2,899 research outputs found
Proximal groupoid patterns In digital images
The focus of this article is on the detection and classification of patterns
based on groupoids. The approach hinges on descriptive proximity of points in a
set based on the neighborliness property. This approach lends support to image
analysis and understanding and in studying nearness of image segments. A
practical application of the approach is in terms of the analysis of natural
images for pattern identification and classification.Comment: 9 pages, 6 figure
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
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
Strongly Proximal Continuity \& Strong Connectedness
This article introduces strongly proximal continuous (s.p.c.) functions,
strong proximal equivalence (s.p.e.) and strong connectedness. A main result is
that if topological spaces are endowed with compatible strong proximities
and is a bijective s.p.e., then its extension on the
hyperspaces \CL(X) and \CL(Y), endowed with the related strongly hit and
miss hypertopologies, is a homeomorphism. For a topological space endowed with
a strongly near proximity, strongly proximal connectedness implies
connectedness but not conversely. Conditions required for strongly proximal
connectedness are given. Applications of s.p.c. and strongly proximal
connectedness are given in terms of strongly proximal descriptive proximity.Comment: 11 pages, 10 figure
Proximal Planar Shape Signatures. Homology Nerves and Descriptive Proximity
This article introduces planar shape signatures derived from homology nerves,
which are intersecting 1-cycles in a collection of homology groups endowed with
a proximal relator (set of nearness relations) that includes a descriptive
proximity. A 1-cycle is a closed, connected path with a zero boundary in a
simplicial complex covering a finite, bounded planar shape. The signature of a
shape sh A (denoted by sig(sh A)) is a feature vector that describes sh A. A
signature sig(sh A) is derived from the geometry, homology nerves, Betti
number, and descriptive CW topology on the shape sh A. Several main results are
given, namely, (a) every finite, bounded planar shape has a signature derived
from the homology group on the shape, (b) a homology group equipped with a
proximal relator defines a descriptive Leader uniform topology and (c) a
description of a homology nerve and union of the descriptions of the 1-cycles
in the nerve have same homotopy type.Comment: 15 pages; 4 figure
Strongly Near Voronoi Nucleus Clusters
This paper introduces nucleus clustering in Voronoi tessellations of plane
surfaces with applications in the geometry of digital images. A \emph{nucleus
cluster} is a collection of Voronoi regions that are adjacent to a Voronoi
region called the cluster nucleus. Nucleus clustering is a carried out in a
strong proximity space. Of particular interest is the presence of maximal
nucleus clusters in a tessellation. Among all of the possible nucleus clusters
in a Voronoi tessellation, clusters with the highest number of adjacent
polygons are called \emph{maximal nucleus clusters}. The main results in this
paper are that strongly near nucleus clusters are strongly descriptively near
and every collection of Voronoi regions in a tessellation of a plane surface is
a Zelins'kyi-Soltan-Kay-Womble convexity structure.Comment: 7 pages, 2 figure
Descriptive Unions. A Fibre Bundle Characterization of the Union of Descriptively Near Sets
This paper introduces an extension of descriptive intersection and provides a
framework for descriptive unions of nonempty sets. Fibre bundles provide
structures that characterize spatially near as well as descriptively near sets,
their descriptive intersection and their unions. The properties of four
different forms of descriptive unions are given. A main result given in this
paper is the equivalence between ordinary set intersection and a descriptive
union. Applications of descriptive unions are given with respect to Jeffs-Novik
convex unions and descriptive unions in digital images.Comment: 19 pages, 6 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
Hyperconnected Relator Spaces. CW Complexes and Continuous Function Paths that are Hyperconnected
This article introduces proximal cell complexes in a hyperconnected space.
Hyperconnectedness encodes how collections of path-connected sub-complexes in a
Alexandroff-Hopf-Whitehead CW space are near to or far from each other. Several
main results are given, namely, a hyper-connectedness form of CW (Closure
Finite Weak topology) complex, the existence of continuous functions that are
paths in hyperconnected relator spaces and hyperconnected chains with
overlapping interiors that are path graphs in a relator space. An application
of these results is given in terms of the definition of cycles using the
centroids of triangles.Comment: 15 pages, 5 figure
Proximal Planar Shapes. Correspondence between Shapes and Nerve Complexes
This article considers proximal planar shapes in terms of the proximity of
shape nerves and shape nerve complexes. A shape nerve is collection of
2-simplexes with nonempty intersection on a triangulated shape space. A planar
shape is a shape nerve complex, which is a collection of shape nerves that have
nonempty intersection. A main result in this paper is the homotopy equivalence
of a planar shape nerve complex and the union of its nerve sub-complexes.Comment: 13 pages, 5 figures. arXiv admin note: text overlap with
arXiv:1704.0590
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