Pattern Recognition 35 (2002) 651–658 www.elsevier.com/locate/patcog Hypergraph imaging: an overview  Alain Bretto a ; ∗ , Hocine Cheri b , Driss Aboutajdine c a Universit e Jean Monnet. LIGIV, Site G.I.A.T Ind., rue Javelin Pagnon, BP 505, 42007 Saint-Etienne Cedex 1, France b Universit e de Bourgogne, LIRSIA, UFR Sciences et Techniques, BP 47870, 21078 Dijon Cedex, France c Universit e Mohammed V-Agdal, G. S. C, Facult es des Sciences, BP 1014, Rabat, Maroc Received 16 November 2000; received in revised form 9 December 2000; accepted 9 December 2000 Abstract Hypergraph theory as originally developed by Berge (Hypergraphe, Dunod, Paris, 1987) is a theory of nite combi- natorial sets, modeling lot of problems of operational research and combinatorial optimization. This framework turns out to be very interesting for many other applications, in particular for computer vision. In this paper, we are going to survey the relationship between combinatorial sets and image processing. More precisely, we propose an overview of dierent applications from image hypergraph models to image analysis. It mainly focuses on the combinatorial repre- sentation of an image and shows the eectiveness of this approach to low level image processing; in particular to seg- mentation, edge detection and noise cancellation. ? 2001 Pattern Recognition Society. Published by Elsevier Science Ltd. All rights reserved. Keywords: Combinatorial optimization; Image processing; Image model; Segmentation; Edge detection; Noise reduction; Hyper- graph; Graph 1. Introduction In many areas of research such as, psychology, bio- logy, articial intelligence, etc., the relation between ob- jects represented by binary relations. The appropriate mathematical tools to model these relations are graphs. In a graph the vertices correspond to objects and edges represent the interrelations between these objects. The vertices can be valued by some parameters such as gray level value, hereditary-parameters, temperature, etc. The edges generally correspond to some (dis-) similarity mea- sures. The relation between objects that are of interest  This work has been supported by the project ‘Pars CNR no. 36’ and the ‘comit e mixte inter universitaire franco marocain AI no. 166=SI=98’. * Corresponding author. Tel.: +33-47-7-92-30-30; fax: +33-47-7-92-30-39. E-mail addresses: bretto@vision.univ-st-etienne.fr (A. Bretto), cheri@crid.u-bourgogne.fr (H. Cheri), aboutaj@fsr.ac.ma (D. Aboutajdine). depend on the property that is being studied. A digital im- age can also be considered as a graph when the topology (connectivity) of the support grid is taken into account. But it is dicult for the geometry and the topology of an image to be apprehended by a graph. Indeed both the geo- metry and the topology of an image are not necessarily expressed by binary relations. Hypergraph theory has been originally developed by Berge [1] as a generalization of graph theory. The idea of looking at a family of sets from this standpoint took shape around 1960, in regarding sets as generalized edges and in calling the family itself a hypergraph. This concept can model more general types of relations than binary relations. These mathematical framework can be used to model networks, data structures, process scheduling, computations, and a variety of other systems where the relations (not necessarily binary) between the objects in the system play a dominant role. We will consider hyper- graphs from several perspectives: as mathematical enti- ties with a rich and extensive theory; as model for image analysis. 0031-3203/01/$22.00 ? 2001 Pattern Recognition Society. Published by Elsevier Science Ltd. All rights reserved. PII:S0031-3203(01)00067-X