On the Space Between Critical Points

  • Walter G. KropatschEmail author
  • Rocio M. Casablanca
  • Darshan Batavia
  • Rocio Gonzalez-Diaz
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11414)


The vertices of the neighborhood graph of a digital picture P can be interpolated to form a 2-manifold M with critical points (maxima, minima, saddles), slopes and plateaus being the ones recognized by local binary patterns (LBPs). Neighborhood graph produces a cell decomposition of M: each 0-cell is a vertex in the neighborhood graph, each 1-cell is an edge in the neighborhood graph and, if P is well-composed, each 2-cell is a slope region in M in the sense that every pair of s in the region can be connected by a monotonically increasing or decreasing path. In our previous research, we produced superpixel hierarchies (combinatorial graph pyramids) that are multiresolution segmentations of the given picture. Critical points of P are preserved along the pyramid. Each level of the pyramid produces a slope complex which is a cell decomposition of M preserving critical points of P and such that each 2-cell is a slope region. Slope complexes in different levels of the pyramid are always homeomorphic. Our aim in this research is to explore the configuration at the top level of the pyramid which consists of a slope complex with vertices being only the critical points of P. We also study the number of slope regions on the top.


  1. 1.
    Gonzalez-Diaz, R., Ion, A., Iglesias-Ham, M., Kropatsch, W.G.: Invariant representative cocycles of cohomology generators using irregular graph pyramids. Comput. Vis. Image Underst. 115(7), 1011–1022 (2011)CrossRefGoogle Scholar
  2. 2.
    Cerman, M., Gonzalez-Diaz, R., Kropatsch, W.: LBP and irregular graph pyramids. In: Azzopardi, G., Petkov, N. (eds.) CAIP 2015. LNCS, vol. 9257, pp. 687–699. Springer, Cham (2015). Scholar
  3. 3.
    Cerman, M., Janusch, I., Gonzalez-Diaz, R., Kropatsch, W.G.: Topology-based image segmentation using LBP pyramids. Mach. Vis. Appl. 27(8), 1161–1174 (2016)CrossRefGoogle Scholar
  4. 4.
    Edelsbrunner, H., Harer, J., Zomorodian, A.: Hierarchical morse - smale complexes for piecewise linear 2-manifolds. Discrete Comput. Geom. 30(1), 87–107 (2003)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
  6. 6.
    Kropatsch, W.G.: Building irregular pyramids by dual-graph contraction. IEEE Proceedings on Vision Image and Signal Processing, vol. 142, no. 6, pp. 366–374 (1995)CrossRefGoogle Scholar
  7. 7.
    Kropatsch, W.G., Haxhimusa, Y., Pizlo, Z., Langs, G.: Vision pyramids that do not grow too high. Pattern Recognit. Lett. 26(3), 319–337 (2005)CrossRefGoogle Scholar
  8. 8.
    Latecki, L., Eckhardt, U., Rosenfeld, A.: Well-composed sets. Comput. Vis. Image Underst. 61, 70–83 (1995)CrossRefGoogle Scholar
  9. 9.
    Reinhard, D.: Graph Theory. Graduate Texts in Maths. Springer, Heidelberg (1997)zbMATHGoogle Scholar
  10. 10.
    Ojala, T., Pietikainen, M., Harwood, D.: A comparative study of texture measures with classification based on featured distributions. Pattern Recognit. 29(1), 51–59 (1996)CrossRefGoogle Scholar
  11. 11.
    Pietikäinen, M., Hadid, A., Zhao, G., Ahonen, T.: Computer Vision Using Local Binary Patterns, vol. 40. Springer, London (2011). Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Walter G. Kropatsch
    • 1
    Email author
  • Rocio M. Casablanca
    • 2
  • Darshan Batavia
    • 1
  • Rocio Gonzalez-Diaz
    • 2
  1. 1.Pattern Recognition and Image Processing Group 193/03TU WienViennaAustria
  2. 2.Applied Math IUniversity of SevilleSevilleSpain

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