In this paper we investigate how to establish a hypergraph model for characterizing object structures and how to embed this model into a low-dimensional pattern space. Each hyperedge of the hypergraph model is derived from a seed feature point of the object and embodies those neighbouring feature points that satisfy a similarity constraint. We show how to construct the Laplacian matrix of the hypergraph. We adopt the spectral method to construct pattern vectors from the hypergraph Laplacian. We apply principal component analysis (PCA) to the pattern vectors to embed them into a low-dimensional space. Experimental results show that the proposed scheme yields good clusters of distinct objects viewed from different directions.


Feature Point Bipartite Graph Adjacency Matrix Incidence Matrix Laplacian Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    IEEE TPAMI: Special Section on Graph Algorithms and Computer Vision, 23 10401151.1 (2001)Google Scholar
  2. 2.
    Agarwal, S., Branson, K., Belongie, S.: Higher-Order Learning with Graphs. In: ICML, pp. 17–23 (2006)Google Scholar
  3. 3.
    Agarwal, S., Lim, J., Zelnik-Manor, L., Perona, P., Kriegman, D., Belongie, S.: Beyond pairwise clustering. CVPR 2, 838–845 (2005)Google Scholar
  4. 4.
    Bretto, A., Cherifi, H., Aboutajdine, D.: Hypergraph imaging: an overview. Pattern Recognition 35(3), 651–658 (2002)CrossRefzbMATHGoogle Scholar
  5. 5.
    Chung, F.K.: The Laplacian of a Hypergraph. AMS DIMACS Series in Discrete Mathematics and Theoretical Computer Science 10, 21–36 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gibson, D., Kleinberg, J., Raghavan, P.: Clustering categorical data: An approach based on dynamical systems. The VLDB Journal 8(3-4), 222–236 (2000)CrossRefGoogle Scholar
  7. 7.
    Hagen, L., Kahng, A.B.: New spectral methods for ratio cut partitioning and clustering. IEEE TCAD 11(9), 1074–1085 (1992)Google Scholar
  8. 8.
    Harris, C.G., Stephens, M.J.: A combined corner and edge detector. In: Proceedings of Fourth Alvey Vision Conference, pp. 147–151 (1994)Google Scholar
  9. 9.
    Li, W., Sole, P.: Spectra of Regular Graphs and Hypergraphs and Orthogonal Polynomials. European Journal of Combinatorics 17, 461–477 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Luo, B., Wilson, R.C., Hancock, E.R.: Higher-Order Learning with Graphs. Pattern Recognition 36(10), 2213–2223 (2003)CrossRefzbMATHGoogle Scholar
  11. 11.
    Rodriguez, J.A.: On The Laplacian Eigenvalues and Metric Parameters of Hypergraphs. Linear and Multilinear Algebra 51, 285–297 (2003)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Storm, C.K.: The Zeta Function of a Hypergraph. EJC 13 (2006)Google Scholar
  13. 13.
    Wilson, R.C., Hancock, E.R., Luo, B.: Pattern Vectors from Algebraic Graph Theory. IEEE TPAMI 27(7), 1112–1124 (2005)CrossRefGoogle Scholar
  14. 14.
    Zhou, D., Huang, J., Scholkopf, B.: Learning with Hypergraphs: Clustering, Classification, and Embedding. NIPS 19, 17–24 (2007)Google Scholar
  15. 15.
    Zhu, P., Wilson, R.C.: A Study of Graph Spectra for Comparing Graphs and Trees. Pattern Recognition (to appear, 2008)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Peng Ren
    • 1
  • Richard C. Wilson
    • 1
  • Edwin R. Hancock
    • 1
  1. 1.Department of Computer ScienceThe University of YorkYorkUK

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