Complex Fiedler Vectors for Shape Retrieval

  • Reinier H. van Leuken
  • Olga Symonova
  • Remco C. Veltkamp
  • Raffaele de Amicis
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5342)


Adjacency and Laplacian matrices are popular structures to use as representations of shape graphs, because their sorted sets of eigenvalues (spectra) can be used as signatures for shape retrieval. Unfortunately, the descriptiveness of these spectra is limited, and handling graphs of different size remains a challenge. In this work, we propose a new framework in which the shapes (3D models in our test corpus) are represented by multi-labeled graphs. A Hermitian matrix is associated to each graph, in which the entries are defined such that they contain all information stored in the graph edges. Additional constraints ensure that this Hermitian matrix mimics the well-studied spectral behaviour of the Laplcian matrix. We therefore use the Hermitian Fiedler vector as shape signature during retrieval. To deal with graphs of different size, we efficiently reuse the calculated Fiedler vector to decompose the graph into a limited number of non-overlapping, meaningful subgraphs. Retrieval results are based on both complete matching and subgraph matching.


Shape Descriptor Shape Index Hermitian Matrix Laplacian Matrix Retrieval Result 
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.
    Biasotti, S., Falcidieno, B., Frosini, P., Giorgi, D., Landi, C., Marini, S., Patane, G., Spagnuolo, M.: 3D shape description and matching based on properties of real functions. In: Eurographics - Tutorials (2007)Google Scholar
  2. 2.
    Tangelder, J.W., Veltkamp, R.C.: A survey of content based 3d shape retrieval methods. In: Shape Modeling International (2004)Google Scholar
  3. 3.
    Biasotti, S., Giorgi, D., Spagnuolo, M., Falcidieno, B.: Reeb graphs for shape analysis and applications. Theoretical Computer Science (2007) (to appear)Google Scholar
  4. 4.
    Hilaga, M., Shinagawa, Y., Kohmura, T., Kunii, T.L.: Topology matching for fully automatic similarity estimation of 3d shapes. In: SIGGRAPH (2001)Google Scholar
  5. 5.
    Marini, S., Spagnuolo, M., Falcidieno, B.: From exact to approximate maximum common subgraph. In: Graph-based Representations in Pattern Recognition (2005)Google Scholar
  6. 6.
    Tierny, J., Vandeborre, J.P., Daoudi, M.: Reeb chart unfolding based 3D shape signatures. In: Eurographics (2007)Google Scholar
  7. 7.
    Tung, T., Schmitt, F.: Augmented reeb graphs for content-based retrieval of 3d mesh models. In: Shape Modeling International (2004)Google Scholar
  8. 8.
    Demirci, M.F., van Leuken, R.H., Veltkamp, R.C.: Indexing through laplacian spectra. Computer Vision and Image Understanding 110(3), 312–325 (2008)CrossRefGoogle Scholar
  9. 9.
    Sengupta, K., Boyer, K.L.: Modelbase partitioning using property matrix spectra. Computer Vision and Image Understanding 70(2), 177–196 (1998)CrossRefzbMATHGoogle Scholar
  10. 10.
    Shokoufandeh, A., Macrini, D., Dickinson, S., Siddiqi, K., Zucker, S.: Indexing hierarchical structures using graph spectra. Pattern Analysis and Machine Intelligence 27(7) (2005)Google Scholar
  11. 11.
    Wilson, R.C., Hancock, E.R., Luo, B.: Pattern vectors from algebraic graph theory. Pattern Analysis and Machine Intelligence 27, 1112–1124 (2005)CrossRefGoogle Scholar
  12. 12.
    Zhu, P., Wilson, R.C.: A study of graph spectra for comparing graphs. In: British Machine Vision Conference (2005)Google Scholar
  13. 13.
    Biasotti, S., Falcidieno, B., Spagnuolo, M.: Extended reeb graphs for surface understanding and description. In: Discrete Geometry for Computer Imagery (2000)Google Scholar
  14. 14.
    Mortara, M., Patane, G., Spagnuolo, M., Falcidieno, B., Rossignac, J.: Blowing bubbles for multi-scale analysis and decomposition of triangle meshes. Algorithmica 38(1), 227–248 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Symonova, O., De Amicis, R.: Shape analysis for augmented topological shape descriptor. In: Eurographics (2007)Google Scholar
  16. 16.
    Mohar, B.: The laplacian spectrum of graphs. Graph Theory, Combinatorics and Applications 2, 871–898 (1991)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Merris, R.: Laplacian matrices of graphs: a survey. Linear Algebra and its Applications 197(1), 143–176 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Qiu, H., Hancock, E.R.: Graph partition for matching. In: Graph-based Representations in Pattern Recognition (2003)Google Scholar
  19. 19.
    Veltkamp, R.C., ter Haar, F.B.: SHREC2007: 3D Shape Retrieval Contest. Technical Report UU-CS-2007-015, Utrecht University (2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Reinier H. van Leuken
    • 1
  • Olga Symonova
    • 2
  • Remco C. Veltkamp
    • 1
  • Raffaele de Amicis
    • 2
  1. 1.Universiteit UtrechtThe Netherlands
  2. 2.Fondazione GraphitechTrentoItaly

Personalised recommendations