Abstract cell complex

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. Abstract cell complexes play an important role in image analysis and computer graphics. 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. Abstract cell complexes play an important role in image analysis and computer graphics. The idea of abstract cell complexes (also named abstract cellular complexes) relates to J. Listing (1862) und E. Steinitz (1908). Also A.W Tucker (1933), K. Reidemeister (1938), P.S. Aleksandrov (1956) as well as R. Klette und A. Rosenfeld (2004) have described abstract cell complexes. E. Steinitz has defined an abstract cell complex as C = ( E , B , d i m ) {displaystyle C=(E,B,dim)} 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 B ( a , b ) {displaystyle B(a,b)} , then d i m ( a ) < d i m ( b ) {displaystyle dim(a)<dim(b)} . V. Kovalevsky (1989) described abstract cell complexes for 3D and higher dimensions. He also suggested numerous applications to image analysis. In his book (2008) 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. 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 no simplices: An n-dimensional element of an abstract complexes must not have n+1 zero-dimensional sides, and not each subset of the set of zero-dimensional sides of a cell is 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.