Topic
Abstract cell complex
About: Abstract cell complex is a research topic. Over the lifetime, 9 publications have been published within this topic receiving 75 citations.
Papers
TL;DR: It is demonstrated that in an n-dimensional digital space only those of the (a, b)-adjacencies commonly used in computer imagery have analogs among the LF spaces, in which a and b are different and one of the adjacencies is the “maximal” one, corresponding to 3n− 1 neighbors.
Abstract: The paper presents a new set of axioms of digital topology, which are easily understandable for application developers. They define a class of locally finite (LF) topological spaces. An important property of LF spaces satisfying the axioms is that the neighborhood relation is antisymmetric and transitive. Therefore any connected and non-trivial LF space is isomorphic to an abstract cell complex. The paper demonstrates that in an n-dimensional digital space only those of the (a, b)-adjacencies commonly used in computer imagery have analogs among the LF spaces, in which a and b are different and one of the adjacencies is the "maximal" one, corresponding to 3 n ? 1 neighbors. Even these (a, b)-adjacencies have important limitations and drawbacks. The most important one is that they are applicable only to binary images. The way of easily using LF spaces in computer imagery on standard orthogonal grids containing only pixels or voxels and no cells of lower dimensions is suggested.
47 citations
TL;DR: This theoretical work focuses on the problem of finding efficient algorithmic solutions for topological interrogation of a 2D digital object of interest D of a presegmented digital image I, using 4-adjacency between pixels of D.
16 citations
15 Jan 2003
TL;DR: The paper presents a new method of investigating topological properties of three-dimensional manifolds by means of computers based on subdividing the given set into blocks of cells in such a way that a k-dimensional block be homeomorphic to aK-dimensional ball.
Abstract: The paper presents a new method of investigating topological properties of three-dimensional manifolds by means of computers. Manifolds are represented as block complexes. The paper contains definitions and a theorem necessary to transfer some basic knowledge of the classical topology to finite topological spaces. The method is based on subdividing the given set into blocks of cells in such a way that a k-dimensional block be homeomorphic to a k-dimensional ball. The block structure is described by the data structure known as "cell list" which is generalized here for the multidimensional case. Results of computer experiments are presented.
12 citations
19 Jun 2017
TL;DR: A theoretical parallel algorithm is designed 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 p ACC associated to I and based on the application of elementary homotopy operations to activate the pACC processing units.
Abstract: 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 \(D\subset I\) 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.
6 citations
TL;DR: In this article, the authors studied certain low-level separation axioms of a geometric topological space, denoted by A(X), induced by a geometric AC-complex X.
Abstract: The present paper studies certain low-level separation axioms of a
topological space, denoted by A(X), induced by a geometric AC-complex X.
After proving that whereas A(X) is an Alexandroff space satisfying the
semi-T1 2 -separation axiom, we observe that it does neither satisfy the pre
T1 2 -separation axiom nor is a Hausdorff space. These are main motivations
of the present work. Although not every A(X) is a semi-T1 space, after
proceeding with an edge to edge tiling (or a face to face crystallization)
of Rn, n N, denoted by T(Rn) as an AC complex, we prove that A(T(Rn)) is a
semi-T1 space. Furthermore, we prove that A(En), induced by an nD Cartesian
AC complex Cn = (En,N,dim), is also a semi-T1 space, n N. The paper deals
with AC-complexes with the locally finite (LF-, for brevity) property, which
can be used in the fields of pure and applied mathematics as well as digital
topology, computational topology, and digital geometry.
5 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2019 | 1 |
| 2017 | 1 |
| 2016 | 1 |
| 2014 | 1 |
| 2011 | 1 |
| 2010 | 1 |