A Generalized Fuzzy Extension Principle and Its Application to Information Fusion

Zadeh's extension principle (ZEP) is a fundamental concept in fuzzy set (FS) theory that enables crisp mathematical operation on FSs. A well-known shortcoming of ZEP is that the membership value of the output is restricted to the minimum of the maximum membership values of the input FSs, i.e., the height of the output FS is determined by the lowest height of the input FSs. In this article, we introduce a generalized extension principle (GEP) that eliminates this weakness and in addition provides flexibility and control over how membership values are mapped from input to output. Furthermore, we provide a point-based representation of an FS that allows efficient computation of the output FS. In light of our new definition, we discuss two approaches to perform aggregation of FSs using the Choquet integral. The resultant integrals generalize prior work and lay a foundation for future extensions. Last, we demonstrate the extended integrals via a combination of synthetic and real-world examples.