On the characterization of simple closed surfaces in three-dimensional digital images

The paper is a continuation of a series on the digital geometry of three-dimensional digital images. In earlier reports, D. Morgenthaler and A. Rosenfeld gave symmetric definitions for simple surface points under the concepts of 6-connectivity and 26-connectivity, and they nontrivially characterized a simple closed surface (i.e., a subset of the image which separates its complement into an “inside” and an “outside”) as a connected collection of “orientable” simple surface points. Later, the author and A. Rosenfeld established that the computationally costly assumption of orientability is unnecessary for 6-connectivity by proving that orientability, a local property, is implicitly guaranteed within the (3 × 3 × 3)-neighborhood definition of a 6-connected simple surface point. However, they also showed that no such guarantee exists for 26-connectivity. In this report, the author completes this investigation of simple closed surfaces by showing that orientability is ensured globally by 26-connectivity. Hence, a simple closed surface may be efficiently characterized as a connected collection of simple surface points regardless of the type of connectivity under consideration.