<title>Digital topology of multicolor images</title>

Abstract

In this paper a solution is presented which guarantees we avoid the connectivity paradoxes related to the Jordan Curve Theorem for all multicolor images. Only one connectedness relation is used for the entire digital image. We use only 4-connectedness (which is equivalent to 8-connectedness) for every component of every color. The idea is not to allow a certain `critical configuration' which can be detected locally to occur in digital pictures; such pictures are called `well-composed.' Well-composed images have very nice topological properties. For example, the Jordan Curve Theorem holds and the Euler characteristic is locally computable. This implies that properties of algorithms used in computer vision can be stated and proved in a clear way, and that the algorithms themselves become simpler and faster.