Papers › Spherical Coordinates from Persistent Cohomology
Spherical Coordinates from Persistent Cohomology
Nikolas C. Schonsheck, Stefan C. Schonsheck
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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 α from any significant feature. From this cocycle, we define an associated map α: VR(X) →S² 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 α, 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.
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