Affinity functions in fuzzy connectedness based image segmentation I: Equivalence of affinities

Fuzzy connectedness (FC) constitutes an important class of image segmentation schemas. Although affinity functions represent the core aspect (main variability parameter) of FC algorithms, they have not been studied systematically in the literature. In this paper, we began filling this gap by introducing and studying the notion of equivalent affinities: if any two equivalent affinities are used in the same FC schema to produce two versions of the algorithm, then these algorithms are equivalent in the sense that they lead to identical segmentations. We give a complete and elegant characterization of the affinity equivalence. We also demonstrate that any segmentation obtained via a relative fuzzy connectedness (RFC) algorithm can be viewed as segmentation obtained via absolute fuzzy connectedness (AFC) algorithm with an automatic and adaptive threshold detection. Since the main goal of the paper is to identify, by formal mathematical arguments, the affinity functions that are equivalent, extensive experimental confirmations are not needed — they show completely identical segmentations — and as such, only relevant examples of the theoretical results are provided.