Cubical Homology Research Articles

Singular cell homology

In the paper we generalize the singular cubical homology theory of digraphs and introduce a collection of cell singular homologies that are parametrized by a natural parameter of the category of digraphs. The natural parameter corresponds to the “length” of a directed line digraph that corresponds to the unit segment in algebraic topology. The introduced earlier singular cubical homology is a particular case of this collection that corresponds to the line digraph with one arrow. We describe relations between these theories for different parameters given by a natural number and study their basic properties. In particular, we prove the functoriality and homotopy invariance of these theories and compare the theory with the path homology theory that was introduced in our previous papers. Then we describe the transfer of the cell singular homology theories to the categories of graphs, quivers, and multigraphs.

Novel topological method to define scar heterogeneity relates to arrhythmia vulnerability

Abstract Funding Acknowledgements Type of funding sources: Public Institution(s). Main funding source(s): KU Leuven BOF, C14/18/079 Background Healing of myocardial infarction (MI) can result in heterogeneous scar structures, resulting in border zone conduction channels with altered electrophysiological properties, that lead to arrhythmia vulnerability. Identification of scar heterogeneity and potential conduction channels based on contrast enhanced MRI has shown to be viable. However, current methods rely on manual identification and systematic approaches are lacking. Objective To provide a systematic methodology to annotate scar heterogeneity on contrast enhanced MRI and to correlate it to arrhythmia vulnerability. Methods Antero-septal MI was induced in 10 domestic pigs by 120-minute occlusion of the left anterior descending artery followed by reperfusion. Late gadolinium enhanced MR imaging was performed one month after infarction. The endocardium and epicardium of the left ventricle were segmented and the voxels inside the left ventricle (LV) were extracted. All LV voxels with signal intensity above the middle of the myocardial signal intensity range were defined to be infarct voxels and otherwise intact voxels. The infarct voxel structure and intact voxel structure were analyzed using a novel computational cubical homology approach that automatically identifies "holes" in the respective structures. The so called topological "holes" were annotated using a minimal length cycle encircling the hole (see Figure 1). This cycle annotation corresponds with scar heterogeneity as it pinpoints regions of intertwined fibrosis and viable myocardium which can potentially act as functional arrhythmia substrate. Scar heterogeneity, annotated as cycles, was compared against two measures of arrhythmia vulnerability. On one hand, the number of premature ventricular complexes (PVCs) were determined during an incremental isoproterenol infusion protocol (ranging from no infusion to 0.04 µg/kg with increments of 0.01µg/kg each 5 minutes). And on the other hand, the ease of induction is defined as the complementary percentage of the steps of the programmed electrical stimulation (PES) protocol needed to induce a ventricular tachycardia. Results Scar heterogeneity cycles were automatically annotated in all animals. In 7 of the 9 animals a ventricular tachycardia was successfully induced during the PES protocol. The ease of induction correlated with the circumference of the myocardial infarct (r = 0.69, p = 0.040). Furthermore, the ease of induction tended to correlate with the circumference of the largest heterogeneity cycle (r = 0.61, p = 0.083). The total number of PVCs throughout the isoproterenol infusion protocol correlated with the circumference of the heterogeneity cycle (r = 0.72, p = 0.031). Conclusion A novel automated and systematic method to identify scar heterogeneity on contrast enhanced MR imaging is introduced. Furthermore, the presence of large heterogeneous scar structures, potentially adding an arrhythmia substrate, correlated with arrhythmia vulnerability.

Cubical Homology-Based Machine Learning: An Application in Image Classification

Persistent homology is a powerful tool in topological data analysis (TDA) to compute, study, and encode efficiently multi-scale topological features and is being increasingly used in digital image classification. The topological features represent a number of connected components, cycles, and voids that describe the shape of data. Persistent homology extracts the birth and death of these topological features through a filtration process. The lifespan of these features can be represented using persistent diagrams (topological signatures). Cubical homology is a more efficient method for extracting topological features from a 2D image and uses a collection of cubes to compute the homology, which fits the digital image structure of grids. In this research, we propose a cubical homology-based algorithm for extracting topological features from 2D images to generate their topological signatures. Additionally, we propose a novel score measure, which measures the significance of each of the sub-simplices in terms of persistence. In addition, gray-level co-occurrence matrix (GLCM) and contrast limited adapting histogram equalization (CLAHE) are used as supplementary methods for extracting features. Supervised machine learning models are trained on selected image datasets to study the efficacy of the extracted topological features. Among the eight tested models with six published image datasets of varying pixel sizes, classes, and distributions, our experiments demonstrate that cubical homology-based machine learning with the deep residual network (ResNet 1D) and Light Gradient Boosting Machine (lightGBM) shows promise with the extracted topological features.

Digital homotopy relations and digital homology theories

In this paper we prove results relating to two homotopy relations and four homology theories developed in the topology of digital images.<br /><br />We introduce a new type of homotopy relation for digitally continuous functions which we call ``strong homotopy.'' Both digital homotopy and strong homotopy are natural digitizations of classical topological homotopy: the difference between them is analogous to the difference between digital 4-adjacency and 8-adjacency in the plane.<br /><br />We also consider four different digital homology theories: a simplicial homology theory by Arslan et al which is the homology of the clique complex, a singular simplicial homology theory by D. W. Lee, a cubical homology theory by Jamil and Ali, and a new kind of cubical homology for digital images with $c_1$-adjacency which is easily computed, and generalizes a construction by Karaca \& Ege. We show that the two simplicial homology theories are isomorphic to each other, but distinct from the two cubical theories.<br /><br />We also show that homotopic maps have the same induced homomorphisms in the cubical homology theory, and strong homotopic maps additionally have the same induced homomorphisms in the simplicial theory.

Isomorphism of the cubical and categorical cohomology groups of a higher-rank graph

We use category-theoretic techniques to provide two proofs showing that for a higher-rank graph Λ \Lambda , its cubical (co-)homology and categorical (co-)homology groups are isomorphic in all degrees, thus answering a question of Kumjian, Pask and Sims in the positive. Our first proof uses the topological realization of a higher-rank graph, which was introduced by Kaliszewski, Kumjian, Quigg, and Sims. In our more combinatorial second proof, we construct, explicitly and in both directions, maps on the level of (co-)chain complexes that implement said isomorphism. Along the way, we extend the definition of cubical (co-)homology to allow arbitrary coefficient modules.

On the Vanishing of Discrete Singular Cubical Homology for Graphs

We prove that if $G$ is a graph without 3-cycles and 4-cycles, then the discrete cubical homology of $G$ is trivial in dimension $d$ for all $d\ge 2$. We also construct a sequence $\{G_d\}$ of grap...

Some remarks on hypergestural homology of spaces and its relation to classical homology

Classical homology of a topological space provides invariants of the space by means of triangulation or squaring made up from singular simplices (simplicial homology) or singular cubes (cubical homology) in the space. In much the same way, Mazzola's hypergestural homology intends to associate invariants to topological categories and, in particular, topological spaces by means of approximation with hypergestures playing the role of singular simplices and singular cubes. In this article, we locate Mazzola's hypergestural homology as a special kind of abstract cubical homology and propose two variations of Mazzola's construction, corresponding to simple geometric and physical interpretations of boundaries of hypergestures. Moreover, we discuss the relationship between hypergestural homology and classical cubical homology and prove that in many cases, one of our hypergestural homologies is invariant under homotopy equivalence of spaces, which is the main result of the article. Also, based on some examples, several structural improvements of hypergestural homology are suggested. However, one of these examples suggests that hypergestural homology could provide combinatorial information about a topological space beyond classical homology. Our computations are based on an explicit presentation of hypergestures, not included in previous works on gesture theory. This article has an Online Supplement, in which we expose some technical details, including the proof of the main result.

Labeled homology of higher-dimensional automata

We construct labeling homomorphisms on the cubical homology of higher-dimensional automata and show that they are natural with respect to cubical dimaps and compatible with the tensor product of HDAs. We also indicate two possible applications of labeled homology in concurrency theory.

Limit Theorems for Random Cubical Homology

This paper studies random cubical sets in \(\mathbb {R}^d\). Given a cubical set \(X \subset \mathbb {R}^d\), a random variable \(\omega _Q\in [0,1]\) is assigned for each elementary cube Q in X, and a random cubical set X(t) is defined by the sublevel set of X consisting of elementary cubes with \(\omega _Q\le t\) for each \(t\in [0,1]\). Under this setting, the main results of this paper show the limit theorems (law of large numbers and central limit theorem) for Betti numbers and lifetime sums of random cubical sets and filtrations. In addition to the limit theorems, the positivity of the limiting Betti numbers is also shown.

Computation of cubical homology, cohomology, and (co)homological operations via chain contraction

We introduce algorithms for the computation of homology, cohomology, and related operations on cubical cell complexes, using the technique based on a chain contraction from the original chain complex to a reduced one that represents its homology. This work is based on previous results for simplicial complexes, and uses Serre's diagonalization for cubical cells. An implementation in C++ of the introduced algorithms is available at http://www.pawelpilarczyk.com/chaincon/ together with some examples. The paper is self-contained as much as possible, and is written at a very elementary level, so that basic knowledge of algebraic topology should be sufficient to follow it.

Homology of representable sets

We generalize the notion of cubical homology to the class of locally compact representable sets in order to propose a new convenient method of reducing the complexity of a set while computing its homology.

Cubical homology and the Leech dimension of free partially commutative monoids

The paper is devoted to problems arising when applying homological algebra to computer science. It is proved that the Leech dimension of a free partially commutative monoid is equal to the least upper bound of the cardinalities of finite subsets of pairwise commuting generators of the monoid. For an arbitrary free partially commutative monoid in which every subset of pairwise commuting generators is finite and for any contravariant natural system on we construct a semicubical set with a homological system on this set such that the Leech homology groups are isomorphic to the cubical homology groups . Complexes of Abelian groups are also constructed enabling one to obtain (under additional finiteness conditions) algorithms for computing the Leech homology groups and homology groups with coefficients in right -modules.Bibliography: 16 titles.

Homology algorithm based on acyclic subspace

We present a new reduction algorithm for the efficient computation of the homology of a cubical set. The algorithm is based on constructing a possibly large acyclic subspace, and then computing the relative homology instead of the plain homology. We show that the construction of acyclic subspace may be performed in linear time. This significantly reduces the amount of data that needs to be processed in the algebraic way, and in practice it proves itself to be significantly more efficient than other available cubical homology algorithms.

Ideas from Zariski Topology in the Study of Cubical Homology

AbstractCubical sets and their homology have been used in dynamical systems as well as in digital imaging. We take a fresh look at this topic, following Zariski ideas from algebraic geometry. The cubical topology is defined to be a topology in ℝd in which a set is closed if and only if it is cubical. This concept is a convenient frame for describing a variety of important features of cubical sets. Separation axioms which, in general, are not satisfied here, characterize exactly those pairs of points which we want to distinguish. The noetherian property guarantees the correctness of the algorithms. Moreover, maps between cubical sets which are continuous and closed with respect to the cubical topology are precisely those for whom the homology map can be defined and computed without grid subdivisions. A combinatorial version of the Vietoris–Begle theorem is derived. This theorem plays the central role in an algorithm computing homology of maps which are continuous with respect to the Euclidean topology.

On the detection of simple points in higher dimensions using cubical homology

Simple point detection is an important task for several problems in discrete geometry, such as topology preserving thinning in image processing to compute discrete skeletons. In this paper, the approach to simple point detection is based on techniques from cubical homology, a framework ideally suited for problems in image processing. A (d-dimensional) unitary cube (for a d-dimensional digital image) is associated with every discrete picture element, instead of a point in epsilon(d) (the d-dimensional Euclidean space) as has been done previously. A simple point in this setting then refers to the removal of a unitary cube without changing the topology of the cubical complex induced by the digital image. The main result is a characterization of a simple point p (i.e., simple unitary cube) in terms of the homology groups of the (3d - 1) neighborhood of p for arbitrary, finite dimensions

On the Evolution of Vector Distance Functions of Closed Curves.

Inspired by the work by Gomes et al., we describe and analyze a vector distance function approach for the implicit evolution of closed curves of codimension larger than one. The approach is set up in complete generality, and then applied to the evolution of dynamic geometric active contours in [Formula: see text] (codimension three case). In order to carry this out one needs an explicit expression for the zero level set for which we propose a discrete connectivity method. This leads us to make connections with the new theory of cubical homology. We provide some explicit simulation results in order to illustrate the methodology.

Generalizations of Baxter's Theorem and cubical homology

Generalizations of Baxter's Theorem and cubical homology

Acyclic models and fibre spaces

Introduction. Most of the recent rapid development of the homology or cohomology theory of fibre spaces is based on the use of spectral sequences, as introduced by J. Leray [1]. In order to use spectral sequences successfully, it is almost always necessary to have a good deal of information about the term usually called for cohomology E2 or for homology E2 of the spectral sequence. In his original work, Leray proved that the term E2 in the spectral sequence for the Cech cohomology of a fibre space is naturally isomorphic to the cohomology of the base space with coefficients in the cohomology of the fibre. The appropriate analogues of this result for cubical homology, and cohomology were proved by J.-P. Serre [3 ], and are fundamental in applications of homology theory to homotopy theory. The main object of this paper is to calculate the term E2 in the homology spectral sequence of a fibre space by the use of the theory of acyclic models of Eilenberg and MacLane [4]. Notice that we said the homology spectral sequence of a fibre space, and did not specify whether we meant simplicial or cubical theory. One of our results asserts essentially that there is a homology spectral sequence of a fibre space, and that it may be obtained by using either simplicial or cubical theory. Further, we show that in cubical homology theory a filtration somewhat different from the one used by Serre may be used to obtain the spectral sequence. This filtration is symmetric in all coordinates, and consequently much more convenient to use when dealing with fibre spaces wherein a multiplication is defined in the base space and total space, and such that projection map is a homomorphism. This fact will be exploited elsewhere. In the course of our work, it has been necessary to make a fairly extensive axiomatic investigation of the notion of a singular theory, and of the notion of local One outcome of this work is a proof of the fact that cubical homology and simplicial homology coincide even with coefficients. For ordinary coefficients this was proved by Eilenberg and MacLane [4]. Since much of our work is quite abstract, the general theory will be interspersed with examples from cubical theory. Technically this paper is divided into chapters, and eachi chapter into sections. The notation 11.3.4 refers to the 4th numbered statement in the