The perimeter of a fuzzy set
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TL;DR
This paper proposes a definition of perimeter for fuzzy subsets of the plane and shows that it reduces to the standard definition if the fuzzy subset is an ordinary subset.
Abstract
In pattern recognition one often wants to measure the perimeters of regions in images. This is straightforward if the region is crisply defined, but if it is fuzzy, it is not obvious how its perimeter can be measured. This paper proposes a definition of perimeter for fuzzy subsets of the plane and shows that it reduces to the standard definition if the fuzzy subset is an ordinary subset. The isoperimetric inequality does not generalize to fuzzy subsets, but certain properties of the perimeters of convex sets do generalize to fuzzy perimeters of convex fuzzy subsets.