Multicolor well-composed pictures

Abstract

As was noted early in the history of computer vision, using the same adjacency relation for the entire digital picture leads to so-called “paradoxes” related to the Jordan Curve Theorem. The most popular idea to avoid these paradoxes in binary images was using different adjacency relations for the foreground and the background: 8-adjacency for black points and 4-adjacency for white points, or vice versa. This idea cannot be extended in a straightforward way to multicolor pictures. In this paper a solution is presented which guarantees us to avoid the connectivity paradoxes related to the Jordan Curve Theorem for all multicolor pictures. Only one connectedness relation will be used for the entire digital picture, i.e. 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 will be called “well-composed”. Well-composed pictures 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. Moreover, if a digitization process is guaranteed to preserve topology, then the obtained digital pictures must be well-composed.