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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 it is the case in Euclidean and [[CW complex]]es. Abstract cell complexes play an important role in [[image analysis]] and [[computer graphics]].

==History==

The idea of abstract cell complexes <ref>Reinhard Klette: Cell complexes through time. http://spie.org/Publications/Proceedings/Paper/10.1117/12.404813</ref> (also named abstract cellular complexes) relates to [[Johann Benedict Listing|J. Listing]] (1862) <ref>Listing J.: "Der Census raeumlicherräumlicher Complexe". ''Abhandlungen der KoeniglichenKöniglichen Gesellschaft der Wissenschaften zu GoettingenGöttingen'', v. 10, GoettingenGöttingen, 1862, pp. 97–182.</ref> and [[Ernst Steinitz|E. Steinitz]] (1908).<ref>Steinitz E.: "BeitraegeBeiträge zur Analysis". ''Sitzungsbericht Berliner Mathematischen Gesellschaft'', vBand. 7, 1908, pp. 29–49.</ref> Also A.W Tucker (1933) ,<ref> Tucker A.W.: "An abstract approach to manifolds", Annals Mathematics, v. 34, pp.1933, 191-243. </ref>, K. ReidemacherReidemeister (1938) ,<ref> ReidemacherReidemeister K.: "Topologie der Polyeder und kombinatorische Topologie der Komplexe". Akademische Verlagsgesellschaft Geest & Portig, Leipzig, 1938 (second edition 1953) </ref>, P.S. Aleksandrov (1956) <ref> Aleksandrov P.S.: Combinatorial Topology, Graylock Press, Rochester, 1956, </ref> as well as R. Klette and A. Rosenfeld (2004) <ref> Klette R. and Rosenfeld. A.: "Digital Geometry", Elsevier, 2004. </ref> have described abstract cell complexes. E. Steinitz has defined an abstract cell complex as <math> C=(E,B,dim)</math> 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-negative integer to each element of ''E'' in such a way that if <math>B(a, b)</math>, then <math>dim(a)<dim(b)</math>.
V. [[Vladimir Antonovich Kovalevsky|Kovalevsky]] (1989) <ref>Kovalevsky, V.: "Finite Topology as Applied to Image Analysis", ''Computer Vision, Graphics and Image Processing'', v. 45, No. 2, 1989, pp. 141–161.</ref> described abstract cell complexes for 3D and higher dimensions. He also suggested numerous applications to image analysis. In his book (2008) <ref>{{cite web |url=http://www.geometry.kovalevsky.de./ |title=Home |website=geometry.kovalevsky.de.}}</ref> he has suggested an axiomatic theory of locally finite [[topological spaces]] which are generalization of abstract cell complexes. The book contains among others new definitions of topological balls and spheres independent of [[metric (mathematics)|metric]], a new definition of [[combinatorial manifold]]s and many algorithms useful for image analysis.

==Basic results==

The topology of abstract cell complexes is based on a [[partial order]] in the set of its points or cells.

The topology of abstract complexes is based on a [[partial order]] in the set of its points or cells. E. Steinitz has considered the asymmetric, irreflexive and transitive [[binary relation]] among the cells called '''bounding relation'''. He defined an abstract cell complex as: <math> C=(E, B, \dim) </math> where ''E'' is an abstract set of cells, ''B'' is the bounding relation and dim is a function assigning a non-negative integer number to each cell in such a way that if the cell ''a'' bounds the cell ''b'' then: <math> \dim(a)<\dim(b) </math>.

The notion of an abstract cell complex is closely related to that of an [[abstract simplicial complex]]. A non-simplicial complex is a generalization which allows the introduction of coordinates: there are non-simplicial complexs being Cartesian products of one-dimensional complexes. Only such complexes make it possible to introduce coordinates. An abstract cell complex whose cells look like squares in the 2D case or like cubes in the 3D case makes it possible to introduce coordinates of cells which is impossible in the case of simplicial complexs. Abstract complexes make also possible to introduce a classical topology (Aleksandrov topology) in grids which are the basis of digital image processing. This is the great advantage of abstract cell complexes: It becomes possible to exactly define the notions of connetedness and that of boundary of a subset in digital image processing. The dimension of cells and of complexes is in the general case defined differntly as for simplicial complexes (see below).

V. Kovalevsky regards in his book <ref>V. Kovalevsky: "Geometry of Locally Finite Spaces". Publishing House Dr. Baerbel Kovalevski, Berlin 2008. ISBN 978-3-9812252-0-4.</ref> the theory of '''locally finite spaces''' which are generalization of abstract cell complexes. A locally finite space ''S'' is a set of points in which a subset of ''S'' containing a limited number of points and called the '''smallest neighborhood''' is defined for each point of ''S''. V. Kovalevsky defined the binary '''neighborhood relation''' in the set of points of a locally finite space S: The element (point) ''b'' is in the neighborhood relation to the element ''a'' if ''b'' belongs to the smallest neighborhood of the element ''a''. He formulated new axioms of a locally finite space and proved that the space ''S'' satisfies the axioms if and only if the neighborhood relation is antisymmetric and transitive. The neighborhood relation is the reflexive hull of the inverse bounding relation. He showed that classical axioms of topology can be deduced from the new axioms as theorems. Therefore a locally finite space satisfying the new axioms is a particular case of a classical topological space. Its topology is a [[poset topology]] or [[Alexandrov topology]].

An abstract cell complex is a particular case of a locally finite space in which the dimension is defined for each point. V. Kovalevsky has shown that the dimension of a cell ''c'' of an abstract cell complex is equal to the length (number of cells minus 1) of the maximum bounding path (or neighborhood path) leading from ''c'' to a cell ''c''⁰ which is bounded by no other cells or which does not belong to the smallest neighborhood of some cell different from ''c''.

The book contains the theory of digital straight segments in 2D complexes, numerous algorithms for tracing boundaries in 2D and 3D, for economically encoding the boundaries and for exactly reconstructing a subset from the code of its boundary.

The notion of the abstract cell complex defined by E. Steinitz is related to the notion of an [[abstract simplicial complex]] and it differs from a [[simplicial complex]] by the property that its elements are not [[simplex|simplices]]: An ''n''-dimensional element of an abstract complexes does not necessarily have ''n''+1 zero-dimensional sides, and not each subset of the set of zero-dimensional sides of a cell is necessarily a cell. This is important since the notion of an abstract cell complexes can be applied to the two- and three-dimensional grids used in image processing, which is not true for simplicial complexes. A non-simplicial complex is a generalization which makes the introduction of cell coordinates possible: There are non-simplicial complexes which are Cartesian products of such "linear" one-dimensional complexes where each zero-dimensional cell, besides two of them, bounds exactly two one-dimensional cells. Only such Cartesian complexes make it possible to introduce such coordinates that each cell has a set of coordinates and any two different cells have different coordinate sets. The coordinate set can serve as a name of each cell of the complex which is important for processing complexes.

== References ==

{{Reflist}}

Abstract complexes allow the introduction of classical topology (Alexandrov-topology) in grids being the basis of digital image processing. This possibility defines the great advantage of abstract cell complexes: It becomes possible to exactly define the notions of connectivity and of the boundary of subsets. The definition of dimension of cells and of complexes is in the general case different from that of simplicial complexes (see below).

== External links ==

* [http://www.kovalevsky.de |Prof. Dr. Vladimir Kovalevsky]

The notion of an abstract cell complex differs essentially from that of a CW-complex because an abstract cell complex is not [[Hausdorff space|Hausdorff]]. This is important from the point of view of computer science since it is impossible to explicitly represent a non-discrete Hausdorff space in a computer. (The neighborhood of each point in such a space must have infinitely many points).

V.The Kovalevskybook regardsby in[[Vladimir hisAntonovich bookKovalevsky|V. Kovalevsky]]<ref>V. Kovalevsky: "Geometry of Locally Finite Spaces". PublishingEditing Househouse Dr. BaerbelBärbel Kovalevski, Berlin 2008. {{ISBN |978-3-9812252-0-4}}.</ref> contains the description of the theory of '''[[locally finite spaces'''space]]s which are a generalization of abstract cell complexes. A locally finite space ''S'' is a set of points in whichwhere a subset of ''S'' is defined for each point ''P'' of ''S''. This subset containing a limited number of points andis called the '''smallest neighborhood''' is defined for each point of ''SP''. V. Kovalevsky defined theA binary '''neighborhood relation''' is defined in the set of points of athe locally finite space ''S'': The element (point) ''b'' is in the neighborhood relation towith the element ''a'' if ''b'' belongs to the smallest neighborhood of the element ''a''. He formulated newNew axioms of a locally finite space have been formulated, and provedit was proven that the space ''S'' satisfiesis thein axiomsaccordance ifwith andthe axioms only if the neighborhood relation is antisymmetricanti-symmetric and transitive. The neighborhood relation is the reflexive hull of the inverse bounding relation. HeIt showedwas shown that classical axioms of the topology can be deduced as theorems from the new axioms as theorems. Therefore, a locally finite space satisfying the new axioms is a particular case of a classical topological space. Its topology is a [[poset topology]] or [[Alexandrov topology]].

An abstract cell complex is a particular case of a locally finite space in which the dimension is defined for each point. It was demonstrated that the dimension of a cell ''c'' of an abstract cell complex is equal to the length (number of cells minus 1) of the maximum bounding path leading from any cell of the complex to the cell ''c''. The bounding path is a sequence of cells in which each cell bounds the next one. The book contains the theory of digital straight segments in 2D complexes, numerous algorithms for tracing boundaries in 2D and 3D, for economically encoding the boundaries and for exactly reconstructing a subset from the code of its boundary. Using the abstract cell complexes, efficient algorithms for tracing, coding and polygonization of boundaries, as well as for the edge detection, are developed and described in the book <ref>Kovalevsky, V., Image Processing with Cellular Topology, Springer 2021, ISBN 978-981-16-5771-9.</ref>

==Abstract Cell Complex Digital Image Representation==

[[File:Digital Image ACC Decomposition.PNG|thumb|A 3x4 digital image decomposed into its Abstract Cell Complex dimensional constituents.]]

A digital image may be represented by a 2D Abstract Cell Complex (ACC) by decomposing the image into its ACC dimensional constituents: points (0-cell), cracks/edges (1-cell), and pixels/faces (2-cell).

[[File:Digital Image ACC Coordinate Assignment.PNG|thumb|Digital Image ACC Coordinate Assignment]]

This decomposition together with a coordinate assignment rule to unambiguously assign coordinates from the image pixels to the dimensional constituents permit certain image analysis operations to be carried out on the image with elegant algorithms such as crack [[boundary tracing]], [[digital straight segment]] subdivision, etc. One such rule maps the points, cracks, and faces to the top left coordinate of the pixel. These dimensional constituents require no explicit translation into their own data structures but may be implicitly understood and related to the 2D array which is the usual data structure representation of a digital image. This coordinate assignment rule and the renderings of each cell incident to this image is depicted in the image at right.

== See also ==

{{Portal|Mathematics}}

* [[Simplicial complex]]

* [[Cubical complex]]

== References ==

{{Reflist}}

[[Category:Topology]]