Computing the Component-Labeling and the Adjacency Tree of a Binary Digital Image in Near Logarithmic-Time

Connected component labeling (CCL) of binary images is one of the fundamental operations in real time applications. The adjacency tree (AdjT) of the connected components offers a region-based representation where each node represents a region which is surrounded by another region of the opposite color. In this paper, a fully parallel algorithm for computing the CCL and AdjT of a binary digital image is described and implemented, without the need of using any geometric information. The time complexity order for an image of m × n pixels under the assumption that a processing element exists for each pixel is near O(log(m+ n)). Results for a multicore processor show a very good scalability until the so-called memory bandwidth bottleneck is reached. The inherent parallelism of our approach points to the direction that even better results will be obtained in other less classical computing architectures.Ministerio de Economía y Competitividad MTM2016-81030-PMinisterio de Economía y Competitividad TEC2012-37868-C04-0

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

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

Cell AT-models for digital volumes

In [4], given a binary 26-adjacency voxel-based digital volume V, the homological information (that related to n-dimensional holes: connected components, ”tunnels” and cavities) is extracted from a linear map (called homology gradient vector field) acting on a polyhedral cell complex P(V) homologically equivalent to V. We develop here an alternative way for constructing P(V) based on homological algebra arguments as well as a new more efficient algorithm for computing a homology gradient vector field based on the contractibility of the maximal cells of P(V)

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 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

Homological Region Adjacency Tree for a 3D Binary Digital Image via HSF Model

Given a 3D binary digital image I, we define and compute an edge-weighted tree, called Homological Region Tree (or Hom-Tree, for short). It coincides, as unweighted graph, with the classical Region Adjacency Tree of black 6-connected components (CCs) and white 26- connected components of I. In addition, we define the weight of an edge (R, S) as the number of tunnels that the CCs R and S “share”. The Hom-Tree structure is still an isotopic invariant of I. Thus, it provides information about how the different homology groups interact between them, while preserving the duality of black and white CCs. An experimentation with a set of synthetic images showing different shapes and different complexity of connected component nesting is performed for numerically validating the method.Ministerio de Economía y Competitividad MTM2016-81030-

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

Searching high order invariants in computer imagery

In this paper, we present a direct computational application of Homological Perturbation Theory (HPT, for short) to computer imagery. More precisely, the formulas of the A ∞–coalgebra maps Δ 2 and Δ 3 using the notion of AT-model of a digital image, and the HPT technique are implemented. The method has been tested on some specific examples, showing the usefulness of this computational tool for distinguishing 3D digital images