Tracking Temporal Evolution of Topological Features in Image Data

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Abstract
Topological Data Analysis (TDA) can be used to detect and characterize holes in an image, such as zero-dimensional holes (connected components) or one-dimensional holes (loops). However, there is currently no widely accepted statistical framework for modeling spatiotemporal dependence in the evolution of topological features, such as holes, within a time series of images. We propose a hypothesis testing framework to identify statistically significant topological features of images in space and time, simultaneously. This addition of time may induce higher-dimensional topological features which can be used to establish temporal connections between the lower-dimensional features at each point in time. The temporal evolution of these lower-dimensional features is then represented on a zigzag persistence diagram, as a topological summary statistic focused on time dynamics. We demonstrate that the method effectively captures the emergence and progression of topological features in a study of a series of images of a wounded cell as it repairs. The proposed method outperforms a current approach in a simulation study that includes features of the wound healing process. Since, the wounded cell images exhibit nonlinear, dynamic, spatial, and temporal structures during single-cell repair, they provide a good application for this method.
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