A Modification of Diffusion Distance for Clustering and Image Segmentation

  • Eduard Sojka
  • Jan Gaura
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8192)


Measuring the distances is an important problem in many image-segmentation algorithms. The distance should tell whether two image points belong to a single or, respectively, to two different image segments. The simplest approach is to use the Euclidean distance. However, measuring the distances along the image manifold seems to take better into account the facts that are important for segmentation. Geodesic distance, i.e. the shortest path in the corresponding graph or k shortest paths can be regarded as the simplest way how the distances along the manifold can be measured. At a first glance, one would say that the resistance and diffusion distance should provide the properties that are even better since all the paths along the manifold are taken into account. Surprisingly, it is not often true. We show that the high number of paths is not beneficial for measuring the distances in image segmentation. On the basis of analysing the problems of diffusion distance, we introduce its modification, in which, in essence, the number of paths is restricted to a certain chosen number. We demonstrate the positive properties of this new metrics.


Image segmentation diffusion distance geodesic distance 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Babić, D., Klein, D.J., Lukovits, I., Nikolić, S., Trinajstić, N.: Resistance-Distance Matrix: A Computational Algorithm and Its Applications. Int. J. Quant. Chem. 90, 166–176 (2002)CrossRefGoogle Scholar
  2. 2.
    Coifman, R., Lafon, S.: Diffusion Maps. Applied and Computational Harmonic Analysis 21, 5–30 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Eppstein, D.: Finding the k Shortest Paths. J. Comp. 28(2), 652–673 (1999)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Fiorio, C., Mercat, C., Rieux, F.: Adaptive Discrete Laplace Operator. In: International Symposium on Visual Computing, pp. 567–577 (2011)Google Scholar
  5. 5.
    Fouss, F., Pirotte, A., Renders, J.M., Saerens, M.: Random-Walk Computation of Similarities between Nodes of a Graph with Application to Collaborative Recommendation. Trans. on Knowledge and Data Engineering 19, 355–369 (2007)CrossRefGoogle Scholar
  6. 6.
    Grady, L.: Random Walks for Image Segmentation. TPAMI 28(11), 1768–1783 (2006)CrossRefGoogle Scholar
  7. 7.
    Huang, H., Yoo, S., Qin, H., Yu, D.: A Robust Clustering Algorithm Based on Aggregated Heat Kernel Mapping. In: International Conference on Data Mining, pp. 270–279 (2011)Google Scholar
  8. 8.
    Klein, D., Randić, J.M.: Resistance Distance. J. Mat. Chem. 12, 81–95 (1993)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Lafon, S., Lee, A.B.: Diffusion Maps and Coarse-Graining: a Unified Framework for Dimensionality Reduction, Graph Partitioning, and Data Set Parameterization. TPAMI 28(9), 1393–1403 (2006)CrossRefGoogle Scholar
  10. 10.
    Lipman, Y., Rustamov, R.M., Funkhouser, T.A.: Biharmonic Distance. ACM Transactions on Graphics 29, 1–11 (2010)Google Scholar
  11. 11.
    Martin, D., Fowlkes, C., Tal, D., Malik, J.: A Database of Human Segmented Natural Images and its Application to Evaluating Segmentation Algorithms and Measuring Ecological Statistics. In: International Conference of Computer Vision, pp. 416–423 (2001)Google Scholar
  12. 12.
    Nadler, B., Lafon, S., Coifman, R.R., Kevrekidis, I.G.: Diffusion Maps, Spectral Clustering and Eigenfunctions of Fokker-Planck Operators. Advances in Neural Information Processing Systems 18, 955–962 (2005)Google Scholar
  13. 13.
    Perona, P., Malik, J.: Scale-Space and Edge Detection Using Anisotropic Diffusion. TPAMI 12(7), 629–639 (1990)CrossRefGoogle Scholar
  14. 14.
    Qiu, H., Hancock, E.R.: Clustering and Embedding Using Commute Times. TPAMI 29(11), 1873–1890 (2007)CrossRefGoogle Scholar
  15. 15.
    Tenenbaum, J.B., de Silva, V., Langford, J.C.: A Global Geometric Framework for Nonlinear Dimensionality Reduction. Scienc. 290, 2319–2323 (2010)CrossRefGoogle Scholar
  16. 16.
    Yen, L., Fouss, F., Decaestecker, C., Francq, P., Saerens, M.: Graph Nodes Clustering Based on the Commute-Time Kernel. In: Pacific-Asia Conference on Knowledge Discovery and Data Mining, pp. 1037–1045 (2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Eduard Sojka
    • 1
  • Jan Gaura
    • 1
  1. 1.Faculty of Electrical Engineering and Computer Science, Department of Computer ScienceVŠB - Technical University of OstravaOstrava-PorubaCzech Republic

Personalised recommendations