Robust similarity between hypergraphs based on valuations and mathematical morphology operators

Bloch, Isabelle; Bretto, Alain; Leborgne, Aurélie

doi:10.1016/j.dam.2014.08.013

Cited by 12 publications

(4 citation statements)

References 14 publications

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“…Until now, this link between MM and logic has been studied in the set frame-work (with extensions to fuzzy sets). Since then MM has been extended to a large family of algebraic structures such as graphs [27,28,46,62], hypergraphs [17,18], simplicial complexes [29], various logics, etc. All these extensions proved useful for knowledge representation and reasoning, taking into account low level information (points or neighborhood of points), structural information (e.g.…”

Section: Bmentioning

confidence: 99%

Morpho-logic from a Topos Perspective: Application to symbolic AI

Aiguier¹,

Bloch²,

Salim³

et al. 2023

Preprint

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Modal logics have proved useful for many reasoning tasks in symbolic artificial intelligence (AI), such as belief revision, spatial reasoning, among others. On the other hand, mathematical morphology (MM) is a theory for non-linear analysis of structures, that was widely developed and applied in image analysis. Its mathematical bases rely on algebra, complete lattices, topology. Strong links have been established between MM and mathematical logics, mostly modal logics. In this paper, we propose to further develop and generalize this link between mathematical morphology and modal logic from a topos perspective, i.e. categorial structures generalizing space, and connecting logics, sets and topology. Furthermore, we rely on the internal language and logic of topos. We define structuring elements, dilations and erosions as morphisms. Then we introduce the notion of structuring neighborhoods, and show that the dilations and erosions based on them lead to a constructive modal logic, for which a sound and complete proof system is proposed. We then show that the modal logic thus defined (called morpho-logic here), is well adapted to define concrete and efficient operators for revision, merging, and abduction of new knowledge, or even spatial reasoning.

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Section: Bmentioning

confidence: 99%

Morpho-logic from a Topos Perspective: Application to symbolic AI

Aiguier¹,

Bloch²,

Salim³

et al. 2023

Preprint

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show abstract

“…4 This is the way we will consider structuring elements in this paper, as done in previous work, in particular for mathematical morphology on graphs (see e.g. [8,13,25], among others) or logics (see e.g. [1,2,3,9,6,10,17]).…”

Section: Mathematical Morphology On Setsmentioning

confidence: 99%

“…Hence, for graphs and hypergraphs, we can define two kinds of forgetful functors, one which forgets on vertices and the other which forgets on edge and hyperedges 8 . In presheaf topos of sets, we can associate a unique kind of forgetful functor U : SubpF q Ñ Set which, given a subobject rαs, yields a set S 1 such that the functor F 1 : ‚ Þ Ñ S 1 is isomorphic to dompαq.…”

Section: Presheaf Toposes As Generators Of Morpho-categoriesmentioning

confidence: 99%

“…Since then, mathematical morphology was mainly applied in the field of image processing [27,28]. The first formalization of mathematical morphology was made within the framework of set theory, and more recently it has been extended to a larger family of algebraic structures such as graphs [14,13,25,29], hypergraphs [7,8] and simplicial complexes [16]. The two basic operations of mathematical morphology are erosion and dilation.…”

Section: Introductionmentioning

confidence: 99%

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Abstract Mathematical morphology based on structuring element: Application to morpho-logic

Aiguier,

Bloch,

Pino-Pérez

2020

Preprint

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A general definition of mathematical morphology has been defined within the algebraic framework of complete lattice theory. In this framework, dealing with deterministic and increasing operators, a dilation (respectively an erosion) is an operation which is distributive over supremum (respectively infimum). From this simple definition of dilation and erosion, we cannot say much about the properties of them. However, when they form an adjunction, many important properties can be derived such as monotonicity, idempotence, and extensivity or anti-extensivity of their composition, preservation of infimum and supremum, etc. Mathematical morphology has been first developed in the setting of sets, and then extended to other algebraic structures such as graphs, hypergraphs or simplicial complexes. For all these algebraic structures, erosion and dilation are usually based on structuring elements. The goal is then to match these structuring elements on given objects either to dilate or erode them. One of the advantages of defining erosion and dilation based on structuring elements is that these operations are adjoint. Based on this observation, this paper proposes to define, at the abstract level of category theory, erosion and dilation based on structuring elements. We then define the notion of morpho-category on which erosion and dilation are defined. We then show that topos and more precisely topos of presheaves are good candidates to generate morpho-categories. However, topos do not allow taking into account the notion of inclusion between substructures but rather are defined by monics up to domain isomorphism. Therefore we define the notion of morpholizable category which allows generating morpho-categories where substructures are defined along inclusion morphisms. A direct application of this framework is to generalize modal morpho-logic to other algebraic structures than simple sets.

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A New Entropy for Hypergraphs

Bloch¹,

Bretto²

2019

Discrete Geometry for Computer Imagery

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Robust similarity between hypergraphs based on valuations and mathematical morphology operators

Cited by 12 publications

References 14 publications

Morpho-logic from a Topos Perspective: Application to symbolic AI

Morpho-logic from a Topos Perspective: Application to symbolic AI

Abstract Mathematical morphology based on structuring element: Application to morpho-logic

A New Entropy for Hypergraphs

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