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Digital Topology
Digital topology deals with properties and features of two-dimensional (2D) or three-dimensional (3D) digital images that correspond to topological properties (e.g., connectedness) or topological features (e.g., boundaries) of objects. Concepts and results of digital topology are used to specify and justify important (low-level) image analysis algorithms, including algorithms for thinning, border or surface tracing, counting of components or tunnels, or region-filling. History Digital topology was first studied in the late 1960s by the computer image analysis researcher Azriel Rosenfeld (1931–2004), whose publications on the subject played a major role in establishing and developing the field. The term "digital topology" was itself invented by Rosenfeld, who used it in a 1973 publication for the first time. A related work called the grid cell topology, which could be considered as a link to classic combinatorial topology, appeared in the book of Pavel Alexandrov and Hein ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Two-dimensional
In mathematics, a plane is a Euclidean (flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as subspaces of some higher-dimensional space, as with one of a room's walls, infinitely extended, or they may enjoy an independent existence in their own right, as in the setting of two-dimensional Euclidean geometry. Sometimes the word ''plane'' is used more generally to describe a two-dimensional surface, for example the hyperbolic plane and elliptic plane. When working exclusively in two-dimensional Euclidean space, the definite article is used, so ''the'' plane refers to the whole space. Many fundamental tasks in mathematics, geometry, trigonometry, graph theory, and graphing are performed in a two-dimensional space, often in the plane. Euclidean geometry Euclid set forth the first great landmark of mathematical thought, an axiomatic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Combinatorial Topology
In mathematics, combinatorial topology was an older name for algebraic topology, dating from the time when topological invariants of spaces (for example the Betti numbers) were regarded as derived from combinatorial decompositions of spaces, such as decomposition into simplicial complexes. After the proof of the simplicial approximation theorem this approach provided rigour. The change of name reflected the move to organise topological classes such as cycles-modulo-boundaries explicitly into abelian groups. This point of view is often attributed to Emmy Noether, and so the change of title may reflect her influence. The transition is also attributed to the work of Heinz Hopf, who was influenced by Noether, and to Leopold Vietoris and Walther Mayer, who independently defined homology. A fairly precise date can be supplied in the internal notes of the Bourbaki group. While topology was still ''combinatorial'' in 1942, it had become ''algebraic'' by 1944. This corresponds also to the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Digital Geometry
Digital geometry deals with discrete sets (usually discrete point sets) considered to be digitized models or images of objects of the 2D or 3D Euclidean space. Simply put, digitizing is replacing an object by a discrete set of its points. The images we see on the TV screen, the raster display of a computer, or in newspapers are in fact digital images. Its main application areas are computer graphics and image analysis. Main aspects of study are: * Constructing digitized representations of objects, with the emphasis on precision and efficiency (either by means of synthesis, see, for example, Bresenham's line algorithm or digital disks, or by means of digitization and subsequent processing of digital images). * Study of properties of digital sets; see, for example, Pick's theorem, digital convexity, digital straightness, or digital planarity. * Transforming digitized representations of objects, for example (A) into simplified shapes such as (i) skeletons, by repeated removal of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent. It is commonly denoted by \chi ( Greek lower-case letter chi). The Euler characteristic was originally defined for polyhedra and used to prove various theorems about them, including the classification of the Platonic solids. It was stated for Platonic solids in 1537 in an unpublished manuscript by Francesco Maurolico. Leonhard Euler, for whom the concept is named, introduced it for convex polyhedra more generally but failed to rigorously prove that it is an invariant. In modern mathematics, the Euler characteristic arises from homology and, more abstractly, homological algebra. Polyhedra The Euler characteristic \chi was classically defined for the surfaces of polyhedra, acc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of n-dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not lemniscates. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. The concept has applications in computer-graphics given the need to associate pictures with coordinates (e.g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauss–Bonnet Theorem
In the mathematical field of differential geometry, the Gauss–Bonnet theorem (or Gauss–Bonnet formula) is a fundamental formula which links the curvature of a surface to its underlying topology. In the simplest application, the case of a triangle on a plane, the sum of its angles is 180 degrees. The Gauss–Bonnet theorem extends this to more complicated shapes and curved surfaces, connecting the local and global geometries. The theorem is named after Carl Friedrich Gauss, who developed a version but never published it, and Pierre Ossian Bonnet, who published a special case in 1848. Statement Suppose is a compact two-dimensional Riemannian manifold with boundary . Let be the Gaussian curvature of , and let be the geodesic curvature of . Then :\int_M K\,dA+\int_k_g\,ds=2\pi\chi(M), \, where is the element of area of the surface, and is the line element along the boundary of . Here, is the Euler characteristic of . If the boundary is piecewise smooth, then ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplicial Complexes
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 \ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Piecewise Linear Manifold
In mathematics, a piecewise linear (PL) manifold is a topological manifold together with a piecewise linear structure on it. Such a structure can be defined by means of an atlas, such that one can pass from chart to chart in it by piecewise linear functions. This is slightly stronger than the topological notion of a triangulation. An isomorphism of PL manifolds is called a PL homeomorphism. Relation to other categories of manifolds PL, or more precisely PDIFF, sits between DIFF (the category of smooth manifolds) and TOP (the category of topological manifolds): it is categorically "better behaved" than DIFF — for example, the Generalized Poincaré conjecture is true in PL (with the possible exception of dimension 4, where it is equivalent to DIFF), but is false generally in DIFF — but is "worse behaved" than TOP, as elaborated in surgery theory. Smooth manifolds Smooth manifolds have canonical PL structures — they are uniquely ''triangulizable,'' by Whitehead's theorem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pixels
In digital imaging, a pixel (abbreviated px), pel, or picture element is the smallest addressable element in a raster image, or the smallest point in an all points addressable display device. In most digital display devices, pixels are the smallest element that can be manipulated through software. Each pixel is a sample of an original image; more samples typically provide more accurate representations of the original. The intensity of each pixel is variable. In color imaging systems, a color is typically represented by three or four component intensities such as red, green, and blue, or cyan, magenta, yellow, and black. In some contexts (such as descriptions of camera sensors), ''pixel'' refers to a single scalar element of a multi-component representation (called a ''photosite'' in the camera sensor context, although '' sensel'' is sometimes used), while in yet other contexts (like MRI) it may refer to a set of component intensities for a spatial position. Etymology Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pixel Connectivity
In image processing, pixel connectivity is the way in which pixels in 2-dimensional (or hypervoxels in n-dimensional) images relate to their neighbors. Formulation In order to specify a set of connectivities, the dimension N and the width of the neighborhood n , must be specified. The dimension of a neighborhood is valid for any dimension n\geq1 . A common width is 3, which means along each dimension, the central cell will be adjacent to 1 cell on either side for all dimensions. Let M_N^n represent a N-dimensional hypercubic neighborhood with size on each dimension of n=2k+1, k\in\mathbb Let \vec represent a discrete vector in the first orthant from the center structuring element to a point on the boundary of M_N^n. This implies that each element q_i \in \ ,\forall i \in \ and that at least one component q_i = k Let S_N^d represent a N-dimensional hypersphere with radius of d=\left \Vert \vec \right \Vert. Define the amount of elements on the hypersphere S_N^d with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Digital Manifold
In mathematics, a digital manifold is a special kind of combinatorial manifold which is defined in digital space i.e. grid cell space. A combinatorial manifold is a kind of manifold which is a discretization of a manifold. It usually means a piecewise linear manifold made by simplicial complexes. Concepts Parallel-move is used to extend an i-cell to (i+1)-cell. In other words, if A and B are two i-cells and A is a parallel-move of B, then is an (i+1)-cell. Therefore, k-cells can be defined recursively. Basically, a connected set of grid points M can be viewed as a digital k-manifold if: (1) any two k-cells are (k-1)-connected, (2) every (k-1)-cell has only one or two parallel-moves, and (3) M does not contain any (k+1)-cells. See also *Digital geometry *Digital topology *Topological data analysis *Topology *Discrete mathematics Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, havi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
