Spherical optimal transportation.

Optimal mass transportation (OT) problem aims at finding the most economic way to transform one probability measure to the other, which plays a fundamental role in many fields, such as computer graphics, computer vision, machine learning, geometry processing and medical imaging. Most existing algorithms focus on searching the optimal transportation map in Euclidean space, based on Kantorovich theory or Brenier theory. This work introduces a novel theoretic framework and computational algorithm to compute the optimal transportation map on the sphere. Constructing with a variational principle approach, our spherical OT map is carried out by solving a convex energy minimization problem and building a spherical power diagram.