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Point Clouds and Heatmaps: A Practical Approach to Multidimensional Persistent Homology for Robust Shape Recognition

Banff International Research Station for Mathematical Innovation and Discovery

Persistent homology is a technique in algebraic and computational topology useful in recovering the underlying topology of a given dataset, and has a wide range of applications in computer vision, statistics, genomics, and beyond. This technique suffers, however, through its inability to simultaneously consider multiple parameters describing a dataset—a mathematically difficult and unsolved problem. In particular, this prevents persistent homology from distinguishing between distinct yet topologically-equivalent shapes, such as circles and ellipses, that could otherwise be differentiated by examining both scale and curvature. In this presentation, we put forth a novel extension of persistent homology to two parameters, which we call Heatmap Pseudo-Bifiltration. Furthermore, we develop a robust statistical test to detect differences between point-cloud datasets on the basis of scale, curvature, and topological structure. The effect of sampling variability and noise on the results of this technique will be examined, particularly in the context of point-cloud curvature estimation in arbitrary dimensions.

Mathematics; Statistics

video/mp4

Author affiliation: University of Alberta

2017-03-05

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/60795

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/