Fuzzy mathematical morphology

Abstract

The original extension of binary mathematical morphology to the gray scale is based upon the lattice-theoretic supremum and infimum operations, its geometric genesis being framed in terms of the umbra transform. Abstract formulation of the mathematical theory is set in the context of complete lattices; nonetheless, as applied to the Euclidean gray scale, it remains true to the umbra formulation. In distinction to the ordinary extension of the binary theory to the gray scale, the present paper provides a generalization based on fuzzy set theory. Images are modeled as fuzzy subsets of the Euclidean plane or Cartesian grid, and the morphological operations are defined in terms of a fuzzy index function. This approach leads to a general algebraic paradigm for fuzzy morphological algebras. More specifically, the paper investigates in depth a fuzzy morphology grounded on a fuzzy fitting characterization. Although the resulting algebras reduce to ordinary binary morphology when sets are crisp, the extension is not equivalent to the umbra-modeled approach, and binary morphology is embedded within fuzzy morphology by treating images as {0, 1}-valued rather than {−∞, 0}-valued. As opposed to the usual gray-scale extension, the fuzzy extension closely maintains the notion of erosion being a marker, albeit a fuzzy marker. The present paper discusses fuzzy modeling (via a suitable index function), the fundamental fyzzy morphological operations, and the corresponding fuzzy Minkowski algebra.