On topology as applied to image analysis

Abstract

We discuss the recently published claim of V. A. Kovalevsky that the topology of cellular complexes is the only appropriate topology for image analysis. In some sense we confirm this claim and even generalize it from the finite domain to an infinite one. We prove some results which can be interpreted to show that the class of partially ordered sets is strictly equivalent to a class of topological spaces which is certainly powerful enough to handle all of image analysis. However, such equivalence does not carry over when the partially ordered sets are complemented with a dimension function so as to form cellular complexes. In fact, it remains unclear whether the subclass of cellular complexes which use the assignment of dimension which is standard in image analysis is indeed powerful enough to encompass all problems of image analysis.