2013
DOI: 10.1016/j.cviu.2012.08.016
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Morphological filtering on graphs

Abstract: We study some basic morphological operators acting on the lattice of all subgraphs of an arbitrary (unweighted) graph G. To this end, we consider two dual adjunctions between the edge set and the vertex set of G. This allows us (i) to recover the classical notion of a dilation/erosion of a subset of the vertices of G and (ii) to extend it to subgraphs of G. Afterward, we propose several new openings, closings, granulometries and alternate filters acting (i) on the subsets of the edge and vertex set of G and (i… Show more

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Cited by 51 publications
(106 citation statements)
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“…We finish this section with the presentation of an adjunction that is known for playing the role of a building block for morphology on graphs [19]. It will be useful for expressing several properties in the sequel of this article.…”
Section: Graphsmentioning
confidence: 99%
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“…We finish this section with the presentation of an adjunction that is known for playing the role of a building block for morphology on graphs [19]. It will be useful for expressing several properties in the sequel of this article.…”
Section: Graphsmentioning
confidence: 99%
“…We denote by δ the operator that maps to any subset X of E the subset of V that contains every vertex in V which belongs to an edge in X, i.e., δ(X) = ∪{{x, y} ∈ X}. The pair ( , δ) is an adjunction [19]. Let V ⊆ V and E ⊆ E. Using usual graph terminology, the graphs (V , (V )) and (δ(E ), E) are called the graph induced by V and the graph induced by E respectively.…”
Section: Graphsmentioning
confidence: 99%
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“…On the one hand, we note that in the literature, one can find some works about nonlinear filters on graphs and hypergraphs, particularly mathematical morphology operators in the algebraic sense [3,4,5,6,7,8,9], where the couple of nonlinear operator (dilation/erosion) are maps from two different lattices, i.e., they are maps "from nodes to edges" or "edges to nodes". On the other hand, some regularisation techniques and nonlinear operators have been introduced on function evaluated on graph via directional derivative [10] [11] [12] or discrete version of the p-Laplacian [13].…”
Section: Introductionmentioning
confidence: 99%