Fuzzy Entropy Measures Based on Distance Transformations

Entropy has been a subject of study in Fuzzy Set Theory ever since the pioneering proposal by Zadeh in 1968. Over the years, different interpretations and models for entropy were produced within the community, often tailored for specific generalizations of Fuzzy Sets. Such multiplicity of operators and definitions is suboptimal, although understandable due to the heterogeneous nature of membership degrees in different generalizations of Fuzzy Sets. One of the most difficult tasks in the context of Set-Valued Fuzzy Sets is to find suitable definitions of theoretical notions which can deal with finite and infinite cardinalities simultaneously. In this work, a novel visually representable entropy measure for Set-Valued Fuzzy Sets is proposed, embracing both finite and infinite cardinalities. This entropy measure is hence applicable to Fuzzy Sets or Interval-Valued Fuzzy Sets, allowing for a common interpretation of entropy across different generalizations of Fuzzy Sets. The behaviour of this measure is theoretically studied and its usefulness is demonstrated through a case study in the context of image processing.