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Keywords
(8)
Complete Lattice
Fuzzy Mathematical Morphology
Fuzzy Set
lexicographic order
Mathematical Morphology
Morphological Operation
Point of View
Partial Order
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Lattices of fuzzy sets and bipolar fuzzy sets, and mathematical morphology
Lattices of fuzzy sets and bipolar fuzzy sets, and mathematical morphology,10.1016/j.ins.2010.03.019,Information Sciences,Isabelle Bloch
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Lattices of fuzzy sets and bipolar fuzzy sets, and mathematical morphology
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Citations: 4
)
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Isabelle Bloch
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 extension concerns bipolar fuzzy sets, where both positive information and negative information are handled, along with their imprecision. We detail these extensions from the
point of view
of the underlying lattice structure. In the case of bipolarity, its twocomponents nature raises the question of defining a proper partial ordering. In this paper, we consider Pareto (componentwise) and lexicographic orderings.
Journal:
Information Sciences  ISCI
, vol. 181, no. 10, pp. 20022015, 2011
DOI:
10.1016/j.ins.2010.03.019
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Citation Context
(3)
...The unifying potential of LT for disparate data unification/fusion has been, partly, recognized in MM. For instance, a number of authors have established connections between mathematical morphology and fuzzy set theory [7, 9, 11, 13, 39, 53] including recent extensions of fuzzy MM such as intervalvalued fuzzy, intuitionistic (bipolar) fuzzy and Lfuzzy MM [
8
, 40, 56, 60, 61]...
Vassilis G. Kaburlasos
,
et al.
Binary Image 2D Shape Learning and Recognition Based on LatticeComput...
...In some recent papers, Bloch has established links between these two approaches for the special cases of fuzzy sets and bipolar fuzzy sets [12,
13
]...
...2 and 3 are also valid for other partial ordering schemes that lead to complete lattices such as the lexicographical partial order on bipolar (intuitionistic) fuzzy sets [
13
]...
...Another interpretation of bipolar FMM (i.e., intuitionistic FMM) was provided by I. Bloch who has presented applications to spatial reasoning in image processing [11,
13
]...
Peter Sussner
,
et al.
IntervalValued and Intuitionistic Fuzzy Mathematical Morphologies as ...
...The lexicographic ordering was considered too in [
5
]...
...Based on these concepts, we can now propose a general definition for morphological erosions and dilations, thus extending our previous work in [2,3,
5
]...
...This group has then proposed some extensions in [7], still for the specific case of Pareto ordering, which closely follow our previous results in [2,3,
5
]...
...Examples of connectives and derived morphological operators, along with their properties, can be found for the Pareto ordering and for the lexicographic ordering in our previous work [2,3,
5
]...
...The case of Pareto ordering and lexicographic ordering have been detailed in [2,3,
5
], showing different properties, behaviors and interpretations...
Isabelle Bloch
.
Fuzzy Bipolar Mathematical Morphology: A General Algebraic Setting
References
(45)
On bipolarity in argumentation frameworks
(
Citations: 18
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Leila Amgoud
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Claudette Cayrol
,
Mariechristine Lagasquieschiex
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Journal:
International Journal of Intelligent Systems  IJIS
, vol. 23, no. 10, pp. 10621093, 2008
Morphological colour operators in totally ordered lattices based on distances: Application to image filtering, enhancement and analysis
(
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Jesús Angulo
Journal:
Computer Vision and Image Understanding  CVIU
, vol. 107, no. 12, pp. 5673, 2007
A comparative study on multivariate mathematical morphology
(
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Erchan Aptoula
,
Sébastien Lefèvre
Journal:
Pattern Recognition  PR
, vol. 40, no. 11, pp. 29142929, 2007
On lexicographical ordering in multivariate mathematical morphology
(
Citations: 6
)
Erchan Aptoula
,
Sébastien Lefèvre
Journal:
Pattern Recognition Letters  PRL
, vol. 29, no. 2, pp. 109118, 2008
Intuitionistic fuzzy sets
(
Citations: 982
)
K ATANASSOV
Journal:
Fuzzy Sets and Systems  FSS
, vol. 20, no. 1, pp. 8796, 1986
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Citations
(4)
Morphological perceptrons with competitive learning: Latticetheoretical framework and constructive learning algorithm
(
Citations: 12
)
Peter Sussner
,
Estevão Laureano Esmi
Journal:
Information Sciences  ISCI
, vol. 181, no. 10, pp. 19291950, 2011
Binary Image 2D Shape Learning and Recognition Based on LatticeComputing (LC) Techniques
Vassilis G. Kaburlasos
,
S. E. Papadakis
,
Angelos Amanatiadis
Journal:
Journal of Mathematical Imaging and Vision  JMIV
, pp. 116, 2011
IntervalValued and Intuitionistic Fuzzy Mathematical Morphologies as Special Cases of Fuzzy Mathematical Morphology
Peter Sussner
,
Mike Nachtegael
,
Tom Mélange
,
Glad Deschrijver
,
Estevão Esmi
,
Etienne Kerre
Journal:
Journal of Mathematical Imaging and Vision  JMIV
, pp. 122
Fuzzy Bipolar Mathematical Morphology: A General Algebraic Setting
Isabelle Bloch