The importance of topological connectedness properties in processing digital pictures is well known. A natural way to begin a theory for this is to give
a definition of connectedness for subsets of a digital plane which allows one
to prove a Jordan curve theorem. The generally accepted approach to this
has been a non-topological Jordan curve theorem which requires two different definitions, 4-connectedness, and 8-connectedness, one for the curve and
the other for its complement.In [KKM] we introduced a purely topological context for a digital plane
and proved a Jordan curve theorem. The present paper gives a topological
proof of the non-topological Jordan curve theorem mentioned above and
extends our previous work by considering some questions associated with
image processing:How do more complicated curves separate the digital plane into connected
sets? Conversely given a partition of the digital plane into connected sets,
what are the boundaries like and how can we recover them? Our construction
gives a unified answer to these questions.The crucial step in making our approach topological is to utilize a natural
connected topology on a finite, totally ordered set; the topologies on the
digital spaces are then just the associated product topologies. Furthermore,
this permits us to define path, arc, and curve as certain continuous functions
on such a parameter interval.
A Jordan Curve Theorem for 2-dimensional Tilings
