Continuous Representations of Digital Images

A 2D digital image S is represented conventionally by the union of grid squares containing pixels of S which we denote by F(S). This gives the correct topology for S with 8-adjacency, and with a little imagination, 4-adjacency can also be properly handled. However, one encounters difficulty in extending basic 2D results to 3D digital images. The last few years have seen the need for better methods which give a closer link with well developed continuous topology, especially with the advent of digital surface theory. We define a new continuous model F(S) by refining F(S). We show that this gives a better bridge between the two subjects, digital and continuous topologies. We also show how this space F(S) is related to two other continuous models. Although we concentrate only on 2D images in this paper, the concepts and general ideas extend to 3D images. A 3D version of this paper is in preparation.