  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.
