Computing equivalent affinity classes in a fuzzy connectedness framework

Abstract

The equivalence of affinities in fuzzy connectedness (FC) is a novel concept which gives us the ability to study affinity functions and their precise connection with FC algorithms. Two seminal papers by Ciesielski and Udupa create a strong theoretical background and provide some useful practical examples. Our intention here is to investigate this concept further because from a practical viewpoint if we are able to determine the equivalence classes for a given set of affinity functions and narrow it down to a much smaller set of nonequivalent affinities, then the set can be used more effectively in an optimization framework which searches for the best affinity function or parameters for a special task. In other words, we can find the best configuration for a set of given hardware or an image set with special characteristics. From a theoretical perspective, we are interested in the complexity of this problem, i.e. determining equivalence classes. Here, an affinity operator is used which is a function of a given parameter and maps different parameter values for different affinity functions. Our first questions, namely how many different meaningful, non-equivalent affinities there are and how we can enumerate them, led us to a general problem of how the equivalent affinities partition the parameter's domain and how the corresponding equivalence classes can be determined. We will provide a general algorithm schema to construct special algorithms which are able to compute the equivalence classes. We will also analyze a special but very common scenario of when the affinity operator combines two affinities (e.g. a homogeneity and an object feature-based affinity) using an aggregation operator (e.g. weighted average) and the particular parameter defines the weights of the affinities. Based on the general algorithm schema, we propose algorithms for this special case and we determine their complexity as well. These algorithms will be tested on two sets of medical images, namely, 25 digital dermoscopy images 1280 x 1024 pixels in size and 3 x 25 simulated brain MRI slices 181 x 217 in size.