2D) binary images, we develop an algorithm for computing a spatiotemporal 0–barcode encoding lifetime of connected components on the image sequence over time. This information may not coincide with the one provided by the 0–barcode encoding the 0–persistent homology, since the latter does not respect the principle that it is not possible to move backwards in time. A cell complex K is computed from the given sequence, being the cells of K classified as spatial or temporal depending on whether they connect two consecutive frames or not. A spatiotemporal path is defined as a sequence of edges of K forming a path such that two edges of the path cannot connect the same two consecutive frames. In our algorithm, for each vertex v ∈ K, a spatiotemporal path from v to the “oldest” spatiotemporally-connected vertex is computed and the corresponding spatiotemporal 0–bar is added to the spatiotemporal 0–barcode.Junta de Andalucía FQM-369Ministerio de Economía y Competitividad MTM2012-3270

Efficiently Storing Well-Composed Polyhedral Complexes Computed Over 3D Binary Images

A 3D binary image I can be naturally represented by a combinatorial-algebraic structure called cubical complex and denoted by Q(I ), whose basic building blocks are vertices, edges, square faces and cubes. In Gonzalez-Diaz et al. (Discret Appl Math 183:59–77, 2015), we presented a method to “locally repair” Q(I ) to obtain a polyhedral complex P(I ) (whose basic building blocks are vertices, edges, specific polygons and polyhedra), homotopy equivalent to Q(I ), satisfying that its boundary surface is a 2D manifold. P(I ) is called a well-composed polyhedral complex over the picture I . Besides, we developed a new codification system for P(I ), encoding geometric information of the cells of P(I ) under the form of a 3D grayscale image, and the boundary face relations of the cells of P(I ) under the form of a set of structuring elements. In this paper, we build upon (Gonzalez-Diaz et al. 2015) and prove that, to retrieve topological and geometric information of P(I ), it is enough to store just one 3D point per polyhedron and hence neither grayscale image nor set of structuring elements are needed. From this “minimal” codification of P(I ), we finally present a method to compute the 2-cells in the boundary surface of P(I ).Ministerio de Economía y Competitividad MTM2015-67072-

3D well-composed polyhedral complexes

A binary three-dimensional (3D) image II is well-composed if the boundary surface of its continuous analog is a 2D manifold. In this paper, we present a method to locally “repair” the cubical complex Q(I)Q(I) (embedded in R3R3) associated to II to obtain a polyhedral complex P(I)P(I) homotopy equivalent to Q(I)Q(I) such that the boundary surface of P(I)P(I) is a 2D manifold (and, hence, P(I)P(I) is a well-composed polyhedral complex). For this aim, we develop a new codification system for a complex KK, called ExtendedCubeMap (ECM) representation of KK, that codifies: (1) the information of the cells of KK (including geometric information), under the form of a 3D grayscale image gPgP; and (2) the boundary face relations between the cells of KK, under the form of a set BPBP of structuring elements that can be stored as indexes in a look-up table. We describe a procedure to locally modify the ECM representation EQEQ of the cubical complex Q(I)Q(I) to obtain an ECM representation of a well-composed polyhedral complex P(I)P(I) that is homotopy equivalent to Q(I)Q(I). The construction of the polyhedral complex P(I)P(I) is accomplished for proving the results though it is not necessary to be done in practice, since the image gPgP (obtained by the repairing process on EQEQ) together with the set BPBP codify all the geometric and topological information of P(I)P(I)

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

Topological tracking of connected components in image sequences

Persistent homology provides information about the lifetime of homology classes along a filtration of cell complexes. Persistence barcode is a graphi- cal representation of such information. A filtration might be determined by time in a set of spatiotemporal data, but classical methods for computing persistent homology do not respect the fact that we can not move back- wards in time. In this paper, taking as input a time-varying sequence of two-dimensional (2D) binary digital images, we develop an algorithm for en- coding, in the so-called spatiotemporal barcode, lifetime of connected compo- nents (of either the foreground or background) that are moving in the image sequence over time (this information may not coincide with the one provided by the persistence barcode). This way, given a connected component at a specific time in the sequence, we can track the component backwards in time until the moment it was born, by what we call a spatiotemporal path. The main contribution of this paper with respect to our previous works lies in a new algorithm that computes spatiotemporal paths directly, valid for both foreground and background and developed in a general context, setting the ground for a future extension for tracking higher dimensional topological features in nD binary digital image sequences.Ministerio de Economía y Competitividad MTM2015-67072-

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

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|

Integral operators for computing homology generators at any dimension

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

Modelo álgebro-topológico de representación de imágenes digitales en nD

Para introducir este trabajo es necesario hablar de TOPOLOG´IA. La topología es la rama de las matemáticas que estudia las propiedades de los objetos que no cambian bajo homeomorfismos (aplicaciones biyectivas y bicontinuas). Un problema fundamental en Topología es la clasificación de objetos salvo homeomorfismos