Topology-Driven Trajectory Synthesis with an Example on Retinal Cell Motions

  • Chen Gu
  • Leonidas Guibas
  • Michael Kerber
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8701)


We design a probabilistic trajectory synthesis algorithm for generating time-varying sequences of geometric configuration data. The algorithm takes a set of observed samples (each may come from a different trajectory) and simulates the dynamic evolution of the patterns in O(n 2 logn) time. To synthesize geometric configurations with indistinct identities, we use the pair correlation function to summarize point distribution, and α-shapes to maintain topological shape features based on a fast persistence matching approach. We apply our method to build a computational model for the geometric transformation of the cone mosaic in retinitis pigmentosa — an inherited and currently untreatable retinal degeneration.


trajectory pair correlation function alpha shapes persistent homology retinitis pigmentosa 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Carlsson, G.E.: Topology and data. Bulletin of the American Mathematical Society 46, 255–308 (2009)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Caroli, M., Teillaud, M.: Computing 3D periodic triangulations. In: Proceedings of the European Symposium on Algorithms, pp. 59–70 (2009)Google Scholar
  3. 3.
    Devillers, O., Meiser, S., Teillaud, M.: Fully dynamic Delaunay triangulation in logarithmic expected time per operation. Computational Geometry: Theory and Applications 2(2), 55–80 (1992)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Edelsbrunner, H.: Alpha shapes — a survey. Tessellations in the Sciences (2011)Google Scholar
  5. 5.
    Edelsbrunner, H., Harer, J.L.: Computational Topology. An Introduction. American Mathematical Society (2010)Google Scholar
  6. 6.
    Edelsbrunner, H., Morozov, D.: Persistent homology: theory and practice. In: Proceedings of the European Congress of Mathematics, pp. 31–50 (2012)Google Scholar
  7. 7.
    Illian, J., Penttinen, A., Stoyan, H., Stoyan, D.: Statistical Analysis and Modelling of Spatial Point Patterns. Wiley Interscience (2008)Google Scholar
  8. 8.
    Ji, Y., Zhu, C.L., Grzywacz, N.M., Lee, E.-J.: Rearrangement of the cone mosaic in the retina of the rat model of retinitis pigmentosa. The Journal of Comparative Neurology 520(4), 874–888 (2012)CrossRefGoogle Scholar
  9. 9.
    Kirkpatrick, S., Gelatt, C.D., Vecchi, M.P.: Optimization by simulated annealing. Science 220(4598), 671–680 (1983)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Kruithof, N.: 2D periodic triangulations. CGAL User and Reference Manual (2009)Google Scholar
  11. 11.
    Kuhn, H.W.: The Hungarian method for the assignment problem. Naval Research Logistics 2(1-2), 83–97 (1955)CrossRefGoogle Scholar
  12. 12.
    Lagae, A., Dutre, P.: A comparison of methods for generating Poisson disk distributions. Computer Graphics Forum 27(1), 114–129 (2008)CrossRefGoogle Scholar
  13. 13.
    Lee, E.-J., Ji, Y., Zhu, C.L., Grzywacz, N.M.: Role of muller cells in cone mosaic rearrangement in a rat model of retinitis pigmentosa. Glia 59(7), 1107–1117 (2011)CrossRefGoogle Scholar
  14. 14.
    Oztireli, A.C., Gross, M.: Analysis and synthesis of point distributions based on pair correlation. Transactions on Graphics 31(6), 170 (2012)CrossRefGoogle Scholar
  15. 15.
    Schlomer, T., Deussen, O.: Accurate spectral analysis of two-dimensional point sets. Journal of Graphics, GPU, and Game Tools 15(3), 152–160 (2011)CrossRefGoogle Scholar
  16. 16.
    Vicsek, T., Czirok, A., Ben-Jacob, E., Cohen, I., Shochet, O.: Novel type of phase transition in a system of self-driven particles. Physical Review Letters 75(6), 1226–1229 (1995)CrossRefGoogle Scholar
  17. 17.
    Wilkinson, L., Anand, A., Grossman, R.: Graph-theoretic scagnostics. In: Proceedings of the Symposium on Information Visualization, pp. 157–164 (2005)Google Scholar
  18. 18.
    Yanoff, M., Duker, J.S.: Ophthalmology. Elsevier (2009)Google Scholar
  19. 19.
    Yellott, J.I.: Spectral consequences of photoreceptor sampling in the rhesus retina. Science 221(4608), 382–385 (1983)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Chen Gu
    • 1
  • Leonidas Guibas
    • 2
  • Michael Kerber
    • 3
  1. 1.Google Inc.USA
  2. 2.Stanford UniversityUSA
  3. 3.Max-Planck-Institute for InformaticsGermany

Personalised recommendations