Mesh Learning Using Persistent Homology on the Laplacian Eigenfunctions

TL;DR

This paper introduces a novel shape descriptor for triangulated 2-manifolds by combining persistent homology with Laplacian eigenfunctions, enabling effective shape similarity analysis.

Contribution

It proposes a new method that leverages persistent homology and Laplacian eigenfunctions to create shape descriptors for 2-manifolds, enhancing shape comparison techniques.

Findings

Effective shape similarity measurement demonstrated

Descriptors encode rich geometric information

Method utilizes accessible topological tools

Abstract

We use persistent homology along with the eigenfunctions of the Laplacian to study similarity amongst triangulated 2-manifolds. Our method relies on studying the lower-star filtration induced by the eigenfunctions of the Laplacian. This gives us a shape descriptor that inherits the rich information encoded in the eigenfunctions of the Laplacian. Moreover, the similarity between these descriptors can be easily computed using tools that are readily available in Topological Data Analysis. We provide experiments to illustrate the effectiveness of the proposed method.