Estimating Temporal Evolution of Topological Features in Image Data

Music City Center 

One popular technique in Topological Data Analysis (TDA) called persistent homology (PH) is used to describe holes in an image through their dimension and the functional values (e.g., thresholds or scales) at which they are created and filled. While TDA has been successfully applied to identifying shape in a static image through its hole structure, estimating the changes in that hole structure within a time-evolving image set is relatively understudied. We develop a method which first identifies statistically significant topological features in the spatial and temporal dimensions simultaneously. These higher-dimensional topological features are then used to establish temporal connections between the lower-dimensional features they are built from, effectively separating spatial connections from temporal connections. The spatial structure of the lower-dimensional features can be analyzed at each time-point separately and their temporal evolution represented on a ZigZag diagram (topological summary statistic focused on time dynamics). The method's effectiveness in capturing the emergence and progression of topological features is tested on a time series of images from cell wounds.

Image Processing

, Spatiotemporal Analysis

Topological Data Analysis

High Dimensional Statistics 

Section on Statistics in Imaging