page_title: Abstract cell complex
section_title: Abstract Cell Complex Digital Image Representation
hierarchical_section_title: Abstract cell complex / Abstract Cell Complex Digital Image Representation
caption_attribution_description: English: A 3x4 digital image decomposed into its Abstract Cell Complex dimensional constituents.
context_page_description: 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.
context_section_description: 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. This coordinate assignment rule and the renderings of each cell incident to this image is depicted in the image at right.
caption: A 3x4 digital image decomposed into its Abstract Cell Complex dimensional constituents.
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