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论文类型：期刊论文

发表时间：2019-10-01

发表刊物：COMPUTER-AIDED DESIGN

收录刊物：SCIE、EI

卷号：115

页面范围：181-193

ISSN号：0010-4485

关键字：Optimal transport; Area-preserving mapping; Spherical geometry; Surface parameterization

摘要：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. (C) 2019 Elsevier Ltd. All rights reserved.