  We describe a method to obtain spherical parameterizations of arbitrary data
through the use of persistent cohomology and variational optimization. We begin
by computing the second-degree persistent cohomology of the filtered
Vietoris-Rips (VR) complex of a data set $X$ and extract a cocycle $\alpha$
from any significant feature. From this cocycle, we define an associated map
$\alpha: VR(X) \to S^2$ and use this map as an infeasible initialization for a
variational model, which we show has a unique solution (up to rigid motion). We
then employ an alternating gradient descent/M\"{o}bius transformation update
method to solve the problem and generate a more suitable, i.e., smoother,
representative of the homotopy class of $\alpha$, preserving the relevant
topological feature. Finally, we conduct numerical experiments on both
synthetic and real-world data sets to show the efficacy of our proposed
approach.
