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

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.

[1]  J. Yellott.  Spectral consequences of photoreceptor sampling in the rhesus retina. , 1983, Science.

[2]  C. D. Gelatt,et al.  Optimization by Simulated Annealing , 1983, Science.

[3]  Olivier Devillers,et al.  Fully Dynamic Delaunay Triangulation in Logarithmic Expected Time Per Operation , 1992, Comput. Geom..

[4]  Vicsek,et al.  Novel type of phase transition in a system of self-driven particles. , 1995, Physical review letters.

[5]  Alexander Russell,et al.  Computational topology: ambient isotopic approximation of 2-manifolds , 2003, Theor. Comput. Sci..

[6]  R. Grossman,et al.  Graph-theoretic scagnostics , 2005, IEEE Symposium on Information Visualization, 2005. INFOVIS 2005..

[7]  D. Stoyan,et al.  Statistical Analysis and Modelling of Spatial Point Patterns , 2008 .

[8]  Ares Lagae,et al.  A Comparison of Methods for Generating Poisson Disk Distributions , 2008, Comput. Graph. Forum.

[9]  H. Edelsbrunner.  Alpha Shapes — a Survey , 2009 .

[10]  Monique Teillaud,et al.  Computing 3D Periodic Triangulations , 2009, ESA.

[11]  Gunnar E. Carlsson,et al.  Topology and data , 2009 .

[12]  Herbert Edelsbrunner,et al.  Computational Topology - an Introduction , 2009 .

[13]  Harold W. Kuhn,et al.  The Hungarian method for the assignment problem , 1955, 50 Years of Integer Programming.

[14]  N. Grzywacz,et al.  Role of Müller cells in cone mosaic rearrangement in a rat model of retinitis pigmentosa , 2011, Glia.

[15]  Oliver Deussen,et al.  Accurate Spectral Analysis of Two-Dimensional Point Sets , 2011, J. Graphics, GPU, & Game Tools.

[16]  N. Grzywacz,et al.  Rearrangement of the cone mosaic in the retina of the rat model of retinitis pigmentosa , 2012, The Journal of comparative neurology.

[17]  Markus Gross,et al.  Analysis and synthesis of point distributions based on pair correlation , 2012, ACM Trans. Graph..

[18]  Herbert Edelsbrunner,et al.  Persistent Homology: Theory and Practice , 2013 .