A fast approximate skeleton with guarantees for any cloud of points in a Euclidean space

TL;DR

This paper introduces a fast, guaranteed approximation algorithm for reconstructing a minimal-vertex tree (skeleton) from any point cloud in Euclidean space, with applications in material science and data analysis.

Contribution

It presents a novel Approximate Skeleton algorithm with theoretical guarantees, near-linear computation time, and improved accuracy over previous methods.

Findings

Algorithm computes skeletons close to optimal size.

Runs in near-linear time relative to point cloud size.

Outperforms previous methods on real and synthetic datasets.

Abstract

The tree reconstruction problem is to find an embedded straight-line tree that approximates a given cloud of unorganized points in $\mathbb{R}^m$ up to a certain error. A practical solution to this problem will accelerate a discovery of new colloidal products with desired physical properties such as viscosity. We define the Approximate Skeleton of any finite point cloud $C$ in a Euclidean space with theoretical guarantees. The Approximate Skeleton ASk$(C)$ always belongs to a given offset of $C$, i.e. the maximum distance from $C$ to ASk$(C)$ can be a given maximum error. The number of vertices in the Approximate Skeleton is close to the minimum number in an optimal tree by factor 2. The new Approximate Skeleton of any unorganized point cloud $C$ is computed in a near linear time in the number of points in $C$. Finally, the Approximate Skeleton outperforms past skeletonization algorithms on the size and accuracy of reconstruction for a large dataset of real micelles and random clouds.