Jordan surfaces in simply connected digital spaces

Abstract

Certain tasks in multidimensional digital image analysis, in particular surface detection and volume estimation, lead us to the study of “surfaces” in the digital environment. It is desirable that these surfaces should have a “Jordan” property analogous to that of simple closed curves in two dimensions-namely, that they should partition the underlying space into an inside and an outside which are disconnected from each other. A version of this property, called near-Jordanness, has been previously defined and has been shown to be useful within a general theory of surfaces in digital spaces. The definition of a near-Jordan surface is global: it demands that all paths from the interior to the exterior cross the surface. This makes it difficult in many practical applications to check whether surfaces are near-Jordan. The work reported in this paper is motivated by the desire for a “local” condition, such that if a surface satisfies this condition at each of its elements, then it is guaranteed to be near-Jordan. In the search for such a condition, we were led to a concept of “simple connectedness” of a digital space, which resembles simple connectedness in ordinary topological spaces. We were then able to formulate the desired local condition for simply connected digital spaces. Many digital spaces, in particular those based on the commonly-studied tessellations of n-dimensional Euclidean space, are shown to be simply connected and thus our theory yields general sufficient conditions for boundaries in binary pictures to be near-Jordan.