Homological optimality in Discrete Morse Theory through chain homotopies

Morse theory is a fundamental tool for analyzing the geometry and topology of smooth manifolds. This tool was translated by Forman to discrete structures such as cell complexes, by using discrete Morse functions or equivalently gradient vector fields. Once a discrete gradient vector field has been defined on a finite cell complex, information about its homology can be directly deduced from it. In this paper we introduce the foundations of a homology-based heuristic for finding optimal discrete gradient vector fields on a general finite cell complex K. The method is based on a computational homological algebra representation (called homological spanning forest or HSF, for short) that is an useful framework to design fast and efficient algorithms for computing advanced algebraic-topological information (classification of cycles, cohomology algebra, homology A(∞)-coalgebra, cohomology operations, homotopy groups, …). Our approach is to consider the optimality problem as a homology computation process for a chain complex endowed with an extra chain homotopy operator

Towards optimality in discrete Morse Theory through chain homotopies

Once a discrete Morse function has been defined on a finite cell complex, information about its homology can be deduced from its critical elements. The main objective of this paper is to define optimal discrete gradient vector fields on general finite cell complexes, where optimality entails having the least number of critical elements. Our approach is to consider this problem as a homology computation question for chain complexes endowed with extra algebraic nilpotent operator

Generating Second Order (Co)homological Information within AT-Model Context

In this paper we design a new family of relations between (co)homology classes, working with coefficients in a field and starting from an AT-model (Algebraic Topological Model) AT(C) of a finite cell complex C These relations are induced by elementary relations of type “to be in the (co)boundary of” between cells. This high-order connectivity information is embedded into a graph-based representation model, called Second Order AT-Region-Incidence Graph (or AT-RIG) of C. This graph, having as nodes the different homology classes of C, is in turn, computed from two generalized abstract cell complexes, called primal and dual AT-segmentations of C. The respective cells of these two complexes are connected regions (set of cells) of the original cell complex C, which are specified by the integral operator of AT(C). In this work in progress, we successfully use this model (a) in experiments for discriminating topologically different 3D digital objects, having the same Euler characteristic and (b) in designing a parallel algorithm for computing potentially significant (co)homological information of 3D digital objects.Ministerio de Economía y Competitividad MTM2016-81030-PMinisterio de Economía y Competitividad TEC2012-37868-C04-0

Homological Spanning Forests for Discrete Objects

Computing and representing topological information form an important part in many applications such as image representation and compression, classification, pattern recognition, geometric modelling, etc. The homology of digital objects is an algebraic notion that provides a concise description of their topology in terms of connected components, tunnels and cavities. The purpose of this work is to develop a theoretical and practical frame- work for efficiently extracting and exploiting useful homological information in the context of nD digital images. To achieve this goal, we intend to combine known techniques in algebraic topology, and image processing. The main notion created for this purpose consists of a combinatorial representation called Homological Spanning Forest (or HSF, for short) of a digital object or a digital image. This new model is composed of a set of directed forests, which can be constructed under an underlying cell complex format of the image. HSF’s are based on the algebraic concept of chain homotopies and they can be considered as a suitable generalization to higher dimensional cell complexes of the topological meaning of a spanning tree of a geometric graph. Based on the HSF representation, we present here a 2D homology-based framework for sequential and parallel digital image processing.Premio Extraordinario de Doctorado U

Homological spanning forest framework for 2D image analysis

A 2D topology-based digital image processing framework is presented here. This framework consists of the computation of a flexible geometric graph-based structure, starting from a raster representation of a digital image I. This structure is called Homological Spanning Forest (HSF for short), and it is built on a cell complex associated to I. The HSF framework allows an efficient and accurate topological analysis of regions of interest (ROIs) by using a four-level architecture. By topological analysis, we mean not only the computation of Euler characteristic, genus or Betti numbers, but also advanced computational algebraic topological information derived from homological classification of cycles. An initial HSF representation can be modified to obtain a different one, in which ROIs are almost isolated and ready to be topologically analyzed. The HSF framework is susceptible of being parallelized and generalized to higher dimensions

Triangle mesh compression and homological spanning forests

Triangle three-dimensional meshes have been widely used to represent 3D objects in several applications. These meshes are usually surfaces that require a huge amount of resources when they are stored, processed or transmitted. Therefore, many algorithms proposing an efficient compression of these meshes have been developed since the early 1990s. In this paper we propose a lossless method that compresses the connectivity of the mesh by using a valence-driven approach. Our algorithm introduces an improvement over the currently available valence-driven methods, being able to deal with triangular surfaces of arbitrary topology and encoding, at the same time, the topological information of the mesh by using Homological Spanning Forests. We plan to develop in the future (geo-topological) image analysis and processing algorithms, that directly work with the compressed data

Toward Parallel Computation of Dense Homotopy Skeletons for nD Digital Objects

An appropriate generalization of the classical notion of abstract cell complex, called primal-dual abstract cell complex (pACC for short) is the combinatorial notion used here for modeling and analyzing the topology of nD digital objects and images. Let D ⊂ I be a set of n-xels (ROI) and I be a n-dimensional digital image.We design a theoretical parallel algorithm for constructing a topologically meaningful asymmetric pACC HSF(D), called Homological Spanning Forest of D (HSF of D, for short) starting from a canonical symmetric pACC associated to I and based on the application of elementary homotopy operations to activate the pACC processing units. From this HSF-graph representation of D, it is possible to derive complete homology and homotopy information of it. The preprocessing procedure of computing HSF(I) is thoroughly discussed. In this way, a significant advance in understanding how the efficient HSF framework for parallel topological computation of 2D digital images developed in [2] can be generalized to higher dimension is made.Ministerio de Economía y Competitividad TEC2016-77785-PMinisterio de Economía y Competitividad MTM2016-81030-

Effective homology of k-D digital objects (partially) calculated in parallel

In [18], a membrane parallel theoretical framework for computing (co)homology information of fore- ground or background of binary digital images is developed. Starting from this work, we progress here in two senses: (a) providing advanced topological information, such as (co)homology torsion and effi- ciently answering to any decision or classification problem for sum of k -xels related to be a (co)cycle or a (co)boundary; (b) optimizing the previous framework to be implemented in using GPGPU computing. Discrete Morse theory, Effective Homology Theory and parallel computing techniques are suitably com- bined for obtaining a homological encoding, called algebraic minimal model, of a Region-Of-Interest (seen as cubical complex) of a presegmented k -D digital image

Connectivity calculus of fractal polyhedrons

The paper analyzes the connectivity information (more precisely, numbers of tunnels and their homological (co)cycle classification) of fractal polyhedra. Homology chain contractions and its combinatorial counterparts, called homological spanning forest (HSF), are presented here as an useful topological tool, which codifies such information and provides an hierarchical directed graph-based representation of the initial polyhedra. The Menger sponge and the Sierpiński pyramid are presented as examples of these computational algebraic topological techniques and results focussing on the number of tunnels for any level of recursion are given. Experiments, performed on synthetic and real image data, demonstrate the applicability of the obtained results. The techniques introduced here are tailored to self-similar discrete sets and exploit homology notions from a representational point of view. Nevertheless, the underlying concepts apply to general cell complexes and digital images and are suitable for progressing in the computation of advanced algebraic topological information of 3-dimensional objects

Towards a certified computation of homology groups for digital images

International audienceIn this paper we report on a project to obtain a verified computation of homology groups of digital images. The methodology is based on program- ming and executing inside the COQ proof assistant. Though more research is needed to integrate and make efficient more processing tools, we present some examples partially computed in COQ from real biomedical images