Spherical optimal transportation. (October 2019)

• Record Type: Journal Article
• Authors: Cui, Li; Qi, Xin; Wen, Chengfeng; Lei, Na; Li, Xinyuan; Zhang, Min; Gu, Xianfeng
• Is Part Of: Computer aided design. Volume 115(2019)
• Page Start: 181
• Page End: 193
• Publication Date: 2019-10
• Subjects: Optimal transport -- Area-preserving mapping -- Spherical geometry -- Surface parameterization; Computer-aided design -- Periodicals; Engineering design -- Data processing -- Periodicals; Computer graphics -- Periodicals; Conception technique -- Informatique -- Périodiques; Infographie -- Périodiques; Computer graphics; Engineering design -- Data processing; Periodicals; Electronic journals; 620.00420285
• DOI: 10.1016/j.cad.2019.05.024
• Languages: English
• ISSNs: 0010-4485

Abstract

Abstract: 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. In theory, we prove the existence and the uniqueness of the spherical optimal transportation map; in practice, we present an efficient algorithm using the variational framework and Newton's method. Comparing to the existing approaches, this work is more rigorous, efficient, robust and intrinsic to the spherical geometry. It can be generalized to the hyperbolic geometry or to higher dimensions. Our experimental results on a variety of models demonstrate efficacy and efficiency of the proposed method. At the same time, our method generates diffeomorphic, area-preserving, and seamless spherical parameterization results. Highlights: Spherical optimal transport theory is introduced via spherical power diagram. The existence and the uniqueness of the solutions is proved. Area preserving mapping is constructed from topological spheres to unit spheres. The mapping is diffeomorphic and unique under normalization.