Interactive Visualization of 2-D Persistence Modules

TL;DR

This paper introduces RIVET, a software tool that enables interactive visualization of 2-D persistence modules, extending persistent homology analysis to higher dimensions with efficient computation and novel data structures.

Contribution

We develop RIVET, a practical tool for visualizing 2-D persistence modules, including mathematical foundations, an efficient data structure, and algorithms for fast slice analysis.

Findings

RIVET allows interactive visualization of 2-D persistence modules.

The tool computes and visualizes Betti numbers and vector space dimensions.

A novel data structure based on planar line arrangements enables fast queries.

Abstract

The goal of this work is to extend the standard persistent homology pipeline for exploratory data analysis to the 2-D persistence setting, in a practical, computationally efficient way. To this end, we introduce RIVET, a software tool for the visualization of 2-D persistence modules, and present mathematical foundations for this tool. RIVET provides an interactive visualization of the barcodes of 1-D affine slices of a 2-D persistence module $M$. It also computes and visualizes the dimension of each vector space in $M$ and the bigraded Betti numbers of $M$. At the heart of our computational approach is a novel data structure based on planar line arrangements, on which we can perform fast queries to find the barcode of any slice of $M$. We present an efficient algorithm for constructing this data structure and establish bounds on its complexity.