The algebraic basis of mathematical morphology I. Dilations and erosions

Heimans, H. J. A. M.; Ronse, Christian

doi:10.1016/0734-189x(90)90148-o

Cited by 367 publications

(174 citation statements)

References 10 publications

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“…As a matter of fact, image analysis techniques on intensity regular mesh, like classical bitmap intensity images are quite mastered. Some works deal with general graph morphology operators focusing on vertices in the theoretical framework of complete lattices (Serra, 1988) (Heijmans & Ronse, 1990) (Dougherty, 1993). But the equivalent operators for geometrical irregular mesh have not been explored extensively.…”

Section: Introductionmentioning

confidence: 99%

Morphological mesh filtering and -objects

Loménie

Stamon

2008

Pattern Recognition Letters

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Since image analysis techniques have come to maturity, mesh analysis has remained challenging requiring more and more efforts for elaborating an effective theoretical model. In this article, following algebraic mesh operators, we introduce algorithms that perform morphological transformations on unorganized point sets connected by their Delaunay triangulations. We show that these algorithms correspond to morphological operators like erosion, dilation or opening, acting as "shape filters" on meshes. More theoretically, a link is established between these algorithms and the formalisms of edge algebra and α-objects. Then, the mesh operators are defined in terms of complete lattices. These algorithms are applied to the problem -among others -of scene reconstruction by stereoscopy in which objects are represented by unstructured and noisy clouds of 3D points. Rapid prototyping should also benefit from these algorithms.

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Section: Introductionmentioning

confidence: 99%

Morphological mesh filtering and -objects

Loménie

Stamon

2008

Pattern Recognition Letters

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“…Let (L, ) and (L , ) be two complete lattices (which do not need to be equal). All the following definitions and results are common to the general algebraic framework of mathematical morphology in complete lattices [10,11,15,18]. Different terminologies can be found in different lattice theory related contexts (refer to [16] for equivalence tables).…”

Section: Mathematical Morphologymentioning

confidence: 99%

Mathematical Morphology Operators over Concept Lattices

Atif

Bloch

Distel

et al. 2013

Formal Concept Analysis

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Abstract. Although mathematical morphology and formal concept analysis are two lattice-based data analysis theories, they are still developed in two disconnected research communities. The aim of this paper is to contribute to fill this gap, beyond the classical relationship between the Galois connections defined by the derivation operators and the adjunctions underlying the algebraic mathematical morphology framework. In particular we define mathematical morphology operators over concept lattices, based on distances, valuations, or neighborhood relations in concept lattices. Their properties are also discussed. These operators provide new tools for reasoning over concept lattices.

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“…While the lattice theoretic basis of set-based mathematical morphology has long been understood [8,3], this is not the case when we move to graphs and hypergraphs.…”

Section: Introductionmentioning

confidence: 99%

Symmetric Heyting relation algebras with applications to hypergraphs

Stell

2015

Journal of Logical and Algebraic Methods in Programming

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Article:Stell, JG (2014) Symmetric Heyting relation algebras with applications to hypergraphs. Journal of Logical and Algebraic Methods in Programming, 84. pp. 440-455. ISSN 2352-2208 https://doi.org/10.1016/j.jlamp.2014.12.001 eprints@whiterose.ac.uk https://eprints.whiterose.ac.uk/ Reuse Unless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version -refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher's website. TakedownIf you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. AbstractA relation on a hypergraph is a binary relation on the set consisting of all the nodes and the edges, and which satisfies a constraint involving the incidence structure of the hypergraph. These relations correspond to join preserving mappings on the lattice of sub-hypergraphs. This paper introduces a generalization of a relation algebra in which the Boolean algebra part is replaced by a Heyting algebra that supports an order-reversing involution. A general construction for these symmetric Heyting relation algebras is given which includes as a special case the algebra of relations on a hypergraph. A particular feature of symmetric Heyting relation algebras is that instead of an involutory converse operation they possess both a left converse and a right converse which form an adjoint pair of operations. Properties of the converses are established and used to derive a generalization of the well-known connection between converse, complement, erosion and dilation in mathematical morphology. This provides part of the foundation necessary to develop mathematical morphology on hypergraphs based on relations on hypergraphs.

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The algebraic basis of mathematical morphology I. Dilations and erosions

Cited by 367 publications

References 10 publications

Morphological mesh filtering and -objects

Morphological mesh filtering and -objects

Mathematical Morphology Operators over Concept Lattices

Symmetric Heyting relation algebras with applications to hypergraphs

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