Abstract
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 , 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 of . This paper also studies the concept of morphological adjunction on hypergraphs for which both the input and the output are hypergraphs.
1. Introduction
Mathematical morphology, appeared in 1960s, is a theory of nonlinear information processing [1–4]. 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 [7–10] and simplicial complexes [11], but little work has been done on hypergraphs [12–15].
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, .
| (a) |
| (b) |
| (c) Dilation |
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 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) is the result of the dilation of the blue subhypergraph in Figure 1(b) proposed in this paper. Here the resultant subhypergraph in Figure 1(c) is not induced by its vertex set.
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.
2. 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].
3. 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 . A vertex in is called an isolated vertex of if . The empty hypergraph is the hypergraph such that and . The partial hypergraph of generated by is the hypergraph ) where and . A hypergraph is called a subhypergraph of , denoted by , if and .
Let and where such that . We denote by (resp., ) by the complementary set of (resp., ). Let and , respectively, denote the hypergraphs and .
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.
3.2. Mathematical Morphology
Now let us briefly recall some algebraic tools that are fundamental in mathematical morphology [5–7, 19]. Given two lattices and , any operator that distributes over the supremum and preserves the least element is called a dilation (i.e., . Similarly an operator that distributes over the infimum and preserves the greatest element is called an erosion.
Two operators and form an adjunction , if for any and any , we have , where and denote the order relations in and , respectively [19]. Given two operators and , if the pair is an adjunction, then is an erosion and is a dilation. If , , and are three lattices and if , , , and are four operators such that and are adjunctions, then the pair is also an adjunction.
Given two complemented lattices, and , two operators and are dual with respect to the complement of each other, if for each , we have . If and are dual of each other, then is an erosion whenever is a dilation.
4. 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 , and of,respectively, all subsets of , all subsets of , and all subhypergraphs of .
The set of all subhypergraphs of a hypergraph forms a complete lattice [15]. is not a Boolean algebra as the complement of a subhypergraph of needs not be a subhypergraph of . But and 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 , , and .
Definition 1 (vertex-hyperedge correspondence). We define the operators , from into and the operators , from into 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.
| (a) |
| (b) |
| (c) |
| (d) |
| (e) |
| (f) |
| (g) |
| (h) |
| (i) |
| (j) |
| (k) |
| (l) |
Property 1. For any and any , where such that (1) is such that ;(2) is such that ;(3) is such that ;(4) is such that .
Proof. (1) and (2) follow from the definition of and .
(3) . Thus,
(4) . Thus, .
Note that such that 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 2 (dilation, erosion, adjunction, and duality). (1) Operators and (resp., and ) are dual of each other.
(2) Both and are adjunctions.
(3) Operators and are erosions.
(4) Operators and are dilations.
Proof. (1) We will prove that and :
Thus, and are duals:
Therefore, and are duals.
(2) Suppose that . Then,
Therefore, .
Conversly, if , then,
Thus, . Therefore, is an adjunction:
Therefore, is an adjunction.
Properties (3) and (4) follow from the dilation/erosion property of adjunctions.
Definition 2 (vertex dilation, vertex erosion). We define and that act on by and .
Property 3. For any , (1) such that ;(2) such that .
Proof. Consider the following:
(1)
(2)
Definition 3 (hyper-edge dilation, hyper-edge erosion). We define and that act on by and .
Property 4. For any , :(1) such that ;(2).
Proof. Consider the following:
(1)
(2)
Remark 4. Being the compositions of, respectively, dilations and erosions, and are, respectively, a dilation and an erosion [19]. Moreover by composition of adjunctions and dual operators, and are dual and is an adjunction. In a similar manner is also an adjunction.
Definition 5 (hypergraph dilation, hypergraph erosion). We define the operators and by, respectively, and , for any .
Theorem 6. The operators and are, respectively, a dilation and an erosion acting on the lattice .
Proof. We will prove that for every , . implies that there exists some such that . But , since . Thus, . Therefore, . This implies .
If , then for every , and so :
Therefore, .
Theorem 7. is an adjunction.
Proof. Let and be two hypergraphs in . The following statements are equivalent: Thus the pair is an adjunction, which implies that is an erosion and is a dilation.
5. Conclusion
This paper investigates the lattice of all subhypergraphs of a hypergraph and provides it with morphological operators. By the composition of the operators presented in this paper, we can define other adjunctions on hypergraphs. The proposed framework can be extended to morphological filtering on hypergraphs and is left to the future work.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The first author is on deputation from Mar Athanasius College, Kothamangalam, India, and is supported by the University Grants Commission (UGC), Government of India, under the FDP scheme.