© 1996, Topology Atlas

According to the Jordan curve theorem, simple closed curves in the plane have an interior and an exterior which between them contain all the points not on the curve and both of which are connected, but are disconnected from each other. Generalization to surfaces in digital spaces is useful when displaying (only the exterior of a surface is visible from any direction) or analyzing (the interior volume is well-defined).

In its most abstract representation, a digital space is a (possibly infinite) directed graph. (A concrete interpretation in a Euclidean space is that the space is tessellated into polyhedra --- which are the nodes of the graph --- and an arc represents the oriented face between two polyhedra which have a face in common.) A nonempty set of arcs is said to be near-Jordan surface in the digital space if any path from an initial node of an arc in S to a terminal node of an arc in S must contain an element of S. It can be shown that a minimally near-Jordan surface partitions the set of nodes into an interior (which contains all the initial nodes of arcs in S) and an exterior (which contains all the terminal nodes of arcs in S) such that both of them are connected (in the sense that from any element there is a path to any other element which does not contain an element of S), but any path from any interior node to any exterior node must contain an element of S. Thus, minimally near-Jordan surfaces are digital space analogues of simple closed curves in the plane.

The above and many other related results are presented in [1]. An already published paper which introduces the basic concepts (and contains some early results on near-Jordan surfaces) is [2].

[1] R. Aharoni, G. T. Herman, and M. Loebl. Jordan graphs. Graphical Models and Image Processing, to appear.

[2] G. T. Herman. Oriented surfaces in digital spaces. CVGIP: Graphical Models and Image Processing, 55 (1993), 381--396.