Abstract cell complex

Abstract cell complexes play an important role in image analysis and computer graphics.

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

In his book (2008) [9] he suggested an axiomatic theory of locally finite topological spaces which are generalization of abstract cell complexes.

Abstract complexes allow the introduction of classical topology (Alexandrov-topology) in grids being the basis of digital image processing.

The book by V. Kovalevsky[10] contains the description of the theory of locally finite spaces which are a generalization of abstract cell complexes.

An abstract cell complex is a particular case of a locally finite space in which the dimension is defined for each point.

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 [11] 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).

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.