Abstract
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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Carlsson, G.E.: Topology and data. Bulletin of the American Mathematical Society 46, 255–308 (2009)
Caroli, M., Teillaud, M.: Computing 3D periodic triangulations. In: Proceedings of the European Symposium on Algorithms, pp. 59–70 (2009)
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)
Edelsbrunner, H.: Alpha shapes — a survey. Tessellations in the Sciences (2011)
Edelsbrunner, H., Harer, J.L.: Computational Topology. An Introduction. American Mathematical Society (2010)
Edelsbrunner, H., Morozov, D.: Persistent homology: theory and practice. In: Proceedings of the European Congress of Mathematics, pp. 31–50 (2012)
Illian, J., Penttinen, A., Stoyan, H., Stoyan, D.: Statistical Analysis and Modelling of Spatial Point Patterns. Wiley Interscience (2008)
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)
Kirkpatrick, S., Gelatt, C.D., Vecchi, M.P.: Optimization by simulated annealing. Science 220(4598), 671–680 (1983)
Kruithof, N.: 2D periodic triangulations. CGAL User and Reference Manual (2009)
Kuhn, H.W.: The Hungarian method for the assignment problem. Naval Research Logistics 2(1-2), 83–97 (1955)
Lagae, A., Dutre, P.: A comparison of methods for generating Poisson disk distributions. Computer Graphics Forum 27(1), 114–129 (2008)
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)
Oztireli, A.C., Gross, M.: Analysis and synthesis of point distributions based on pair correlation. Transactions on Graphics 31(6), 170 (2012)
Schlomer, T., Deussen, O.: Accurate spectral analysis of two-dimensional point sets. Journal of Graphics, GPU, and Game Tools 15(3), 152–160 (2011)
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)
Wilkinson, L., Anand, A., Grossman, R.: Graph-theoretic scagnostics. In: Proceedings of the Symposium on Information Visualization, pp. 157–164 (2005)
Yanoff, M., Duker, J.S.: Ophthalmology. Elsevier (2009)
Yellott, J.I.: Spectral consequences of photoreceptor sampling in the rhesus retina. Science 221(4608), 382–385 (1983)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Gu, C., Guibas, L., Kerber, M. (2014). Topology-Driven Trajectory Synthesis with an Example on Retinal Cell Motions. In: Brown, D., Morgenstern, B. (eds) Algorithms in Bioinformatics. WABI 2014. Lecture Notes in Computer Science(), vol 8701. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44753-6_24
Download citation
DOI: https://doi.org/10.1007/978-3-662-44753-6_24
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-44752-9
Online ISBN: 978-3-662-44753-6
eBook Packages: Computer ScienceComputer Science (R0)