Grid cell topology

Grid cell topology

The grid cell topology is studied in digital topology for the purpose of providing a 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).

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)ISBN 1-55860-861-3

"Topologie" I, by Paul Alexandroff and Heinz Hopf, Springer, Berlin, 1935, xiii + 636 pp.


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