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Abstract Cell Complexes
In mathematics, an abstract cell complex is an abstract set with Alexandrov topology in which a non-negative integer number called dimension is assigned to each point. The complex is called “abstract” since its points, which are called “cells”, are not subsets of a Hausdorff space as is the case in Euclidean and CW complexes. Abstract cell complexes play an important role in image analysis and computer graphics. History The idea of abstract cell complexes (also named abstract cellular complexes) relates to J. Listing (1862) and E. Steinitz (1908). Also A.W Tucker (1933), K. Reidemeister (1938), P.S. Aleksandrov (1956) as well as R. Klette and A. Rosenfeld (2004) have described abstract cell complexes. E. Steinitz has defined an abstract cell complex as C=(E,B,dim) where ''E'' is an abstract set, ''B'' is an asymmetric, irreflexive and transitive binary relation called the bounding relation among the elements of ''E'' and ''dim'' is a function assigning a non-negati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexandrov Topology
In topology, an Alexandrov topology is a topology in which the intersection of any family of open sets is open. It is an axiom of topology that the intersection of any ''finite'' family of open sets is open; in Alexandrov topologies the finite restriction is dropped. A set together with an Alexandrov topology is known as an Alexandrov-discrete space or finitely generated space. Alexandrov topologies are uniquely determined by their specialization preorders. Indeed, given any preorder ≤ on a set ''X'', there is a unique Alexandrov topology on ''X'' for which the specialization preorder is ≤. The open sets are just the upper sets with respect to ≤. Thus, Alexandrov topologies on ''X'' are in one-to-one correspondence with preorders on ''X''. Alexandrov-discrete spaces are also called finitely generated spaces since their topology is uniquely determined by the family of all finite subspaces. Alexandrov-discrete spaces can thus be viewed as a generalization of finite topolog ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Simplicial Complex
In combinatorics, an abstract simplicial complex (ASC), often called an abstract complex or just a complex, is a family of sets that is closed under taking subsets, i.e., every subset of a set in the family is also in the family. It is a purely combinatorial description of the geometric notion of a simplicial complex. Lee, John M., Introduction to Topological Manifolds, Springer 2011, , p153 For example, in a 2-dimensional simplicial complex, the sets in the family are the triangles (sets of size 3), their edges (sets of size 2), and their vertices (sets of size 1). In the context of matroids and greedoids, abstract simplicial complexes are also called independence systems. An abstract simplex can be studied algebraically by forming its Stanley–Reisner ring; this sets up a powerful relation between combinatorics and commutative algebra. Definitions A collection of non-empty finite subsets of a set ''S'' is called a set-family. A set-family is called an abstract simplicial c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplicial Complex
In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex. To distinguish a simplicial from an abstract simplicial complex, the former is often called a geometric simplicial complex.'', Section 4.3'' Definitions A simplicial complex \mathcal is a set of simplices that satisfies the following conditions: :1. Every face of a simplex from \mathcal is also in \mathcal. :2. The non-empty intersection of any two simplices \sigma_1, \sigma_2 \in \mathcal is a face of both \sigma_1 and \sigma_2. See also the definition of an abstract simplicial complex, which loosely speaking is a simplicial complex without an associated geometry. A simplicial ''k''-complex \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Digital Straight Segment
Digital usually refers to something using discrete digits, often binary digits. Technology and computing Hardware *Digital electronics, electronic circuits which operate using digital signals **Digital camera, which captures and stores digital images ***Digital versus film photography **Digital computer, a computer that handles information represented by discrete values **Digital recording, information recorded using a digital signal Socioeconomic phenomena *Digital culture, the anthropological dimension of the digital social changes *Digital divide, a form of economic and social inequality in access to or use of information and communication technologies * Digital economy, an economy based on computing and telecommunications resources Other uses in technology and computing *Digital data, discrete data, usually represented using binary numbers *Digital marketing, search engine & social media presence booster, usually represented using online visibility. *Digital media, media st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boundary Tracing
Boundary tracing, also known as contour tracing, of a binary digital region can be thought of as a segmentation technique that identifies the boundary pixels of the digital region. Boundary tracing is an important first step in the analysis of that region. Boundary is a topological notion. However, a digital image is no topological space. Therefore, it is impossible to define the notion of a boundary in a digital image mathematically exactly. Most publications about tracing the boundary of a subset S of a digital image I describe algorithms which find a set of pixels belonging to S and having in their direct neighborhood pixels belonging both to S and to its complement I - S. According to this definition the boundary of a subset S is different from the boundary of the complement I – S which is a topological paradox. To define the boundary correctly it is necessary to introduce a topological space corresponding to the given digital image. Such space can be a two-dimensional abst ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Digital Image ACC Coordinate Assignment
Digital usually refers to something using discrete digits, often binary digits. Technology and computing Hardware *Digital electronics, electronic circuits which operate using digital signals **Digital camera, which captures and stores digital images ***Digital versus film photography **Digital computer, a computer that handles information represented by discrete values **Digital recording, information recorded using a digital signal Socioeconomic phenomena *Digital culture, the anthropological dimension of the digital social changes *Digital divide, a form of economic and social inequality in access to or use of information and communication technologies * Digital economy, an economy based on computing and telecommunications resources Other uses in technology and computing *Digital data, discrete data, usually represented using binary numbers *Digital marketing, search engine & social media presence booster, usually represented using online visibility. *Digital media, media st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Digital Image ACC Decomposition
Digital usually refers to something using discrete digits, often binary digits. Technology and computing Hardware *Digital electronics, electronic circuits which operate using digital signals **Digital camera, which captures and stores digital images ***Digital versus film photography **Digital computer, a computer that handles information represented by discrete values **Digital recording, information recorded using a digital signal Socioeconomic phenomena *Digital culture, the anthropological dimension of the digital social changes *Digital divide, a form of economic and social inequality in access to or use of information and communication technologies * Digital economy, an economy based on computing and telecommunications resources Other uses in technology and computing *Digital data, discrete data, usually represented using binary numbers *Digital marketing, search engine & social media presence booster, usually represented using online visibility. *Digital media, media st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poset Topology
In mathematics, the poset topology associated to a poset (''S'', ≤) is the Alexandrov topology (open sets are upper sets) on the poset of finite chains of (''S'', ≤), ordered by inclusion. Let ''V'' be a set of vertices. An abstract simplicial complex Δ is a set of finite sets of vertices, known as faces \sigma \subseteq V, such that ::\forall \rho \, \forall \sigma \!: \ \rho \subseteq \sigma \in \Delta \Rightarrow \rho \in \Delta. Given a simplicial complex Δ as above, we define a (point set) topology on Δ by declaring a subset \Gamma \subseteq \Delta be closed if and only if Γ is a simplicial complex, i.e. ::\forall \rho \, \forall \sigma \!: \ \rho \subseteq \sigma \in \Gamma \Rightarrow \rho \in \Gamma. This is the Alexandrov topology on the poset of faces of Δ. The order complex associated to a poset (''S'', ≤) has the set ''S'' as vertices, and the finite chains of (''S'', ≤) as faces. The poset topology associated to a poset (''S'', ≤) is then the Alexandrov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Finite Space
In the mathematical field of topology, a locally finite space is a topological space in which every point has a finite neighborhood, that is, an open neighborhood consisting of finitely many elements. A locally finite space is Alexandrov. A T1 space is locally finite if and only if it is discrete Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory * Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a .... References * General topology Properties of topological spaces {{topology-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. For example, * a 0-dimensional simplex is a point, * a 1-dimensional simplex is a line segment, * a 2-dimensional simplex is a triangle, * a 3-dimensional simplex is a tetrahedron, and * a 4-dimensional simplex is a 5-cell. Specifically, a ''k''-simplex is a ''k''-dimensional polytope which is the convex hull of its ''k'' + 1 vertices. More formally, suppose the ''k'' + 1 points u_0, \dots, u_k \in \mathbb^ are affinely independent, which means u_1 - u_0,\dots, u_k-u_0 are linearly independent. Then, the simplex determined by them is the set of points : C = \left\ This representation in terms of weighted vertices is known as the barycentric coordinate system. A regular simplex is a simplex that is also a regular polytope. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplicial Complex
In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex. To distinguish a simplicial from an abstract simplicial complex, the former is often called a geometric simplicial complex.'', Section 4.3'' Definitions A simplicial complex \mathcal is a set of simplices that satisfies the following conditions: :1. Every face of a simplex from \mathcal is also in \mathcal. :2. The non-empty intersection of any two simplices \sigma_1, \sigma_2 \in \mathcal is a face of both \sigma_1 and \sigma_2. See also the definition of an abstract simplicial complex, which loosely speaking is a simplicial complex without an associated geometry. A simplicial ''k''-complex \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Order
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation indicating that, for certain pairs of elements in the set, one of the elements precedes the other in the ordering. The relation itself is called a "partial order." The word ''partial'' in the names "partial order" and "partially ordered set" is used as an indication that not every pair of elements needs to be comparable. That is, there may be pairs of elements for which neither element precedes the other in the poset. Partial orders thus generalize total orders, in which every pair is comparable. Informal definition A partial order defines a notion of comparison. Two elements ''x'' and ''y'' may stand in any of four mutually exclusive relationships to each other: either ''x'' ''y'', or ''x'' and ''y'' are ''incompar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
