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than graph partitioning in many cases of practical
interest. For example, in the row-wise decompo-
sition of a sparse matrix for parallel matrix-vector
multiplication, a hypergraph model provides an ex-
act measure of communication cost, whereas a graph
model can only provide an upper bound (Trifunovic
and Knottenbelt, 2004a) (Catalyurek and Aykanat.,
1999). It has been shown that, in general, there does
not exist a graph model that correctly represents the
cut properties of the corresponding hypergraph (Ih-
ler et al., 1993). Recently, several serial and parallel
hypergraph partitioning techniques have been exten-
sively studied (Sanchis, 1989) (Trifunovic and Knot-
tenbelt, 2004a)(Karypis, 2002) and tools support ex-
ists (e.g. hMETIS (Karypis and Kumar, 1998), PaToH
(Catalyurek and Aykanat., 1999) and Parkway (Tri-
funovic and Knottenbelt, 2004b)). These partitioning
techniques showed a very great efficiency in distrib-
uted databases and VLSI circuits fields.
In this paper, we widen the application area of hy-
pergraph partitioning algorithms to image fields and
more particularly to the image segmentation. The ba-
sic idea of this algorithm can be described as follows
and summarize in two steps:
1. It first builds a hypergraph of the image.
2. Then the algorithm partitions this representation
into a set of vertices, representing homogeneous re-
gions.
The aim of the first step is to capture all global
and local properties of the image data and the whole
key information for the segmentation purpose. This
model has proved to be extremely useful for solving
some applications in image processing fields such as
noise removal (Rital et al., 2001) and edge detection
(Rital and Cherifi, 2004). While the second step of
the proposed approach partition this representation to
a homogenous regions. It is done by a fast multilevel
programming algorithm. Throughout this paper, we
will denote the hypergraph of the image by the Image
Neighborhood Hypergraph INH.
In section 2, we briefly review some background on
hypergraph theory. The proposed segmentation ap-
proach is presented in Section 3 and its performance
is illustrated in Section 4. The paper ends with a con-
clusion in Section 5.
2 BACKGROUND
Our main interest in this paper is to use combinatorial
models. We will introduce basic tools that are needed.
A hypergraph H on a set X is a family (E
i
)
i∈I
of
non-empty subsets of X called hyperedges with:
i∈I
E
i
= X, I = {1, 2,...,n},n∈ N.
Given a graph G(X; e), the hypergraph having the
vertices of G as vertices and the neighborhood of
these vertices as hyperedges (including these vertices)
is called the neighborhood hypergraph of G. To each
graph we can associate a neighborhood hypergraph :
H
G
=(X, (E
x
= {x}∪Γ(x))) (4)
where Γ(x)={y ∈ X, (x, y) ∈ e}.
2.1 Multilevel Hypergraph
Partitioning
The goal of the k-way hypergraph partitioning prob-
lem is to partition the vertices of the hypergraph into
k disjoint subsets X
i
,(i =0,...,k − 1), such that
a certain objective functions defined over the hyper-
edges is optimized.
Let us note H(X, E) a hypergraph. We will as-
sume that each vertex and hyperedge has a weight
associated with it, and we will use w(x) to denote
the weight of a vertex x, and w(E) to denote the
weight of a hyperedge E. One of the most com-
monly used objective functions is to minimize the
hyperedge-cut of the partitioning; i.e., the sum of the
weights of the hyperedges that span multiple parti-
tions: cut{A, B} =
E
i
∈A,E
j
∈B
w(E
i
,E
j
),A,B
are two partitions. Another objective that is often used
is to minimize the sum of external degrees (SOED) of
all hyperedges that span multiple partitions (Karypis
et al., 1999).
The most commonly used approach for computing
a k-way partitioning is based on recursive bisection.
In this approach, the overall k-way partitioning is ob-
tained by initially bisecting the hypergraph to obtain a
two-way partitioning. Then, each of these parts is fur-
ther bisected to obtain a four-way partitioning, and so
on. The problem of computing an optimal bisection of
a hypergraph is at least NP-hard (Garey and Johnson,
1979); however, many heuristic algorithms have been
developed. The survey by Alpert and Kahng (Alpert
and Kahng, 1995) provides a detailed description and
comparison of various such schemes.
The key idea behind the multilevel approach for
hypergraph partitioning is fairly simple and straight-
forward. Multilevel partitioning algorithm, instead of
trying to compute the partitioning directly in the orig-
inal hypergraph, partition the hypergraph using three
process (Fig.2):
Coarsening phase: first obtain a sequence of suc-
cessive approximations of the original hypergraph.
Each one of these approximations represents a prob-
lem whose size is smaller than the size of the origi-
nal hypergraph. This process continues until a level
of approximation is reached in which the hypergraph
contains only a few tens of vertices (Fig. 3).
NEIGHBORHOOD HYPERGRAPH PARTITIONING FOR IMAGE SEGMENTATION
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