A fuzzy medial axis transformation based on fuzzy disks

Abstract

A fuzzy disk with center P is a fuzzy set in which membership depends only on distance from P. For any fuzzy set ƒ, there is a maximal fuzzy disk g_p^ƒ≤ƒ centered at every point P, and ƒis the sup of the g_p^ƒ's. (Moreover, if ƒ is fuzzy convex, so is every g_p^ƒ, but not conversely.) We call a set S_ƒ of points ƒ-sufficient if every g_p^ƒ ≤ g_Q^ƒ for some Q in S_ƒ; evidently ƒ is then the sup of the g_Q^ƒ's. In particular, in a digital image, the set of Q's at which g^ƒ is a (nonstrict) local maximum is ƒ-sufficient. This set is called the fuzzy medial axis of ƒ, and the set of g_Q^ƒ's is called the fuzzy medial axis transformation (FMAT) of ƒ. These definitions evidently reduce to the standard one if ƒ is a crisp set. Unfortunately, for an arbitrary ƒ, specifying the FMAT may require more storage space than specifying ƒ itself.