Affinity functions in fuzzy connectedness based image segmentation II: Defining and recognizing truly novel affinities

Affinity functions — the measure of how strongly pairs of adjacent spels in the image hang together — represent the core aspect (main variability parameter) of the fuzzy connectedness (FC) algorithms, an important class of image segmentation schemas. In this paper, we present the first ever theoretical analysis of the two standard affinities, homogeneity and object-feature, the way they can be combined, and which combined versions are truly distinct from each other. The analysis is based on the notion of equivalent affinities, the theory of which comes from a companion Part I of this paper (Ciesielski and Udupa, in this issue) [11]. We demonstrate that the homogeneity based and object feature based affinities are equivalent, respectively, to the difference quotient of the intensity function and Rosenfeld’s degree of connectivity. We also show that many parameters used in the definitions of these two affinities are redundant in the sense that changing their values lead to equivalent affinities. We finish with an analysis of possible ways of combining different component affinities that result in non-equivalent affinities. In particular, we investigate which of these methods, when applied to homogeneity based and object-feature based components lead to truly novel (non-equivalent) affinities, and how this is affected by different choices of parameters. 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 FC segmentations — and as such, only relevant examples of the theoretical results are provided. Instead, we focus mainly on theoretical results within a perspective of the fuzzy connectedness segmentation literature.