Intrinsic Hierarchical Clustering Behavior Recovers Higher Dimensional Shape Information

TL;DR

This paper demonstrates that analyzing hierarchical clustering dynamics of point cloud samples can recover higher-dimensional shape information, linking clustering behavior to topological shape features.

Contribution

It introduces a method to extract higher-dimensional shape information from point clouds by using hierarchical clustering and persistence diagrams, a novel approach in shape analysis.

Findings

Hierarchical clustering captures higher-dimensional shape features.

Bottleneck distance distributions reveal shape artifacts.

Method effectively recovers shape information from point cloud data.

Abstract

We show that specific higher dimensional shape information of point cloud data can be recovered by observing lower dimensional hierarchical clustering dynamics. We generate multiple point samples from point clouds and perform hierarchical clustering within each sample to produce dendrograms. From these dendrograms, we take cluster evolution and merging data that capture clustering behavior to construct simplified diagrams that record the lifetime of clusters akin to what zero dimensional persistence diagrams do in topological data analysis. We compare differences between these diagrams using the bottleneck metric, and examine the resulting distribution. Finally, we show that statistical features drawn from these bottleneck distance distributions detect artefacts of, and can be tapped to recover higher dimensional shape characteristics.