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 log n) time. To synthesize geometric configurations with indistinct identities, we use the pair correlation function to summarize point distribution, and alpha-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.