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Alain Bretto - One of the best experts on this subject based on the ideXlab platform.

  • Random walks in directed Hypergraphs and application to semi-supervised image segmentation
    Computer Vision and Image Understanding (CVIU), 2014
    Co-Authors: Aurélien Ducournau, Alain Bretto
    Abstract:

    In this paper, we introduce for the first time the notion of directed Hypergraphs in image processing and particularly image segmentation. We give a formulation of a random walk in a directed Hypergraph that serves as a basis to a semi-supervised image segmentation procedure that is configured as a machine learning problem, where a few sample pixels are used to estimate the labels of the unlabeled ones. A directed Hypergraph model is proposed to represent the image content, and the directed random walk formulation allows to compute a transition matrix that can be exploited in a simple iterative semi-supervised segmentation process. Experiments over the Microsoft GrabCut dataset have achieved results that demonstrated the relevance of introducing directionality in Hypergraphs for computer vision problems.

  • Mathematical morphology on Hypergraphs, application to similarity and positive kernel
    Computer Vision and Image Understanding, 2013
    Co-Authors: Isabelle Bloch, Alain Bretto
    Abstract:

    The focus of this article is to develop mathematical morphology on Hypergraphs. To this aim, we define lattice structures on Hypergraphs on which we build mathematical morphology operators. We show some relations between these operators and the Hypergraph structure, considering in particular transversals and duality notions. Then, as another contribution, we show how mathematical morphology can be used for classification or matching problems on data represented by Hypergraphs: thanks to dilation operators, we define a similarity measure between Hypergraphs, and we show that it is a kernel. A distance is finally introduced using this similarity notion.

  • Some Particular Hypergraphs
    Hypergraph Theory, 2013
    Co-Authors: Alain Bretto
    Abstract:

    Roughly speaking we introduced the more important concepts about Hypergraphs, we will see a little bit more in the next chapters, but there are very important classes of Hypergraphs. This chapter introduces some particular Hypergraphs which either have good properties, or are very important for applications of the theory. In the sequel we will suppose most of the time that Hypergraph are without repeated hyperedge.

  • DGCI - Mathematical morphology on Hypergraphs: preliminary definitions and results
    Discrete Geometry for Computer Imagery, 2011
    Co-Authors: Isabelle Bloch, Alain Bretto
    Abstract:

    In this article we introduce mathematical morphology on Hypergraphs. We first define lattice structures and then mathematical morphology operators on Hypergraphs. We show some relations between these operators and the Hypergraph structure, considering in particular duality and similarity aspects.

  • mathematical morphology on Hypergraphs preliminary definitions and results
    Discrete Geometry for Computer Imagery, 2011
    Co-Authors: Isabelle Bloch, Alain Bretto
    Abstract:

    In this article we introduce mathematical morphology on Hypergraphs. We first define lattice structures and then mathematical morphology operators on Hypergraphs. We show some relations between these operators and the Hypergraph structure, considering in particular duality and similarity aspects.

Gábor Simonyi - One of the best experts on this subject based on the ideXlab platform.

  • Entropy Splitting Hypergraphs
    Journal of Combinatorial Theory Series B, 1996
    Co-Authors: Gábor Simonyi
    Abstract:

    Hypergraph entropy is an information theoretic functional on a Hypergraph with a probability distribution on its vertex set. It is sub-additive with respect to the union of Hypergraphs. In case of simple graphs, exact additivity for the entropy of a graph and its complement with respect to every probability distribution on the vertex set gives a characterization of perfect graphs. Here we investigate uniform Hypergraphs with an analoguous behaviour of their entropy. The main result is the characterization of 3-uniform Hypergraphs having this entropy splitting property. It is also shown that fork?4 no non-trivialk-uniform Hypergraph has this property.

  • Entropy splitting Hypergraphs
    Proceedings of 1994 IEEE International Symposium on Information Theory, 1994
    Co-Authors: Gábor Simonyi
    Abstract:

    Hypergraph entropy is a sub-additive functional on Hypergraphs. We characterize those uniform Hypergraphs F for which the entropy of F and the entropy of its complement adds up exactly to the entropy of the complete uniform Hypergraph. Hypergraph entropy is an information theoretic functional on a Hypergraph with a probability distribution on its vertex set. It is a generalisation of graph entropy. >

Isabelle Bloch - One of the best experts on this subject based on the ideXlab platform.

Shashi Shekhar - One of the best experts on this subject based on the ideXlab platform.

  • multilevel Hypergraph partitioning applications in vlsi domain
    IEEE Transactions on Very Large Scale Integration Systems, 1999
    Co-Authors: George Karypis, R Aggarwal, Vipin Kumar, Shashi Shekhar
    Abstract:

    In this paper, we present a new Hypergraph-partitioning algorithm that is based on the multilevel paradigm. In the multilevel paradigm, a sequence of successively coarser Hypergraphs is constructed. A bisection of the smallest Hypergraph is computed and it is used to obtain a bisection of the original Hypergraph by successively projecting and refining the bisection to the next level finer Hypergraph. We have developed new Hypergraph coarsening strategies within the multilevel framework. We evaluate their performance both in terms of the size of the hyperedge cut on the bisection, as well as on the run time for a number of very large scale integration circuits. Our experiments show that our multilevel Hypergraph-partitioning algorithm produces high-quality partitioning in a relatively small amount of time. The quality of the partitionings produced by our scheme are on the average 6%-23% better than those produced by other state-of-the-art schemes. Furthermore, our partitioning algorithm is significantly faster, often requiring 4-10 times less time than that required by the other schemes. Our multilevel Hypergraph-partitioning algorithm scales very well for large Hypergraphs. Hypergraphs with over 100 000 vertices can be bisected in a few minutes on today's workstations. Also, on the large Hypergraphs, our scheme outperforms other schemes (in hyperedge cut) quite consistently with larger margins (9%-30%).

  • multilevel Hypergraph partitioning application in vlsi domain
    Design Automation Conference, 1997
    Co-Authors: George Karypis, R Aggarwal, Vipin Kumar, Shashi Shekhar
    Abstract:

    In this paper, we present a new Hypergraph partitioning algorithmthat is based on the multilevel paradigm. In the multilevel paradigm,a sequence of successively coarser Hypergraphs is constructed. Abisection of the smallest Hypergraph is computed and it is used toobtain a bisection of the original Hypergraph by successively projectingand refining the bisection to the next level finer Hypergraph.We evaluate the performance both in terms of the size of the hyper-edgecut on the bisection as well as run time on a number of VLSIcircuits. Our experiments show that our multilevel Hypergraph partitioningalgorithm produces high quality partitioning in relativelysmall amount of time. The quality of the partitionings produced byour scheme are on the average 4% to 23% better than those producedby other state-of-the-art schemes. Furthermore, our partitioning algorithmissignificantly faster, often requiring 4 to 15 times less timethan that required by the other schemes. Our multilevel Hypergraphpartitioning algorithm scales very well for large Hypergraphs. Hypergraphswith over 100,000 vertices can be bisected in a few minuteson today's workstations. Also, on the large Hypergraphs, ourscheme outperforms other schemes (in hyperedge cut) quite consistentlywith larger margins (9% to 30%).

Syama Sundar Rangapuram - One of the best experts on this subject based on the ideXlab platform.

  • The Total Variation on Hypergraphs - Learning on Hypergraphs Revisited
    arXiv: Machine Learning, 2013
    Co-Authors: Matthias Hein, Simon Setzer, Leonardo Jost, Syama Sundar Rangapuram
    Abstract:

    Hypergraphs allow one to encode higher-order relationships in data and are thus a very flexible modeling tool. Current learning methods are either based on approximations of the Hypergraphs via graphs or on tensor methods which are only applicable under special conditions. In this paper, we present a new learning framework on Hypergraphs which fully uses the Hypergraph structure. The key element is a family of regularization functionals based on the total variation on Hypergraphs.

  • NIPS - The Total Variation on Hypergraphs - Learning on Hypergraphs Revisited
    2013
    Co-Authors: Matthias Hein, Simon Setzer, Leonardo Jost, Syama Sundar Rangapuram
    Abstract:

    Hypergraphs allow one to encode higher-order relationships in data and are thus a very flexible modeling tool. Current learning methods are either based on approximations of the Hypergraphs via graphs or on tensor methods which are only applicable under special conditions. In this paper, we present a new learning framework on Hypergraphs which fully uses the Hypergraph structure. The key element is a family of regularization functionals based on the total variation on Hypergraphs.

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