The grid cell topology is studied in
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 an ...
as part of the theoretical basis for (low-level) algorithms in
computer image analysis
Image analysis or imagery analysis is the extraction of meaningful information from images; mainly from digital images by means of digital image processing techniques. Image analysis tasks can be as simple as reading bar coded tags or as sophist ...
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 de ...
.
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 ...
(''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 bina ...
''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 rest ...
(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 simplicia ...
.)
Alexandrov 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 as ...
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 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 piec ...
s.
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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