Grid cell topology
Encyclopedia
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
.
The elements of the n-dimensional grid cell topology
(n ≥ 1) are all n-dimensional grid cubes and their k-dimensional faces ( for 0 ≤ k ≤ n−1); between these a partial order 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
(open sets are up-sets) with respect to this partial order. (See also poset topology
.)
Alexandrov
and Hopf
first introduced the grid cell topology, for the two-dimensional 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
s.
Digital topology
Digital topology deals with properties and features of two-dimensional or three-dimensional digital imagesthat correspond to topological properties or topological features of objects....
as part of the theoretical basis for (low-level) algorithms in computer image analysis or computer graphics
Computer graphics
Computer graphics are graphics created using computers and, more generally, the representation and manipulation of image data by a computer with help from specialized software and hardware....
.
The elements of the n-dimensional grid cell topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
(n ≥ 1) are all n-dimensional grid cubes and their k-dimensional faces ( for 0 ≤ k ≤ n−1); between these a partial order 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
Alexandrov topology
In topology, an Alexandrov space is a topological space 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...
(open sets are up-sets) with respect to this partial order. (See also poset topology
Poset topology
In mathematics, the poset topology associated with a partially ordered set S is the Alexandrov topology on the poset of finite chains of S, ordered by inclusion.Let V be a set of vertices...
.)
Alexandrov
Pavel Sergeevich Alexandrov
Pavel Sergeyevich Alexandrov , sometimes romanized Aleksandroff or Aleksandrov was a Soviet Russian mathematician...
and Hopf
Heinz Hopf
Heinz Hopf was a German mathematician born in Gräbschen, Germany . He attended Dr. Karl Mittelhaus' higher boys' school from 1901 to 1904, and then entered the König-Wilhelm- Gymnasium in Breslau. He showed mathematical talent from an early age...
first introduced the grid cell topology, for the two-dimensional 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
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.-...
s.