Luc Gillibert

Hypergraph-Based image representation

An appropriate image representation induces some good image treatment algorithms. Hypergraph theory is a theory of finite combinatorial sets, modeling a lot of problems of operational research and combinatorial optimization. Hypergraphs are now used in many domains such as chemistry, engineering and image processing. We present an overview of a hypergraph-based picture representation giving much application in picture manipulation, analysis and restoration: the Image Adaptive Neighborhood Hypergraph (IANH). With the IANH it is possible to build powerful noise detection an elimination algorithm, but also to make some edges detection or some image segmentation. IANH has various applications and this paper presents a survey of them.

G-graphs: A new representation of groups

An important part of computer science is focused on the links that can be established between group theory and graph theory and graphs. Cayley graphs, that establish such a link, are useful in a lot of areas of sciences. This paper introduces a new type of graph associated with a group, the G-graphs, and presents many of their properties. We show that various characteristics of a group can be seen on its associated G-graph. We also present an implementation of the algorithm constructing these new graphs, an implementation that will lead to some experimental results. Finally we show that many classical graphs are G-graphs. The relations between G-graphs and Cayley graphs are also studied.

Symmetric and semisymmetric graphs construction using G-graphs

Symmetric and semisymmetric graphs are used in many scientific domains, especially parallel computation and interconnection networks. The industry and the research world make a huge usage of such graphs. Constructing symmetric and semisymmetric graphs is a large and hard problem. In this paper a tool called G-graphs and based on group theory is used. We show the efficiency of this tool for constructing symmetric and semisymmetric graphs and we exhibit experimental results.

Hypergraph lossless image compression

Hypergraphs are a very powerful tool and can represent many problems. In this paper we define a new image representation based on hypergraphs. This representation conducts to a new lossless compression algorithm for images called HLC. We present the algorithm and give some experimental results proving its efficiency. Finally we show that this algorithm can be generalized to three-dimensional images and to parametric lossy compression.

G-graphs: An efficient tool for constructing symmetric and semisymmetric graphs

Symmetric and semisymmetric graphs are used in many scientific domains, especially parallel computation and interconnection networks. The industry and the research world make a huge usage of such graphs. Constructing symmetric and semisymmetric graphs is a large and hard problem. In this paper a tool called G-graphs and based on group theory is used. We show the efficiency of this tool for constructing symmetric and semisymmetric graphs and we exhibit experimental results.

Graphical and Computational Representation of Groups

An important part of the computer science is focused on the links that can be established between group theory and graph theory. Cayley graphs can establish such a link but meet some limitations. This paper introduces a new type of graph associated to a group: the G-graphs. We present an implementation of the algorithm constructing these new graphs. We establish a library of the most common G-graphs, using GAP and the SmallGroups library. We give some experimental results with GAP and we show that many classical graphs are G-graphs.

G-graphs for the cage problem: a new upper bound

Constructing some regular graph with a given girth,a given degree and the fewest possible vertices is a hard problem. This problem is called the cage graph problem and has some links with the error codes theory. In this paper we presents some new graphs, constructed from a group, with a girth of 6 and regular of degree <i>p</i>, for any prime number <i>p</i>. This graphs are of order 2 x <i>p</i>² when the best upper bound known for the (<i>p</i>,6)-cage problem was the Sauer bound, equal to 4(<i>p</i> -- 1)³.

A New Property of Hamming Graphs and Mesh of d-ary Trees

In this article we characterize two well-known graphs used in many applications, particularly in network applications: Hamming graphs and meshes of d-ary trees MT(d,1). More precisely, we show that they are so-called

Hypergraphs for Generic Lossless Image Compression

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Hypergraphs for Near-lossless Volumetric Compression

-graphs.