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.
A rough set-based approach to handling spatial uncertainty in binary images
Abstract
In this paper we consider the problem of detecting binary objects using rough sets. We present a method for constructing a gray-scaled (or, fuzzy) template for use in correlation-based matching of Boolean images. We assume a cause for spatial uncertainty that is quite common in machine vision applications and present a methodology for modeling it indirectly in the construction of the template. Our technique is computationally efficient and is superior to correlation-based techniques, which can be easily fooled and automates the hand-selection of structuring elements for the hit-or-miss transform technique, both of which are usually used to solve this problem.
Invariant characterizations and pseudocharacterizations of finite multidimensional sets based on mathematical morphology
Abstract
Thispaper outlines a methodology for characterizing finite subsets of (eta)d where (eta) is the set of integers and integerd >= 1. The schemes are based on the lattice-basedmathematical morphology and on the notion of a pseudo-complement. Threedifferent schemes for multidimensional objects are presented, two of whichprovide only pseudo- characterizations, while the third scheme provides acomplete characterization of objects. The performance of the proposed pseudo-characterizationshas been shown to be superior to the similar existingalgorithms. The proposed methods lend themselves to efficient implementation onparallel machines.
