Advertisement

A Discrete Scale Space Neighborhood for Robust Deep Structure Extraction

  • Martin Tschirsich
  • Arjan Kuijper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7626)

Abstract

Linear or Gaussian scale space is a well known multi-scale representation for continuous signals. The exploration of its so-called deep structure by tracing critical points over scale has various theoretical applications and allows for the construction of a scale space hierarchy tree. However, implementational issues arise, caused by discretization and quantization errors. In order to develop more robust scale space based algorithms, the discrete nature of computer processed signals has to be taken into account. Aiming at a computationally practicable implementation of the discrete scale space framework, we investigated suitable neighborhoods, boundary conditions and sampling methods. We show that the resulting discrete scale space respects important topological invariants such as the Euler number, a key criterion for the successful implementation of algorithms operating on its deep structure. We discuss promising properties of topological graphs under the influence of smoothing, setting the stage for more robust deep structure extraction algorithms.

Keywords

Deep Structure Scale Space Euler Number Topological Graph Discrete Signal 
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.

References

  1. 1.
    Kanters, F., Florack, L., Duits, R., Platel, B., ter Haar Romeny, B.: Scalespaceviz: a-scale spaces in practice. Pattern Recognition and Image Analysis 17, 106–116 (2007)CrossRefGoogle Scholar
  2. 2.
    Lindeberg, T.: Discrete Scale-Space Theory and the Scale-Space Primal Sketch. PhD thesis, Royal Institute of Technology (1991)Google Scholar
  3. 3.
    Tschirsich, M.: The discrete scale space as a base for robust scale space algorithms. Technical report, Department of Computer Science, Technical University of Darmstadt (June 2012)Google Scholar
  4. 4.
    Scott, P.J.: An algorithm to extract critical points from lattice height data. International Journal of Machine Tools and Manufacture 41(13-14), 1889–1897 (2001)CrossRefGoogle Scholar
  5. 5.
    Takahashi, S., Ikeda, T., Shinagawa, Y., Kunii, T.L., Ueda, M.: Algorithms for extracting correct critical points and constructing topological graphs from discrete geographical elevation data. Computer Graphics Forum 14(3), 181–192 (1995)CrossRefGoogle Scholar
  6. 6.
    Perez, A.: Determining the genus of a graph. HC Mathematics Review 1(2), 4–13 (2007)Google Scholar
  7. 7.
    Kovalevsky, V.A.: Discrete topology and contour definition. Pattern Recognition Letters 2(5), 281–288 (1984)CrossRefGoogle Scholar
  8. 8.
    Kuijper, A.: On detecting all saddle points in 2d images. Pattern Recognition Letters 25(15), 1665–1672 (2004)CrossRefGoogle Scholar
  9. 9.
    Koenderink, J.J.: The structure of images. Biological Cybernetics 50, 363–370 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Weickert, J., Ishikawa, S., Imiya, A.: Scale-space has been discovered in japan. Technical report, Department of Computer Science, University of Copenhagen (August 1997)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Martin Tschirsich
    • 1
  • Arjan Kuijper
    • 1
    • 2
  1. 1.Technische Universität DarmstadtGermany
  2. 2.Fraunhofer IGDDarmstadtGermany

Personalised recommendations