10.6084/m9.figshare.5758542.v1
Chul Moon
Chul
Moon
Noah Giansiracusa
Noah
Giansiracusa
Nicole A. Lazar
Nicole
A. Lazar
Persistence Terrace for Topological Inference of Point Cloud Data
Taylor & Francis Group
2018
Density estimation
Morse theory
Persistent homology
Topological data analysis
2018-01-04 20:58:30
Dataset
https://tandf.figshare.com/articles/dataset/Persistence_Terrace_for_Topological_Inference_of_Point_Cloud_Data/5758542
<p>Topological data analysis (TDA) is a rapidly developing collection of methods for studying the shape of point cloud and other data types. One popular approach, designed to be robust to noise and outliers, is to first use a smoothing function to convert the point cloud into a manifold and then apply persistent homology to a Morse filtration. A significant challenge is that this smoothing process involves the choice of a parameter and persistent homology is highly sensitive to that choice; moreover, important scale information is lost. We propose a novel topological summary plot, called a persistence terrace, that incorporates a wide range of smoothing parameters and is robust, multi-scale, and parameter-free. This plot allows one to isolate distinct topological signals that may have merged for any fixed value of the smoothing parameter, and it also allows one to infer the size and point density of the topological features. We illustrate our method in some simple settings where noise is a serious issue for existing frameworks and then we apply it to a real dataset by counting muscle fibers in a cross-sectional image. Supplementary material for this article is available online.</p>