Fuzzification of set inclusion: Theory and applications

Sinha, Divyendu; Dougherty, Edward R.

doi:10.1016/0165-0114(93)90299-w

Cited by 227 publications

(127 citation statements)

References 7 publications

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“…Extending mathematical morphology to fuzzy sets was proposed in the early 90's, by several teams independently [11,12,13,14,15], and was then largely developed (see e.g. [16,17,18,19,20,21]).…”

Section: Fuzzy Mathematical Morphologymentioning

confidence: 99%

Fuzzy and Bipolar Mathematical Morphology, Applications in Spatial Reasoning

Bloch

2009

Lecture Notes in Computer Science

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Abstract. Mathematical morphology is based on the algebraic framework of complete lattices and adjunctions, which endows it with strong properties and allows for multiple extensions. In particular, extensions to fuzzy sets of the main morphological operators, such as dilation and erosion, can be done while preserving all properties of these operators. Another, more recent, extension, concerns bipolar fuzzy sets. These extensions have numerous applications, two of each being presented here. The first one concerns the definition of spatial relations, for applications in spatial reasoning and model-based recognition of structures in images. The second one concerns the handling of the bipolarity feature of spatial information.

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Section: Fuzzy Mathematical Morphologymentioning

confidence: 99%

Fuzzy and Bipolar Mathematical Morphology, Applications in Spatial Reasoning

Bloch

2009

Lecture Notes in Computer Science

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“…The relative cardinality is both existential and universal. Axioms for comparison indices have been proposed by Dubois and Prade (1982a), and Sinha and Dougherty (1993) specifically for inclusion. The following freely borrows from both.…”

Section: Building Comparison Indicesmentioning

confidence: 99%

“…A inclusion index is a mapping I(F, G) from F(U) 2 to the unit interval, such that Such axioms are freely inspired from (Sinha and Dougherty, 1993) where equality instead of some of the above inequalities is requested. A typical inclusion index for comparing normal fuzzy sets is I(F, G) = e(F→G) for a universal evaluator (like the plinth) and a residuated implication (since a→b = 1 for a ≤ b, and 1→0 = 0).…”

Section: Building Comparison Indicesmentioning

confidence: 99%

Fuzzy Sets: History and Basic Notions

Dubois¹,

Ostasiewicz

Prade³

2000

The Handbooks of Fuzzy Sets Series

139

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Abstract. This paper is an introduction to fuzzy set theory. It has several purposes. First, it tries to explain the emergence of fuzzy sets from an historical perspective. Looking back to the history of sciences, it seems that fuzzy sets were bound to appear at some point in the 20th century. Indeed, Zadeh's works have cristalized and popularized a concern that has appeared in the first half of the century, mainly in philosophical circles. Another purpose of the paper is to scan the basic definitions in the field, that are required for a proper reading of the rest of the volume, as well as the other volumes of the Handbooks of Fuzzy Sets Series. This Chapter also contains a discussion on operational semantics of the generally too abstract notion of membership function. Lastly, a survey of variants of fuzzy sets and related matters is provided.

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“…Extending mathematical morphology to fuzzy sets was proposed in the early nineties, by several teams independently ( [1,9,18,19,41]), and was then largely developed (see for instance [6,10,16,17,20,32,34]). …”

Section: Introductionmentioning

confidence: 99%

On the relation between fuzzy closing morphological operators, fuzzy consequence operators induced by fuzzy preorders and fuzzy closure and co-closure systems

Elorza

Fuentes-González

Bragard

et al. 2013

Fuzzy Sets and Systems

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In a previous paper, Elorza and Burillo explored the coherence property in fuzzy consequence operators. In this paper we show that fuzzy closing operators of mathematical morphology are always coherent operators. We also show that the coherence property is the key to link the four following families: fuzzy closing morphological operators, fuzzy consequence operators, fuzzy preorders and fuzzy closure and co-closure systems. This will allow to translate important well-known properties from the field of approximate reasoning to the field of image processing.

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Fuzzification of set inclusion: Theory and applications

Cited by 227 publications

References 7 publications

Fuzzy and Bipolar Mathematical Morphology, Applications in Spatial Reasoning

Fuzzy and Bipolar Mathematical Morphology, Applications in Spatial Reasoning

Fuzzy Sets: History and Basic Notions

On the relation between fuzzy closing morphological operators, fuzzy consequence operators induced by fuzzy preorders and fuzzy closure and co-closure systems

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