On the role of complete lattices in mathematical morphology: From tool to uncertainty model

Nachtegael, Mike; Sussner, Peter; Mélange, Tom; Kerre, Etienne

doi:10.1016/j.ins.2010.03.009

Cited by 46 publications

(51 citation statements)

References 40 publications

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“…There are also many other applications for lattice in image processing. So it is essential to have a very consistent mathematical theory in order to provide a safe framework to deal with those issues (see [4,5]). …”

Section: Background and Literature Reviewmentioning

confidence: 99%

Some results on extension of lattice-valued QL-implications

2016

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Background: A very important issue in lattice theory is how to extend a given operator preserving its algebraic properties. For lattice-valued fuzzy operators framework, in 2008 Saminger-Platz presented a way to extend t-norms which was generalized by Palmeira et al. (2011) for t-norms, t-conorms, fuzzy negations and implications, considering the scenery provided by the (r, s)-sublattice. Methods: In this paper we investigated how to extend QL-implications and which properties of it are preserved by the extension method via retractions (EMR). Results:As results, we proved that properties (LB), (RB), (CC1), (CC2), (CC3), (CC4), (L-NP), (EP) and (IP) are preserved by EMR. Conclusions: However, the extension method via retractions fails in preserving the important properties (NP), (OP), (IBL), (CP), (P) and (LEM).

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Section: Background and Literature Reviewmentioning

confidence: 99%

Some results on extension of lattice-valued QL-implications

2016

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“…This paper investigates a number of theoretical aspects of L-fuzzy mathematical morphology. Unlike previous papers on interval-valued and intuitionistic fuzzy mathematical morphology [10,11,58,59], this paper treats intervalvalued and intuitionistic FMMs as special cases of L-fuzzy MM. This approach not only allows for a top-down view of the corresponding mathematical frameworks but also for the construction of L-fuzzy MMs for other particular instances of complete lattices L.…”

Section: Introductionmentioning

confidence: 99%

Interval-Valued and Intuitionistic Fuzzy Mathematical Morphologies as Special Cases of $\mathbb{L}$ -Fuzzy Mathematical Morphology

et al. 2011

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Mathematical morphology (MM) offers a wide range of tools for image processing and computer vision. MM was originally conceived for the processing of binary images and later extended to gray-scale morphology. Extensions of classical binary morphology to gray-scale morphology include approaches based on fuzzy set theory that give rise to fuzzy mathematical morphology (FMM). From a mathematical point of view, FMM relies on the fact that the class of all fuzzy sets over a certain universe forms a complete lattice. Recall that complete lattices provide for the most general framework in which MM can be conducted.The concept of L-fuzzy set generalizes not only the concept of fuzzy set but also the concepts of interval-valued fuzzy set and Atanassov's intuitionistic fuzzy set. In addition, the class of L-fuzzy sets forms a complete lattice whenever the underlying set L constitutes a complete lattice. Based on these observations, we develop a general approach towards L-fuzzy mathematical morphology in this paper. Our focus is in particular on the construction of connectives for interval-valued and intuitionistic fuzzy mathematical morphologies that arise as special, isomorphic cases of L-fuzzy MM. As an application of these ideas, we generate a combination of some well-known medical image reconstruction Grants or other notes.Peter Sussner and Estevão Esmi University of Campinas, Department of Applied Mathematics, Campinas, SP, 13083-859, Brazil Tel.: +55-19-3521 5959 Fax: +55-19-3289-5766 E-mail: sussner,ra050652@ime.unicamp.br Mike Nachtegael, Tom Mélange, Glad Deschrijver, and Etienne Kerre Ghent University, Department of Applied Mathematics and Computer Science, Krijgslaan 281 -S9, 9000 Ghent, Belgium Tel.: +32-9-264 4765 Fax: +32-9-264 4995 E-mail: Mike.Nachtegael,Tom.Melange,Glad.Deschrijver,Etienne.Kerre @UGent.be techniques in terms of interval-valued fuzzy image processing.

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“…(33) where "/ S r " denotes the residuum given by (29). Dually, the spherical CIELab dilation of f ∈S X r by an s ∈S Y r , denoted by δ S r s ( f ), is the color image obtained as follows:…”

Section: Remarkmentioning

confidence: 99%

“…From the theoretical point of view, MM can be very well defined in a mathematical structure called complete lattice [22,33,36]. A complete lattice is a partially ordered set in which any subset has both a supremum and an infimum.…”

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confidence: 99%

Mathematical Morphology on the Spherical CIELab Quantale with an Application in Color Image Boundary Detection

Valle

Valente²

2016

J Math Imaging Vis

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Mathematical morphology is a theory with applications in image processing and analysis. This paper presents a quantale-based approach to color morphology based on the CIELab color space in spherical coordinates. The novel morphological operations take into account the perceptual difference between color elements by using a distance-based ordering scheme. Furthermore, the novel approach allows for the use of non-flat structuring elements. An illustrative example reveals that non-flat dilations and erosions may preserve more features of a color image than their corresponding flat operations. Furthermore, the novel non-flat morphological operators yielded promising results on experiments concerning the detection of the boundaries of objects on color images.

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On the role of complete lattices in mathematical morphology: From tool to uncertainty model

Cited by 46 publications

References 40 publications

Some results on extension of lattice-valued QL-implications

Some results on extension of lattice-valued QL-implications

Interval-Valued and Intuitionistic Fuzzy Mathematical Morphologies as Special Cases of $\mathbb{L}$ -Fuzzy Mathematical Morphology

Mathematical Morphology on the Spherical CIELab Quantale with an Application in Color Image Boundary Detection

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