Abstract cell complex: Difference between revisions

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{{COI|date=August 2020}}

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 räumlicher Complexe". ''Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen'', v. 10, Göttingen, 1862, 97–182.</ref> and [[Ernst Steinitz|E. Steinitz]] (1908).<ref>Steinitz E.: "Beiträge zur Analysis". ''Sitzungsbericht Berliner Mathematischen Gesellschaft'', Band. 7, 1908, 29–49.</ref> Also A.W Tucker (1933),<ref>Tucker A.W.: "An abstract approach to manifolds", Annals Mathematics, v. 34, 1933, 191-243.</ref> K. Reidemeister (1938),<ref>Reidemeister 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, 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 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 nonot [[simplex|simplices]]: An ''n''-dimensional element of an abstract complexes mustdoes 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.

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).

The notion of an abstract cell complex differs essentially from that of a CW-complex because an abstract cell complex is nonot [[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).

The book by [[Vladimir Antonovich Kovalevsky|V. Kovalevsky ]]<ref>V. Kovalevsky: "Geometry of Locally Finite Spaces". Editing house Dr. Bärbel Kovalevski, Berlin 2008. {{ISBN|978-3-9812252-0-4}}.</ref> contains the description of the theory of [[locally finite space]]s which are a generalization of abstract cell complexes. A locally finite space ''S'' is a set of points where a subset of ''S'' is defined for each point ''P'' of ''S''. This subset containing a limited number of points is called the '''smallest neighborhood''' of ''P''. A binary neighborhood relation is defined in the set of points of the locally finite space ''S'': The element (point) ''b'' is in the neighborhood relation with the element ''a'' if ''b'' belongs to the smallest neighborhood of the element ''a''. New axioms of a locally finite space have been formulated, and it was proven that the space ''S'' is in accordance with the axioms only if the neighborhood relation is anti-symmetric and transitive. The neighborhood relation is the reflexive hull of the inverse bounding relation. It was shown that classical axioms of the topology can be deduced as theorems from the new axioms. 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==

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* [[Simplicial complex]]

* [[Cubical complex]]

== References ==