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 $K$ of the Euclidean plane. A Cech nerve is a collection of intersecting balls. A Cech complex is a collection of nerves that cover $K$. 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 $K$ 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 $K$.Comment: 11 pages, 2 figures, keywords: Ball, Cech Complex, Cech Nerve, Cover, Homotopic Equivalence, Proximit

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 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 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 $k$-spokes, $k\geq 1$, are a natural extension of nerve complexes. A $k$-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 $\delta$. 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 $X$, 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 $n$-sphere or a region of a normed linear space. In this work, an $n$-sphere surface is covered by a collection of strings. For a strongly proximal continuous function on an $n$-sphere into $n$-dimensional Euclidean space, there exists a pair of antipodal $n$-sphere strings with matching descriptions that map into Euclidean space $\mathbb{R}^n$. Each region $M$ of a string-covered $n$-sphere is a worldsheet. For a strongly proximal continuous mapping from a worldsheet-covered $n$-sphere to $\mathbb{R}^n$, strongly near antipodal worldsheets map into the same region in $\mathbb{R}^n$. 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

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 $\mathcal{B}$ 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 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 ${CL}(X)$, the hyperspace of nonempty closed subsets of $X$, based on the strongly near proximity.Comment: 7 pages, 2 figures. arXiv admin note: substantial text overlap with arXiv:1502.0277

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 $\textrm{CL}(X)$ associated with} ${\mathscr{B}}$, a nonempty family of subsets on the topological space $X$. 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 $\textrm{RCL}(X)$ is the space of regular closed subsets of $X$. To obtain the results, special local families are introduced.Comment: 8 pages, 4 figures. arXiv admin note: text overlap with arXiv:1502.0591