P.: Chain Homotopies for Object Topological Representations

Discrete Applied Mathematics, 2009

This paper presents a set of tools to compute topological information of simplicial complexes, tools that are applicable to extract topological information from digital pictures. A simplicial complex is encoded in a (non-unique) algebraic-topological format called AM-model. An AM-model for a given object K is determined by a concrete chain homotopy and it provides, in particular, integer (co)homology generators of K and representative (co)cycles of these generators. An algorithm for computing an AM-model and the cohomological invariant HB1 (derived from the rank of the cohomology ring) with integer coefficients for a finite simplicial complex in any dimension is designed here, extending the work done in in which the ground ring was a field. A concept of generators which are "nicely" representative is also presented. Moreover, we extend the definition of AM-models to 3D binary digital images and we design algorithms to update the AM-model information after voxel set operations (union, intersection, difference and inverse).

Lecture Notes in Computer Science, 2006

In this paper, algorithms for computing integer (co)homology of a simplicial complex of any dimension are designed, extending the work done in [1, 2, 3]. For doing this, the homology of the object is encoded in an algebraic-topological format (that we call AM-model). Moreover, in the case of 3D binary digital images, having as input AM-models for the images I and J, we design fast algorithms for computing the integer homology of I ∪ J, I ∩ J and I \ J.

Journal of Inequalities and Applications, 2013

In this paper, we recall some definitions and properties from digital topology, and we consider the simplices and simplicial complexes in digital images due to adjacency relations. Then we define the simplicial set and conclude that the simplicial identities are satisfied in digital images. Finally, we construct the group structure in digital images and define the simplicial groups in digital images. Consequently, we calculate the digital homology group of two-dimensional digital simplicial group.

Lecture Notes in Computer Science

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|.

SIAM Conference on Geometric and Physical Modeling (GD/SPM 2011), 2011

We propose a new iterative algorithm for computing the homology of arbitrary shapes discretized through simplicial complexes. We demonstrate how the simplicial homology of a shape can be effectively expressed in terms of the homology of its sub-components. The proposed algorithm retrieves the complete homological information of an input shape including the Betti numbers, the torsion coefficients and the representative homology generators. To the best of our knowledge, this is the first algorithm based on the constructive Mayer-Vietoris sequence, which relates the homology of a topological space to the homologies of its sub-spaces, i.e. the sub-components of the input shape and their intersections. We demonstrate the validity of our approach through a specific shape decomposition, based only on topological properties, which minimizes the size of the intersections between the sub-components and increases the efficiency of the algorithm.

Lecture Notes in Computer Science, 2008

Given a 3D binary voxel-based digital object V , an algorithm for computing homological information for V via a polyhedral cell complex is designed. By homological information we understand not only Betti numbers, representative cycles of homology classes and homological classification of cycles but also the computation of homology numbers related additional algebraic structures defined on homology (coproduct in homology, product in cohomology, (co)homology operations,...). The algorithm is mainly based on the following facts: a) a local 3D-polyhedrization of any 2×2×2 configuration of mutually 26-adjacent black voxels providing a coherent cell complex at global level; b) a description of the homology of a digital volume as an algebraic-gradient vector field on the cell complex (see Discrete Morse Theory [5],AT-model method [7,5]). Saving this vector field, we go further obtaining homological information at no extra time processing cost.

Discrete Applied Mathematics, 2005

We propose a method for computing the cohomology ring of three-dimensional (3D) digital binary-valued pictures. We obtain the cohomology ring of a 3D digital binary-valued picture I, via a simplicial complex K(I) topologically representing (up to isomorphisms of pictures) the picture I. The usefulness of a simplicial description of the "digital" cohomology ring of 3D digital binaryvalued pictures is tested by means of a small program visualizing the different steps of the method. Some examples concerning topological thinning, the visualization of representative (co)cycles of (co)homology generators and the computation of the cup product on the cohomology of simple pictures are showed.

Applied Mathematics & Information Sciences, 2014

The first goal of this paper is to show that the relative cohomology groups of digital images are determined algebraically by the relative homology groups of digital images. Then we state simplicial cup product for digital images and use it to establish ring structure of digital cohomology. Furthermore we give a method for computing the cohomology ring of digital images and give some examples concerning cohomology ring.

In this article we study the digital cubical homology groups of digital images which are based on the cubical homology groups of topological spaces in algebraic topology. We investigate some fundamental properties of cubical homology groups of digital images. We also calculate cubical homology groups of certain 2-dimensional and 3-dimensional digital im- ages. We give a relation between digital simplicial homology groups and digital cubical homology groups. Moreover we show that the Mayer- Vietoris Theorem need not be hold in digital images.

Proceedings 2001 International Conference on Image Processing (Cat. No.01CH37205), 2001

There are a number of tasks in low level vision and image processing that involve computing certain topological characteristics of objects in a given image including connectivity and the number of holes. In this note, we combine a new method combinatorial topology to compute the number of connected components and holes of objects in a given image, and fast segmentation methods to extract the objects.