The grid cell topology is studied in
digital topology as part of the theoretical basis for (low-level) algorithms in
computer image analysis or
computer graphics
Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great deal ...
.
The elements of the ''n''-dimensional grid cell
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...
(''n'' ≥ 1) are all ''n''-dimensional grid cubes and their ''k''-dimensional faces ( for 0 ≤ ''k'' ≤ ''n''−1); between these a
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 ...
''A'' ≤ ''B'' is defined if ''A'' is a subset of ''B'' (and thus also dim(''A'') ≤ dim(''B'')). The grid cell topology is the
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 re ...
(open sets are up-sets) with respect to this partial order. (See also
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 simpli ...
.)
Alexandrov Alexandrov (masculine, also written Alexandrow) or Alexandrova (feminine) may refer to:
* Alexandrov (surname) (including ''Alexandrova''), a Slavic last name
* Alexandrov, Vladimir Oblast, Russia
* Alexandrov (inhabited locality), several inhabite ...
and
Hopf first introduced the grid cell topology, for the
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 ...
case, within an exercise in their text ''Topologie'' I (1935).
A recursive method to obtain ''n''-dimensional grid cells and an intuitive definition for
grid cell manifolds can be found in Chen, 2004. It is related to
digital manifolds.
See also
*
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 ...
References
*''Digital Geometry: Geometric Methods for Digital Image Analysis'', by Reinhard Klette and Azriel Rosenfeld, Morgan Kaufmann Pub, May 2004, (The Morgan Kaufmann Series in Computer Graphics)
*''Topologie'' I, by Paul Alexandroff and Heinz Hopf, Springer, Berlin, 1935, xiii + 636 pp.
*
Digital topology
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