On Hausdorff-like metrics for fuzzy sets

In (Chaudhuri and Rosenfeld, 1996), metrics were developed for fuzzy sets u and v defined on the same support set S. These metrics are based on applying the Hausdorff metric to pairs of membership level sets (u 1 ( [ tk, 1 ] ), v1 ( [ tk, 1 ] ) ), where tk E [0, 1] is a (possible) membership value of u or v. The metrics of (Chaudhuri and Rosenfeld, 1996) give nice comparisons of fuzzy sets by measuring the differences in their images, but suffer the undesirable restriction that fuzzy sets so compared must have the same maximum values. In the current paper, we show how the metrics of (Chaudhuri and Rosenfeld, 1996) may be modified so as to remove this restriction while preserving some of the "feel" of the metrics described in (Chaudhuri and Rosenfeld, 1996). We also present an algorithm for computing our metric that runs in O(mn) time for a 2-dimensional digital picture of n pixels on which fuzzy sets are allowed m distinct membership values. © 1997 Elsevier Science B.V.