# Skeletonisation Algorithms with Theoretical Guarantees for Unorganised Point Clouds with High Levels of Noise

## TL;DR

This paper evaluates algorithms for skeletonising noisy, unorganised point clouds, providing theoretical guarantees for shape reconstruction, including a universal method called HoPeS that adapts across scales.

## Contribution

It introduces a homologically persistent skeleton (HoPeS) algorithm with guarantees for accurate shape reconstruction from noisy point clouds, without requiring extra parameters.

## Key findings

- HoPeS is effective across various noise levels.
- Algorithms can reconstruct underlying shapes with correct homotopy type.
- Maximum noise levels for successful reconstruction are identified.

## Abstract

Data Science aims to extract meaningful knowledge from unorganised data. Real datasets usually come in the form of a cloud of points with only pairwise distances. Numerous applications require to visualise an overall shape of a noisy cloud of points sampled from a non-linear object that is more complicated than a union of disjoint clusters. The skeletonisation problem in its hardest form is to find a 1-dimensional skeleton that correctly represents a shape of the cloud. This paper compares several algorithms that solve the above skeletonisation problem for any point cloud and guarantee a successful reconstruction. For example, given a highly noisy point sample of an unknown underlying graph, a reconstructed skeleton should be geometrically close and homotopy equivalent to (has the same number of independent cycles as) the underlying graph. One of these algorithm produces a Homologically Persistent Skeleton (HoPeS) for any cloud without extra parameters. This universal skeleton contains sub-graphs that provably represent the 1-dimensional shape of the cloud at any scale. Other subgraphs of HoPeS reconstruct an unknown graph from its noisy point sample with a correct homotopy type and within a small offset of the sample. The extensive experiments on synthetic and real data reveal for the first time the maximum level of noise that allows successful graph reconstructions.