Title: Challenges in Topological Object Data Analysis

Author(s): Daniel Osborne, Peter Bubenik, Robert L. Paige and Vic Patrangenaru
Issue: Volume 81 Series A Part 1 Year 2019
Pages: 244 -- 271
Statistical analysis on object data presents many challenges. Basic summaries such as means and variances are difficult to compute. We apply ideas from topology to study object data. We present a framework for using death vectors and persistence landscapes to vectorize object data and perform statistical analysis. We apply this method to some common leaf images that were previously shown to be challenging to compare using a 3D shape techniques. Surprisingly, the most persistent features are shown to be “topological noise” and the statistical analysis depends on the less persistent features which we refer to as the “geometric signal”. We also describe the first steps to a new approach to using topology for object data analysis, which applies topology to distributions on object spaces. We introduce a new Fr´echet- Morse function technique for probability distribution on a compact object space, extending the Fr´echet means lo a larger number of location parameters, including Fr´echet antimeans. An example of 3D data analysis to distinguish two flowers using the new location parameters associated with a Veronese- Whitney (VW) embedding of random projective shapes of 3D configurations extracted from a set of pairs of their digital camera images is also given here.
Primary: 62H35, 62G10, 55N05; Secondary: 57R45, 55N99, 62G99.
Keywords and phrases: VW-means of index r, Topological data analysis, Relative homology, Object data analysis, Persistence landscapes