Abstract cell complex
Abstract cell complex
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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 called “cells” are not subsets of a Hausdorff space as it is the case in Euclidean and CW complex. Abstract cell complexes play an important role in image analysis and computer graphics.
History
The idea of abstract complexes relates to J. Listing (1862) and E. Steinitz(1908). V. Kovalevsky (1989) described abstract cell complexes for 3D and higher dimensions. He also suggested numerous applications to image analysis. In his book (2008) (www.kovalevsky.de) 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, a new definition of combinatorial manifolds and many algorithms useful for image analysis.
Basic Results
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: 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: . V. Kovalevsky regards in his book (2008) 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 (poin) 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 of the maximum bounding path (or neighborhood path) leading from c to a cell c0 which is bounded by no other cells or which does not belong to the smallest neighborhood of some cell different from c. The book (2008) 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.
References
- Kovalevsky, V. (2008). Geometry of Locally Finite Spaces". Publishing house Dr. Baerbel Kovalevski, Berlin. ISBN 978-3-9812252-0-4.
- Kovalevsky, V. (1989). ""Finite Topology as Applied to Image Analysis"". Computer Visdion, Graphics and Image Processing, v. 45, No.2, pp. 141-161.
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External links
VAKovalevsky (talk) 12:10, 27 March 2012 (UTC)