Mathematics has always fascinated us in many ways, new theories and proposals come up each other day which changes the precious theories. In an open family of intersection where there are open sets and the removal of finite restriction is majorly called the Alexandrov topology, the Abstract Cell Complexes or ACC works on this principle. In this a positive or integer of non-negative value is allocated to a point each, such integers are called dimensions. ACC is mainly proposed for 3-D and larger dimension spaces, analysis of images etc. The main disadvantage of a Euclidian geometry is its failure to analyze images of different dimensions and resolutions. In this the dimension is defined per points in a finite space satisfying new axioms depending on the relations of the neighborhood space. The basic relation of the ACC as defined by mathematician E-Steinitz is C= (E, B, dim). E denotes abstract cells, B refers to the relation that is bounding and dim points out the positive integer.

Advantages of ACC

  • Independent of other geometrical theories such as Euclidean
  • Solves lot of image analysis problems
  • Used in the field of computer animations and graphics
  • Proposes new algorithms for image analysis
  • Boundaries could be traced in 2D, 3D or higher dimensions using various algorithms
  • Encoding of images in pixel cells
  • The code of the boundary is used for rebuilding subsets

A surface of a polyhedron which is finite is a classic example of a finite topological surface. The edges and the vertices of the polyhedron is also a part of topology. Pixels and Voxels are categorized based on the image analysis. The basic rule of ACC is the co-ordinate assignment rule; in this finally the image elements are assigned with positive integers in it. For that each pixel is identified and market correctly during this process. 

Download Abstract Cell Complexes (ACC) Seminar Report.