Mathematical Morphology on Hypergraphs Using Vertex-Hyperedge Correspondence

The focus of this paper is to develop computationally efficient mathematical morphology operators on hypergraphs. To this aim we consider lattice structures on hypergraphs on which we build morphological operators. We develop a pair of dual adjunctions between the vertex set and the hyperedge set of a hypergraph H, by defining a vertex-hyperedge correspondence. This allows us to recover the classical notion of a dilation/erosion of a subset of vertices and to extend it to subhypergraphs ofH. This paper also studies the concept of morphological adjunction on hypergraphs for which both the input and the output are hypergraphs.


Introduction
Mathematical morphology, appeared in 1960s, is a theory of nonlinear information processing [1][2][3][4].It is a branch of image analysis based on algebraic, set-theoretic, and geometric principles [5,6].Originally, it is developed for binary images by Matheron and Serra.They are the first to observe that a general theory of mathematical morphology is based on the assumption that the underlying image space is a complete lattice.Most of the morphological theories at this abstract level were developed and presented without making references to the properties of the underlying space.Considering digital objects carrying structural information, mathematical morphology has been developed on graphs [7][8][9][10] and simplicial complexes [11], but little work has been done on hypergraphs [12][13][14][15].
When dealing with a hypergraph , we need to consider the hypergraph induced by the subset  • of vertices of  (see Figures 1(a) and 1(b), where the blue vertices and edges in (b) represent ).We associate with  • the largest subset of hyperedges of  such that the obtained pair is a hypergraph.We denote it by ( • ) (see Section 3.1 and Figure 1(b)).We also consider a hypergraph induced by a subset  × of the edges of , namely, ( × ).
Here we propose a systematic study of the basic operators that are used to derive a set of hyperedges from a set of vertices and a set of vertices from a set of hyperedges.These operators are the hypergraph extension to the operators defined by Cousty et al. [7,8] for graphs.Since a hypergraph becomes a graph when |V()| = 2 for every hyperedge , all the properties of these operators are satisfied for graphs also.We emphasise that the input and output of these operators are both hypergraphs.The blue subhypergraph in Figure 1(c This paper is organized as follows.In Section 2 we recall some related works on graphs and hypergraphs.In Section 3, we recall some preliminary definitions and results on mathematical morphology and hypergraphs.In Section 4, we define the vertex-hyperedge correspondence along with various dilations, erosions, and adjunctions on hypergraphs.The properties of these morphological operators are studied in this section.Section 5 concludes the paper with possible future works in this regard.

Related Works
Graph theoretic methods have found increasing applications in image analysis.Morphological operators are well studied on graphs.Vincent [10] defined morphological operators on a graph  = (, ), where  represents a set of weighted vertices and  represents a set of edges between vertices.The dilation (resp., erosion) replaces the value of each vertex with the maximum (resp., minimum) value of its neighbors.Cousty et al. [7,8] considered a graph as a pair  = ( • ,  × ), where  • is the set of vertices and  × is the edge set of the graph .They define morphological operators on various lattices formed by the graph  by defining an edge-vertex correspondence.This powerful tool allows them to recover the classical notion of a dilation/erosion of a subset of vertices of .This leads them to propose several new openings, closings, and granulometries and alternate sequential filters acting on the subsets of the edge sets, subsets of vertex sets, and the lattice of subgraphs of .These operators are further extended to functions that weight the vertices and edges of  [16] and are found to be useful in image filtering.In this work we aim to develop morphological operators on hypergraphs by defining a vertex-hyperedge correspondence.
The theory of hypergraphs originated as a natural generalisation of graphs in 1960s.In a hypergraph, edges can connect any number of vertices and are called hyperedges.Considering the topological and geometrical aspects of an image, Bretto et al. [17] have proposed a hypergraph model to represent an image.The theory of hypergraphs became an active area of research in image analysis.The study of mathematical morphology operators on hypergraphs started recently, and little work is being reported in this regard.Properties of morphological operators on hypergraphs are studied in [15], in which subhypergraphs are considered as relations on hypergraphs.Recently, Bloch and Bretto [12] introduced mathematical morphology on hypergraphs by forming various lattices on hypergraphs.Similarity and pseudometrics based on mathematical morphology are defined and illustrated in [14].Based on these morphological operators, similarity measures are used for classification of data represented as hypergraphs [13].

Preliminaries
3.1.Hypergraphs.We define a hypergraph [12,18] as a pair  = ( • ,  × ) where  • is a set of points called vertices and  × is composed of a family of subsets of  • called hyperedges.We denote  × by  × = (  ) ∈ where  is a finite set of indices.The set of vertices forming the hyperedge  is denoted by Let  • ⊆  • and  × ⊆  × where  × = (  ),  ∈  such that  ⊆ .We denote by  • (resp.,  × ) by the complementary set of  • (resp.,  × ).Let ( • ) and ( × ), respectively, denote the hypergraphs While dealing with a hypergraph , we consider the subhypergraph induced by a subset  • of vertices of , namely, ( • ), and the subhypergraph induced by a subset  × of hyperedges, namely, ( × ).( • ) is the largest subhypergraph of  with  • as vertex set and ( × ) is the smallest subhypergraph of  with  × as its hyperedge set.

Mathematical
Morphology.Now let us briefly recall some algebraic tools that are fundamental in mathematical morphology [5][6][7]19].Given two lattices L 1 and L 2 , any operator  : L 1 → L 2 that distributes over the supremum and preserves the least element is called a dilation (i.e., ∀ ⊆ L 1 , (∨ 1 ) = ∨ 2 {() |  ∈ }).Similarly an operator that distributes over the infimum and preserves the greatest element is called an erosion.
Two operators  : L 1 → L 2 and  : L 2 → L 1 form an adjunction (, ), if for any  ∈ L 1 and any  ∈ L 2 , we have ()≤ 1  ⇔ ≤ 2 (), where ≤ 1 and ≤ 2 denote the order relations in L 1 and L 2 , respectively [19].Given two operators  and , if the pair (, ) is an adjunction, then  is an erosion and  is a dilation.If L 1 , L 2 , and L 3 are three lattices and if  : and   : L 3 → L 2 are four operators such that (, ) and (  ,   ) are adjunctions, then the pair ( ∘   ,  ∘   ) is also an adjunction.
Given two complemented lattices, L 1 and L 2 , two operators  and  are dual with respect to the complement of each other, if for each  ∈ L 1 , we have () = ().If  and  are dual of each other, then  is an erosion whenever  is a dilation.

Hypergraph Morphology: Dilations, Erosions, and Adjunctions
In a hypergraph , we can consider sets of points as well as sets of hyperedges.Therefore it is convenient to consider operators that go from one kind of sets to the other one.In this section we define such operators and study their morphological properties.Based on these operators, we propose several dilations, erosions, and adjunctions on various lattices formed by .
Hereafter the workspace (see [7,8] for a similar structure defined for graphs) is a hypergraph  = ( • ,  × ) and we consider the sets H • , H × , and H of,respectively, all subsets of  • , all subsets of  × , and all subhypergraphs of H.
The set H of all subhypergraphs of a hypergraph  forms a complete lattice [15].H is not a Boolean algebra as the complement of a subhypergraph of  needs not be a subhypergraph of .But H • and H × are Boolean algebras.We define morphological operators on these lattices.We establish a correspondence between the vertex set and the hyperedge set of .Composing these mappings produces morphological operators on the lattices H • , H × , and H.   Definition 1 (vertex-hyperedge correspondence).We define the operators  • ,  • from H × into H • and the operators  × ,  × from H • into H × as in Table 1.
These operators are illustrated in Figures 2(a)-2(f).The choice of  is in such a way that every hyperedge of  is incident with exactly four vertices, and the choice of  is made to present a representative sample of the different possible configurations on subhypergraphs.Property 1.For any  • ⊆  • and any  × ⊆  × , where  × = (  ),  ∈  such that  ⊆ Proof.
Table 1 Provide its complement with a hypergraph structure for some  ∈ }.This property states that  • ( × ) is the set of all vertices which belong to a hyperedge of  × ,  × ( • ) is the set of all hyperedges whose vertices are composed of vertices of  • ,  • ( × ) is the set of all vertices which do not belong to any edge of  × , and  × ( • ) is the set of all hyperedges in  × with at least one vertex in  • .Therefore the previous property locally characterizes the operators defined in vertex-hyperedge correspondence.This property leads to simple linear time algorithms (with respect to | • | and | × |) to compute  • ,  × ,  • , and  × .
Property 4. For any  × ⊆  × ,  × = (  ) ∈ : ( Proof.Consider the following: ) is the result of the dilation [Δ, ]() of the blue subhypergraph  in Figure1(b) proposed in this paper.Here the resultant subhypergraph in Figure1(c) is not induced by its vertex set.