Interactive Visualization of 2-D Persistence Modules

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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.