scholarly journals RIEMANNIAN METRIC FOR TEXTURE RECOGNITION

2018 ◽  
pp. 23-27
Author(s):  
Makarenko N.G. ◽  
ChoYong-beom ◽  
Esenaliev A. B.
Keyword(s):  
Riemannian Geometry ◽  
Topological Property ◽  
Riemannian Metric ◽  
Topological Properties ◽  
Connected Component ◽  
Rectangular Coordinate System ◽  
Texture Recognition ◽  
Persistence Diagram ◽  
Cloud Of Points ◽  
Texture Class

The article discusses the recognition of textures on digital images by methods of computational topology and Riemannian geometry. Topological properties of patterns are represented by segments (barcodes) obtained by filtering by the level of photometric measure. Beginning of barcode encodes level at which topological property appears (connected component and/or “hole”), and its end - level at which the property disappears. Barcodes are conveniently parameterized by coordinates of their ends in rectangular coordinate system “birth” and “death” of topological property. Such representation in form of a cloud of points on plane is called a persistence diagram (PD). In the article show that texture class recognition results are significantly better compared to other vectorization methods of PD.

Separable Topological Space of Hereditary

NUTA Journal ◽  
2020 ◽  
Vol 7 (1-2) ◽  
pp. 68-70
Author(s):  
Raj Narayan Yadav ◽  
Bed Prasad Regmi ◽  
Surendra Raj Pathak
Keyword(s):  
Topological Space ◽  
Topological Property ◽  
Sufficient Condition ◽  
Topological Properties ◽  
Necessary And Sufficient Condition ◽  
Separable Space ◽  
Separable Topological Space ◽  
Necessary And Sufficient

A property of a topological space is termed hereditary ifand only if every subspace of a space with the property also has the property. The purpose of this article is to prove that the topological property of separable space is hereditary. In this paper we determine some topological properties which are hereditary and investigate necessary and sufficient condition functions for sub-spaces to possess properties of sub-spaces which are not in general hereditary.

A Generalization of Finsler Geometry

1962 ◽  
Vol 14 ◽  
pp. 87-112 ◽  
Author(s):  
J. R. Vanstone
Keyword(s):  
Differential Geometry ◽  
General Theory ◽  
Distance Function ◽  
Riemannian Geometry ◽  
Finsler Geometry ◽  
Riemannian Metric ◽  
Elliptic Partial Differential Equations ◽  
Theory Of Relativity ◽  
General Theory Of Relativity ◽  
Metric Function

Modern differential geometry may be said to date from Riemann's famous lecture of 1854 (9), in which a distance function of the form F(xi, dxi) = (γij(x)dxidxj½ was proposed. The applications of the consequent geometry were many and varied. Examples are Synge's geometrization of mechanics (15), Riesz’ approach to linear elliptic partial differential equations (10), and the well-known general theory of relativity of Einstein.Meanwhile the results of Caratheodory (4) in the calculus of variations led Finsler in 1918 to introduce a generalization of the Riemannian metric function (6). The geometry which arose was more fully developed by Berwald (2) and Synge (14) about 1925 and later by Cartan (5), Busemann, and Rund. It was then possible to extend the applications of Riemannian geometry.

scholarly journals Riemannian Structures on Z 2 n -Manifolds

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1469
Author(s):  
Andrew James Bruce ◽  
Janusz Grabowski
Keyword(s):  
Riemannian Geometry ◽  
Fundamental Theorem ◽  
Scalar Product ◽  
Riemannian Metric ◽  
Theoretical Framework ◽  
Similarities And Differences

Very loosely, Z2n-manifolds are ‘manifolds’ with Z2n-graded coordinates and their sign rule is determined by the scalar product of their Z2n-degrees. A little more carefully, such objects can be understood within a sheaf-theoretical framework, just as supermanifolds can, but with subtle differences. In this paper, we examine the notion of a Riemannian Z2n-manifold, i.e., a Z2n-manifold equipped with a Riemannian metric that may carry non-zero Z2n-degree. We show that the basic notions and tenets of Riemannian geometry directly generalize to the setting of Z2n-geometry. For example, the Fundamental Theorem holds in this higher graded setting. We point out the similarities and differences with Riemannian supergeometry.

scholarly journals Topological properties of manifolds admitting a Yx-Riemannian metric

2010 ◽  
Vol 60 (10) ◽  
pp. 1530-1538 ◽  
Author(s):  
Vladimir Chernov ◽  
Paul Kinlaw ◽  
Rustam Sadykov
Keyword(s):  
Riemannian Metric ◽  
Topological Properties

scholarly journals Idempotent Analog of the Legendre Transformation and lts Application in Digital Processing of Signals

2020 ◽  
pp. 96-101
Author(s):  
M.V. Kurkina ◽  
S.P. Semenov ◽  
V.V. Slavsky ◽  
O.V. Samarina ◽  
O.A. Petuhova ◽  
...  
Keyword(s):  
Convex Functions ◽  
Riemannian Geometry ◽  
Duality Theory ◽  
Digital Processing ◽  
Riemannian Metric ◽  
Theoretical Physics ◽  
Vector Spaces ◽  
Legendre Transformation ◽  
Conformally Flat ◽  
Tropical Mathematics

In recent years, a new area of mathematics — idempotent or “tropical” mathematics — has been intensively developed within the framework of the Sofus Lee international center, which is reflected in the works of V.P. Maslov, G.L. Litvinov, and A.N. Sobolevsky. The Legendre transformation plays an important role in theoretical physics, classical and statistical mechanics, and thermodynamics. In mathematics and its applications, the Legendre transformation is based on the concept of duality of vector spaces and duality theory for convex functions and subsets of a vector space. The purpose of this paper is to go beyond linear vector spaces using similar notions of duality in conformally flat Riemannian geometry and in idempotent algebra.An abstract idempotent analog of the Legendre transformation is constructed in a way similar to the polar transformation of the conformally flat Riemannian metric introduced in the works of E.D. Rodionov and V.V. Slavsky. Its capabilities for digital processing of signals and images are being investigated

scholarly journals Texture recognition by the methods of topological data analysis

2016 ◽  
Vol 6 (1) ◽  
Author(s):  
Nikolay Makarenko ◽  
Maksat Kalimoldayev ◽  
Ivan Pak ◽  
Ainur Yessenaliyeva
Keyword(s):  
Betti Numbers ◽  
Remote Sensing Data ◽  
Large Data ◽  
Topological Data Analysis ◽  
Data Sets ◽  
Texture Recognition ◽  
Topological Approach ◽  
Average Efficiency ◽  
Persistence Diagram ◽  
Persistent Betti Numbers

Abstract High spatial resolution satellite images are different from Gaussian statistics of counts. Therefore, texture recognition methods based on variances become ineffective. The aim of this paper is to study possibilities of completely different, topological approach to problems of structures classification. Persistent Betti numbers are signs of texture recognition. They are not associated with metrics and received directly fromdata in form of so-called persistence diagram (PD). The different structures built on PD are used to get convenient numerical statistics. At the present time, three of such objects are known: topological landscapes, persistent images and rank functions. They have been introduced recently and appeared as an attempt to vectorize PD. Typically, each of the proposed structures was illustrated by the authors with simple examples.However, the practical application of these approaches to large data sets requires to evaluate their efficiency within the frame of the selected task at the same standard database. In our case, such a task is to recognize different textures of the Remote Sensing Data (RSD). We check efficiency of structure, called persistent images in this work. We calculate PD for base containing 800 images of high resolution representing 20 texture classes. We have found out that average efficiency of separate image recognition in the classes is 84%, and in 11 classes, it is not less than 90%. By comparison topological landscapes provide 68% for average efficiency, and only 3 classes of not less than 90%. Reached conclusions are of interest for new methods of texture recognition in RSD.

ON CONNECTED COMPONENT DISTRIBUTION OF RANDOM BLOCK-HIERARCHICAL NETWORKS USING THE AUTOMATED SYSTEM $ \textit{XRANDNET} $

2018 ◽  
Vol 52 (1 (245)) ◽  
pp. 34-40
Author(s):  
A.G. Kocharyan
Keyword(s):  
Network Theory ◽  
Hierarchical Models ◽  
Random Network ◽  
Automated System ◽  
Random Networks ◽  
Topological Properties ◽  
Connected Component ◽  
Hierarchical Networks ◽  
Component Distribution ◽  
Random Block

In the paper the result of a research done by using the automated system xRandNet is presented, which is designed and implemented for generating and analyzing the main topological properties of some hierarchical models of random networks. The research is related to the connected component distribution of random block-hierarchical networks, which are quite new objects in the random network theory.

scholarly journals Semi-topological properties

1992 ◽  
Vol 15 (2) ◽  
pp. 267-272
Author(s):  
Bhamini M. P. Nayar ◽  
S. P. Arya
Keyword(s):  
Topological Property ◽  
Topological Properties

A property preserved under a semi-homeomorphism is said to be a semi-topological property. In the present paper we prove the following results: (1) A topological propertyPis semi-topological if and only if the statement(X,𝒯)hasPif and only if(X,F(𝒯))hasP′is true whereF(𝒯)is the finest topology onXhaving the same family of semi-open sets as(X,𝒯), (2) IfPis a topological property being minimalPis semi-topological if and only if for each minimalPspace(X,𝒯),𝒯=F(𝒯).

scholarly journals Riemannian Geometry on Quantum Spaces

1997 ◽  
Vol 12 (05) ◽  
pp. 923-943 ◽  
Author(s):  
Pei-Ming Ho
Keyword(s):  
Projective Space ◽  
Riemannian Geometry ◽  
Riemannian Metric ◽  
Complex Manifolds ◽  
Quantum Sphere ◽  
Quantum Projective Space ◽  
Metric Distance ◽  
Algebraic Formulation

An algebraic formulation of Riemannian geometry on quantum spaces is presented, where Riemannian metric, distance, Laplacian, connection, and curvature have their counterparts. This description is also extended to complex manifolds. Examples include the quantum sphere, the complex quantum projective space and the two-sheeted space.

scholarly journals Evolutionary Sequence of Spacetime and Intrinsic Spacetime and Associated Sequence of Geometries in Metric Force Fields I

2021 ◽  
pp. 1-20
Author(s):  
O. Akindele Adekugbe Joseph
Keyword(s):  
Metric Space ◽  
Riemannian Geometry ◽  
Metric Spaces ◽  
Three Dimensional ◽  
Riemannian Metric ◽  
Vertical Position ◽  
Evolutionary Sequence ◽  
Time Dimension ◽  
Intrinsic Metric ◽  
Riemann Geometry

Two classes of three-dimensional metric spaces are identified. They are the conventional three-dimensional metric space and a new ‘three-dimensional’ absolute intrinsic metric space. Whereas an initial flat conventional proper metric space IE′3 can transform into a curved three-dimensionalRiemannian metric space IM′3 without any of its dimension spanning the time dimension (or in the absence of the time dimension), in conventional Riemann geometry, an initial flat ‘three-dimensional’ absolute intrinsic metric space ∅IˆE3 (as a flat hyper-surface) along the horizontal, evolves into a curved ‘three-dimensional’ absolute intrinsic metric space ∅IˆM3, which is curved (as a curved hyper-surface) toward the absolute intrinsic metric time ‘dimension’ along the vertical, and it is identified as ‘three-dimensional’ absolute intrinsic Riemannian metric space. It invariantly projects a flat ‘three-dimensional’ absolute proper intrinsic metric space ∅IE′3ab along the horizontal, which is made manifested outwardly in flat ‘three-dimensional’ absolute proper metric space IE′3ab, overlying it, both as flat hyper-surfaces along the horizontal. The flat conventional three-dimensional relative proper metric space IE′3 and its underlying flat three-dimensional relative proper intrinsic metric space ∅IE′3 remain unchanged. The observers are located in IE′3. The projective ∅IE′3ab is imperceptibly embedded in ∅IE′3 and IE′3ab in IE′3. The corresponding absolute intrinsic metric time ‘dimension’ is not curved from its vertical position simultaneously with ‘three-dimensional’ absolute intrinsic metric space. The development of absolute intrinsic Riemannian geometry is commenced and the conclusion that the resulting geometry is more all-encompassing then the conventional Riemannian geometry on curved conventional metric space IM′3 only is reached.

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