Width-Based Interval-Valued Distances and Fuzzy Entropies

In this paper, we study distance measures between interval-valued fuzzy sets and entropies of interval-valued fuzzy sets. These are well-known and widely used notions in the fuzzy sets theory. The novelty of our approach is twofold: on one hand, it considers the width of intervals in order to connect the uncertainty of the output with the uncertainty of the inputs. On the other hand, it makes use of total orders between intervals, instead of partial ones, so that the usefulness of the notions related with some kind of monotonicity is fully recovered in the interval-valued setting. The construction of distance measures and entropies is done by aggregating interval-valued restricted dissimilarity functions and interval-valued normal E_{N} -functions. For this reason, we first study these functions, both in line with the two above stated considerations. Finally, we present an illustrative example in image thresholding using an expression of the proposed interval-valued entropy to show the validity of our approach.