Spectral fuzzy sets and soft thresholding

Abstract

A new concept of spectral fuzzy sets has been introduced for dealing with the uncertainty in subjective assignment of membership value to the elements of a fuzzy set. The concept is characterized by a set of membership functions reflecting an individual's (or several individuals') opinion. The uncertainty due to spectralness has been explained. To demonstrate the necessity and effectiveness of this concept, the problem of image segmentation is considered as an example. Here the uncertainty may arise from differences in opinion in selecting a threshold from an ambiguous flat-valleyed histogram and in selecting ill-defined region boundaries. It has been shown that the new concept can be considered as a useful tool for the management of such uncertainty since it allows the implementation of soft decisions on the basis of various opinions on membership functions resulting in a band of thresholds with their respective uncertainties. A hard (crisp) decision obviously corresponds to one with minimum uncertainty. Besides the introduction of a new concept of spectral fuzzy set, the work also shows a natural and appropriate way that the incertitude arising from settling a definite degree of fuzziness (or from defining a precise membership function) may still lead to soft decisions when the image regions are ill-defined.