6,911 research outputs found
Dimension on Discrete Spaces
In this paper we develop some combinatorial models for continuous spaces. In
this spirit we study the approximations of continuous spaces by graphs,
molecular spaces and coordinate matrices. We define the dimension on a discrete
space by means of axioms, and the axioms are based on an obvious geometrical
background. This work presents some discrete models of n-dimensional Euclidean
spaces, n-dimensional spheres, a torus and a projective plane. It explains how
to construct new discrete spaces and describes in this connection several
three-dimensional closed surfaces with some topological singularities
It also analyzes the topology of (3+1)-spacetime. We are also discussing the
question by R. Sorkin [19] about how to derive the system of simplicial
complexes from a system of open covering of a topological space S.Comment: 16 pages, 8 figures, Latex. Figures are not included, available from
the author upon request. Preprint SU-GP-93/1-1. To appear in "International
Journal of Theoretical Physics
Axiomatic Digital Topology
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 3n\"i1 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
Genus Computing for 3D digital objects: algorithm and implementation
This paper deals with computing topological invariants such as connected
components, boundary surface genus, and homology groups. For each input data
set, we have designed or implemented algorithms to calculate connected
components, boundary surfaces and their genus, and homology groups. Due to the
fact that genus calculation dominates the entire task for 3D object in 3D
space, in this paper, we mainly discuss the calculation of the genus. The new
algorithms designed in this paper will perform:
(1) pathological cases detection and deletion, (2) raster space to point
space (dual space) transformation, (3) the linear time algorithm for boundary
point classification, and (4) genus calculation.Comment: 12 pages 7 figures. In Proceedings of the Workshop on Computational
Topology in image context 2009, Aug. 26-28, Austria, Edited by W. Kropatsch,
H. M. Abril and A. Ion, 200
- …