scholarly journals On the Vanishing of Discrete Singular Cubical Homology for Graphs

2021 ◽  
Vol 35 (1) ◽  
pp. 35-54 ◽  
Author(s):  
Hélène Barcelo ◽  
Curtis Greene ◽  
Abdul Salam Jarrah ◽  
Volkmar Welker
Keyword(s):  
Cubical Homology

Ideas from Zariski Topology in the Study of Cubical Homology

2007 ◽  
Vol 59 (5) ◽  
pp. 1008-1028
Author(s):  
Tomasz Kaczynski ◽  
Marian Mrozek ◽  
Anik Trahan
Keyword(s):  
Algebraic Geometry ◽  
Dynamical Systems ◽  
Digital Imaging ◽  
Zariski Topology ◽  
Euclidean Topology ◽  
Separation Axioms ◽  
Cubical Sets ◽  
Noetherian Property ◽  
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.

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

2006 ◽  
Vol 15 (8) ◽  
pp. 2462-2469 ◽  
Author(s):  
M. Niethammer ◽  
W.D. Kalies ◽  
K. Mischaikow ◽  
A. Tannenbaum
Keyword(s):  
Higher Dimensions ◽  
Cubical Homology ◽  
Simple Points

scholarly journals Cubical homology-based Image Classification - A Comparative Study

2021 ◽  
Author(s):  
◽  
Seungho Choe
Keyword(s):  
Machine Learning ◽  
Image Classification ◽  
Digital Image ◽  
Persistent Homology ◽  
Topological Data Analysis ◽  
Connected Components ◽  
Gradient Boosting ◽  
Topological Features ◽  
Light Gradient ◽  
Cubical Homology

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 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 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 score, which measures the significance of each of the sub-simplices in terms of persistence. Also, gray level co-occurrence matrix (GLCM) and contrast limited adapting histogram equalization (CLAHE) are used as a supplementary method for extracting features. Machine learning techniques are then employed to classify images using the topological signatures. Among the eight tested algorithms with six published image datasets with varying pixel sizes, classes, and distributions, our experiments demonstrate that cubical homology-based machine learning with deep residual network (ResNet 1D) and Light Gradient Boosting Machine (lightGBM) shows promise with the extracted topological features.

scholarly journals Limit Theorems for Random Cubical Homology

2018 ◽  
Vol 60 (3) ◽  
pp. 665-687 ◽  
Author(s):  
Yasuaki Hiraoka ◽  
Kenkichi Tsunoda
Keyword(s):  
Limit Theorems ◽  
Cubical Homology

scholarly journals Generalizations of Baxter's Theorem and cubical homology

1995 ◽  
Vol 69 (2) ◽  
pp. 233-287 ◽  
Author(s):  
Richard Ehrenborg ◽  
Gábor Hetyei
Keyword(s):  
Cubical Homology

scholarly journals Digital homotopy relations and digital homology theories

2021 ◽  
Vol 22 (2) ◽  
pp. 223
Author(s):  
P. Christopher Staecker
Keyword(s):  
Digital Images ◽  
Continuous Functions ◽  
Homology Theory ◽  
Simplicial Homology ◽  
Digitally Continuous ◽  
New Type ◽  
The Difference ◽  
Cubical Homology

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 \&amp; 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.

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