36,866 research outputs found
Proximal Planar Cech Nerves. An Approach to Approximating the Shapes of Irregular, Finite, Bounded Planar Regions
This article introduces proximal Cech nerves and Cech complexes, restricted
to finite, bounded regions of the Euclidean plane. A Cech nerve is a
collection of intersecting balls. A Cech complex is a collection of nerves that
cover . Cech nerves are proximal, provided the nerves are close to each
other, either spatially or descriptively. A Cech nerve has an advantage over
the usual Alexandroff nerve, since we need only identify the center and fixed
radius of each ball in a Cech nerve instead of identifying the three vertices
of intersecting filled triangles (2-simplexes) in an Alexandroff nerve. As a
result, Cech nerves more easily cover and facilitate approximation of the
shapes of irregular finite, bounded planar regions. A main result of this
article is an extension of the Edelsbrunner-Harer Nerve Theorem for descriptive
and non-descriptive Cech nerves and Cech complexes, covering .Comment: 11 pages, 2 figures, keywords: Ball, Cech Complex, Cech Nerve, Cover,
Homotopic Equivalence, Proximit
Proximal Vortex Cycles and Vortex Nerves. Non-Concentric, Nesting, Possibly Overlapping Homology Cell Complexes
This article introduces proximal planar vortex 1-cycles, resembling the
structure of vortex atoms introduced by William Thomson (Lord Kelvin) in 1867
and recent work on the proximity of sets that overlap either spatially or
descriptively. Vortex cycles resemble Thomson's model of a vortex atom,
inspired by P.G. Tait's smoke rings. A vortex cycle is a collection of
non-concentric, nesting 1-cycles with nonempty interiors (i.e., a collection of
1-cycles that share a nonempty set of interior points and which may or may not
overlap). Overlapping 1-cycles in a vortex yield an Edelsbrunner-Harer nerve
within the vortex. Overlapping vortex cycles constitute a vortex nerve complex.
Several main results are given in this paper, namely, a Whitehead CW topology
and a Leader uniform topology are outcomes of having a collection of vortex
cycles (or nerves) equipped with a connectedness proximity and the case where
each cluster of closed, convex vortex cycles and the union of the vortex cycles
in the cluster have the same homotopy type.Comment: 10 figures, 25 page
Proximal Delaunay Triangulation Regions
A main result in this paper is the proof that proximal Delaunay triangulation
regions are convex polygons. In addition, it is proved that every Delaunay
triangulation region has a local Leader uniform topology.Comment: 4 pages, 4 figure
Proximal Nerve Complexes. A Computational Topology Approach
This article introduces a theory of proximal nerve complexes and nerve
spokes, restricted to the triangulation of finite regions in the Euclidean
plane. A nerve complex is a collection of filled triangles with a common
vertex, covering a finite region of the plane. Structures called -spokes,
, are a natural extension of nerve complexes. A -spoke is the union
of a collection of filled triangles that pairwise either have a common edge or
a common vertex. A consideration of the closeness of nerve complexes leads to a
proximal view of simplicial complexes. A practical application of proximal
nerve complexes is given, briefly, in terms of object shape geometry in digital
images.Comment: 16 pages, 9 figure
Voronoi Region-Based Adaptive Unsupervised Color Image Segmentation
Color image segmentation is a crucial step in many computer vision and
pattern recognition applications. This article introduces an adaptive and
unsupervised clustering approach based on Voronoi regions, which can be applied
to solve the color image segmentation problem. The proposed method performs
region splitting and merging within Voronoi regions of the Dirichlet
Tessellated image (also called a Voronoi diagram) , which improves the
efficiency and the accuracy of the number of clusters and cluster centroids
estimation process. Furthermore, the proposed method uses cluster centroid
proximity to merge proximal clusters in order to find the final number of
clusters and cluster centroids. In contrast to the existing adaptive
unsupervised cluster-based image segmentation algorithms, the proposed method
uses K-means clustering algorithm in place of the Fuzzy C-means algorithm to
find the final segmented image. The proposed method was evaluated on three
different unsupervised image segmentation evaluation benchmarks and its results
were compared with two other adaptive unsupervised cluster-based image
segmentation algorithms. The experimental results reported in this article
confirm that the proposed method outperforms the existing algorithms in terms
of the quality of image segmentation results. Also, the proposed method results
in the lowest average execution time per image compared to the existing methods
reported in this article.Comment: 21 pages, 5 figure
Strongly far proximity and hyperspace topology
This article introduces strongly far proximity, which is associated with
Lodato proximity . A main result in this paper is the introduction of a
hit-and-miss topology on \mbox{CL}(X), the hyperspace of nonempty closed
subsets of , based on the strongly far proximity.Comment: 6 pages, 1 figur
Region-Based Borsuk-Ulam Theorem and Wired Friend Theorem
This paper introduces a string-based extension of the Borsuk-Ulam Theorem
(denoted by strBUT). A string is a region with zero width and either bounded or
unbounded length on the surface of an -sphere or a region of a normed linear
space. In this work, an -sphere surface is covered by a collection of
strings. For a strongly proximal continuous function on an -sphere into
-dimensional Euclidean space, there exists a pair of antipodal -sphere
strings with matching descriptions that map into Euclidean space
. Each region of a string-covered -sphere is a worldsheet.
For a strongly proximal continuous mapping from a worldsheet-covered -sphere
to , strongly near antipodal worldsheets map into the same region
in . This leads to a wired friend theorem in descriptive string
theory. An application of strBUT is given in terms of the evaluation of
Electroencephalography (EEG) patterns.Comment: 17 pages, 8 figure
Strongly near proximity \& hyperspace topology
This article introduces strongly near proximity, which represents a new kind
of proximity called \emph{almost proximity}. A main result in this paper is the
introduction of a hit-and-miss topology on , the hyperspace of
nonempty closed subsets of , based on the strongly near proximity.Comment: 7 pages, 2 figures. arXiv admin note: substantial text overlap with
arXiv:1502.0277
Strong Proximities on Smooth Manifolds and Vorono\" i Diagrams
This article introduces strongly near smooth manifolds. The main results are
(i) second countability of the strongly hit and far-miss topology on a family
of subsets on the Lodato proximity space of regular open sets to
which singletons are added, (ii) manifold strong proximity, (iii) strong
proximity of charts in manifold atlases implies that the charts have nonempty
intersection. The application of these results is given in terms of the
nearness of atlases and charts of proximal manifolds and what are known as
Vorono\" i manifolds.Comment: 16 pages, 7 figure
Strongly Hit and Far Miss Hypertopology and Hit and Strongly Far Miss Hypertopology
This article introduces the {\it strongly hit and far-miss as well as hit and
strongly far miss hypertopologies on associated with}
, a nonempty family of subsets on the topological space .
They result from the strong farness and strong nearness proximities. The main
results in this paper stem from the Hausdorffness of (\textrm{CL}(X),
\tau_{\doublevee, \mathscr{B}}) and (\textrm{RCL}(X),
\tau^\doublewedge_\mathscr{B} ) , where is the space of
regular closed subsets of . To obtain the results, special local families
are introduced.Comment: 8 pages, 4 figures. arXiv admin note: text overlap with
arXiv:1502.0591
- …