CONCEPTS OF DIGITAL-TOPOLOGY

In an earlier paper written for a different readership [Computers and Graphics 13(2) (1989) 159-166] the first author defined a digital fundamental group-an analog, for binary digital pictures, of the fundamental group. In general the definition of the digital fundamental group involves continuous deformation. But an alternative, discrete, definition of the digital fundamental group was proposed for the strongly normal digital picture spaces defined in the same paper. The above-mentioned paper also defined a "continuous analog" C(P) for each binary digital picture P on such a DPS (DPS = digital picture space). C(P) is a polyhedron constructed by "filling in the gaps" between black points (1's) of the binary digital picture P in a specific way. Other kinds of continuous analog had previously been used by the first two authors. In seeking the simplest and most efficient algorithms for performing image processing operations, researchers have considered many different combinations of grids and adjacency relations. Almost all of those combinations are isomorphic to special cases of the concept of a strongly normal DPS. The main contribution of the present paper is a proof that the digital fundamental groups of binary digital pictures on a strongly normal DPS are naturally isomorphic to the fundamental groups of the digital pictures' continuous analogs. We use this result to establish that on a strongly normal DPS the discrete and continuous definitions of the digital fundamental group are equivalent, up to a natural group isomorphism. We also show that many topological results which hold in the Euclidean plane or Euclidean 3-space have analogs that hold in every strongly normal DPS. Our results suggest that a strongly normal DPS is a suitable domain for studying topology-related image processing operations such as thinning, border tracking and contour filling. The definitions of digital fundamental group, strongly normal DPS and continuous analog are included in this paper so as to make it self-contained.