Defuzzification using Steiner points

Authors:
Thomas Vetterlein;Mirko Navara
Affiliations:
Department of Computer Sciences 1, University of Dortmund, 44221 Dortmund, Germany;Center for Machine Perception, Department of Cybernetics. Faculty of Electrical Engineering, Czech Technical University, Technická 2, 166 27 Praha 6, Czech Republic
Venue:
Fuzzy Sets and Systems
Year:
2006

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Abstract

A defuzzification function assigns to each fuzzy set a crisp value in a way that this value may intuitively be understood as the ''centre'' of the fuzzy set. In the present paper, this vague concept is put into a mathematically rigorous form. To this end, we proceed analogously to the case of sharply bordered subsets, for which the Steiner point is frequently used. The function assigning to each convex subset its Steiner point is characterised by three properties; here, we study functions whose domains consist of fuzzy sets and which fulfil analogous properties. Although uniqueness can no longer be achieved, we give a complete characterisation of what we call Steiner points of fuzzy sets.