Hypergraphs for Near-lossless Volumetric Compression Luc Gillibert Universit´ e de Caen, GREYC UMR-6072. Bd Marechal Juin BP 5186, 14032 Caen cedex, France. luc.gillibert@info.unicaen.fr Alain Bretto Universit´ e de Caen, GREYC UMR-6072. Bd Marechal Juin BP 5186, 14032 Caen cedex, France. alain.bretto@info.unicaen.fr Abstract A hypergraphs-based image representation is already used for 2D lossless image compression. In this paper we extend this hypergraph representation on 3D-image and we add an α-tolerance. This extended representation conducts to a generalisation of the HLC lossless compression al- gorithm [2] for near-lossless 3D-image compression. We present an algorithm performing that compression and we give some experimental results proving its efficiency. This paper is a detailled version of the poster [3]. 1 Introduction Image compression addresses the problem of reducing amount of data needed to represent a digital image. Lossless or reversible compression refers to compression techniques in which the reconstructed data exactly matches the origi- nal. Near-lossless compression denotes compression meth- ods which give quantitative bounds on the nature of the loss that is introduced. Such compression techniques provide the guarantee that no pixel difference between the original and the compressed image is above a given value. Both lossless and near-lossless compression are used for satellite images, where the data loss is undesirable because of im- age collecting cost, and medical images, where difference in original image and uncompressed one can compromise diagnostic accuracy [5, 6]. In this paper we describe a new method for near-lossless 3D-image compression, based on hypergraphs and called HNLC (Hypergraph Near-Lossless Compression). This method is a generalisation of the HLC method (Hypergraph Lossless Compression). The hyper- graphs are a very interesting generalisation of the graphs. Introduced in 1960 by C. BERGE [8], they are now used in many domains such as chemistry, engineering and image processing [1, 4]. We give an algorithm making the conver- sion between a 3D-matrix-based representation and the hy- pergraph representation. We also present some experimen- tal results proving that HNLC, combined with a PPM-based [11] data compression algorithm, is very efficient. 2 Definitions Let V = {x 1 ,x 2 ,...,x n } be a finite set. A hypergraph on V is a family H = {E 1 ,E 2 ,...,E m } of subsets of V such that: • E i = ∅ for i =1, 2,...,m •∪ m i=1 E i = V The elements x 1 ,x 2 ,...,x n are called the vertices, and the sets E 1 ,E 2 ,...,E m are called the hyper-edges of the hy- pergraph. A partial hypergraph H ′ from a hypergraph H is a hypergraph such that H ′ ⊂ H . If the set of vertices of H ′ is equal to the set of vertices of H we say that H ′ is a partial hypergraph covering. A simple hypergraph is a hyper- graph H = {E 1 ,E 2 ,...,E m } such that E i ⊆ E j ⇒ i = j . 3 Hypergraph representation 3.1 Formal definition Let I be a 3-dimensional matrix represented image (if the size of I is i × j × k, I can be see as a set of k 2D matrix of size i × j ). We build a hypergraph H α (I ), called the extended hypergraph representation of the image for the tolerance α, as it follow: 1