A Generalization of the Hausdorff Metric for Fuzzy Sets

For the description of shape differences between fuzzy objects, such as molecular electron density clouds, it is useful to generalize the Hausdorff metric for fuzzy sets. The ordinary Hausdorff distance, a formal dis- 

For a formal definition of the Hausdorff distance, first we shall review some relevant concepts. We assume that A and B are subsets of a set X, and for points of W a distance function is already defined. For example, if X is the ordinary, three-dimensional Euclidean space and if the points a and b of A are represented by their three Cartesian coordinates [, 2) 3) and respectively, where a and b can be written as 

Note that infinitely many other choices d (a,b) for a distance function can be made, as long as these choices fulfill the conditions for a metric  

After these preparations, the Hausdorff distance h A,B) between two subsets A and B of is defined as 

The Hausdorff distance h A,B) is a proper metric within any family of compact sets, for example, / (/ , B) is zero if and only if the two sets are the same. 

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