Divyendu Sinha

Fuzzification of set inclusion: theory and applications

Abstract Fuzzification of set inclusion for fuzzy sets is developed in terms of an indicator for set inclusion, the indicator giving the degree to which a fuzzy set is a subset of another fuzzy set. To date, such indicators have been called ‘inclusion grades’; however, in contrast to most existing indicators, it is proposed in the present paper that the indicator must be two-valued for crisp sets. The approach, herein, is to begin by postulating desired properties of indicators for fuzzified set inclusion, to then assume a specific mathematical form for such indicators, and then derive the necessary and sufficient conditions under which the specified formula gives rise to indicators possessing the desired properties. The investigation results in a very general class of indicators based on the bold union operation, and, most importantly, in a complete measure-theoretic characterization of this class. The characterization takes the form of a constrained representation providing explicit formulation for all indicators in the class of interest. The paper closes with applications of fuzzified set inclusion to shape recognition via image processing, in particular, mathematical morphology, and to the measurement of fuzziness in fuzzy sets by means of entropy.

Fuzzy mathematical morphology

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 general axiomatic theory of intrinsically fuzzy mathematical morphologies

Intrinsic fuzzification of mathematical morphology is grounded on an axiomatic characterization of subset fuzzification. The result is an axiomatic formulation of fuzzy Minkowski algebra. Part of the Minkowski algebra results solely from the axioms themselves and part results from a specific postulated form of a subsethood indicator function. There exists an infinite number of fuzzy morphologies satisfying the axioms; in particular, there are uncountably many indicators satisfying the postulated form. This paper develops fuzzy Minkowski algebra, with special emphasis on fitting characterizations of fuzzy erosion and opening, examines key properties of the indicator function, and provides fuzzy extensions of the basic binary Matheron representations for openings and increasing, translation-invariant operators.

Design and analysis of fuzzy morphological algorithms for image processing

A general paradigm for lifting binary morphological algorithms to fuzzy algorithms is employed to construct fuzzy versions of classical binary morphological operations. The lifting procedure is based upon an epistemological interpretation of both image and filter fuzzification. Algorithms are designed via the paradigm for various fuzzifications and their performances are analyzed to provide insight into the kind of liftings that produce suitable results. Algorithms are discussed for three image processing tasks: shape detection, edge detection, and clutter removal. Detailed analyses are given for the effect of noise and its mitigation owing to fuzzy approaches. It is demonstrated how the fuzzy hit-or-miss transform can be used in conjunction with a decision procedure to achieve word recognition.

Computational gray-scale mathematical morphology on lattices (a comparator-based image algebra). Part 1: architecture

Abstract Computational mathematical-morphology has been developed to provide a directly computable alternative to classical gray-scale morphology that is range preserving and compatible with the design of statistically optimal filters based on morphological representation. It serves as an image algebra because of the expressive capability of its image-operator representations. Because representations are based on binary comparators used in conjunction with AND and OR operations, it provides a low-level, efficient computational environment that is a direct extension of the finite Boolean operational environment. The paper focuses on development of the comparator-based representation, providing the relevant representation theory for lattice operators and lattice-vector operators. Computational lattice-operator theory represents a comparator-based alternative to the classical morphological lattice theory, one that is directly implementable in logic. For totally ordered valuation spaces, the lattice theory reduces to a simplified form appropriate to straightforward optimization. The present paper treats architectural considerations and a second part considers the implications for latticevalued image operators.

Computational mathematical morphology

Abstract As classically defined via the umbra transform (or via lattice theory), the theory of gray-scale morphology applies to functions possessing the extended real line (or integers) as range. Four interrelated problems arise: (1) binary morphology embeds via {− ∞, 0}-valued functions; (2) finite-range function classes are not preserved; (3) gray-scale filters are not directly expressible in terms of logical variables, as are binary filters and, more generally, stack filters; and (4) the theory of optimal binary filters does not fall out directly as a special case of the gray-scale theory. The present paper discusses a different gray-scale morphology that eliminates the preceding anomalies. Major topics addressed are filter structure, representation of both increasing and nonincreasing operators, and, in particular, the theory of optimal filters.

Fuzzification of Set Inclusion

Fuzzification of set inclusion for fuzzy sets is developed in terms of an indicator for set inclusion, the indicator giving the degree to which a fuzzy set is a subset of another fuzzy set. In contrast to most existing indicators, it is proposed in the present paper that the indicator must be two-valued for crisp sets. The approach is to postulate axioms for the indicators, assume a specific mathematical form for such indicators, and then give necessary and sufficient conditions under which the specified formula gives rise to suitable indicator, in effect, providing a characterization of the indicator.

Computational Gray-scale Mathematical Morphology on Lattices (A Comparator-based Image Algebra) Part II

In this second part of a two-part study, the lattice-based computational representational structure is applied to image operators. The computational representations are first applied to some common non-linear window (vector) operators used in image processing ? for instance, flat erosion, flat dilation, fuzzy erosion, and flat filters, in general. For these, application of the representations is direct. Representations are then developed for gray-to-binary and gray-to-gray image operators. In all cases, images are assumed to be lattice-valued. It is shown that under appropriate circumstances the representations can be viewed as structural generalizations of classical representations. Flat (stack) filters are treated in their own fight (as operators on lattice-valued images) and it is seen that for these the lattice representations can be interpreted in terms of threshold decomposition.

Extensions to the fuzzy pointed set with applications to image processing

In all man-machine systems with image processing functions, an important unsolved problem arises in the treatment of uncertain and incomplete image information. Several frameworks have been suggested for handling uncertain image information including; expert systems, fuzzification, likelihood estimation, and neural networks. In this paper we review those methods. We also present a new method for handling uncertainties by unifying the representations of gray-values and uncertainty into one framework in a way that parallels fuzzy logic. This new framework is based on the application of the extended fuzzy pointed set and an associated algebra to handle uncertain information. We further show how this framework can be used in image processing and artificial intelligence.

Characterization of fuzzy Minkowski algebra

There are various fuzzy morphologies (Minkowski algebras), these depending on the particular fuzzifification of set inclusion that is employed for the definition of erosion. Set-inclusion fuzzification depends upon the choice of an indicator for set inclusion and, based upon a collection of nine axioms, a class of indicators results such that each indicator in the class yields a Minkowski algebra in which a certain core of the ordinary propositions typically associated with mathematical morphology are valid. By going a bit further and postulating a certain mathematical form for the indicator, one obtains fitting characterizations for the basic operators. In ordinary crisp-set binary morphology, certain fundamental representation theorems hold, specifically the Matheron representations for increasing, translation invariant mappings and for T-openings. The definition of a T-opening extends for fuzzy T-openings. There is also a weakened version of the Matheron kernel representation for increasing, translation invariant mappings.