Spherical optimal transportation
Spherical optimal transportation
复制标题
球形最优交通
DOI：
10.1016/j.cad.2019.05.024
复制
发表时间：
2019-10-01
影响因子：
4.3
通讯作者：
Gu, Xianfeng
中科院分区：
文献类型：
--
作者：
Cui, Li;Qi, Xin;Gu, Xianfeng
最优交通问题（Optimized mass transport，OT）是一个以最经济的方式将一种概率测度转换为另一种概率测度的问题，在计算机图形学、计算机视觉、机器学习、几何处理和医学成像等领域具有重要的应用价值。现有的算法大多是基于Kantorovich理论或Brenier理论，在欧氏空间中搜索最优交通地图。这项工作引入了一种新颖的理论框架和计算算法来计算球体上的最优交通地图。利用变分原理的方法，通过求解一个凸能量最小化问题和构造一个球面幂图来构造球面OT映射，在理论上证明了球面最优运输映射的存在唯一性，在实践中利用变分框架和牛顿法给出了一个有效的算法.与现有的方法相比，该方法更加严格、高效、鲁棒，并且是球面几何的固有特征。该方法可以推广到双曲几何或更高维的情形，对多种模型的实验结果表明了该方法的有效性。与此同时，我们的方法生成的几何纯，面积保持，和无缝的球形参数化结果。(C)2019爱思唯尔有限公司版权所有。
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.