基于离散最优传输的网格保测参数化研究

As a fundamental tool for digital geometric processing, mesh parameterization has found its widespread application background and great significance of theoretical research. The existing mesh parameterization methods mainly focus on how to minimize the geometric distortion caused by parameterization, but neglect the measure distortion in a more general sense. This project will do some research on the global measure-preserving parameterization methods for triangle meshes and tetrahedron solids. We will deeply and systematically study the theories, the pivotal techniques as well as the applications of measure preserving parameterization, so that we can build a complete set of theories and algorithms of global measure preserving parameterization, which is credible for models of any topology. The overall research idea of measure preserving parameterization is as follows: firstly, use Ricci curvature flow to push forward the measures to certain constant curvature parameter domain determined by its mesh topology; then refined the measure distortion via discrete optimal mass transport, making them consistent with the target ones. The cores are discrete Ricci flow and discrete optimal mass transport on all kinds of constant curvature space. Concretely, this project will be carried out in four aspects: discrete optimal mass transport theory on constant curvature space, measure preserving parameterization for triangle meshes, measure preserving parameterization for tetrahedron solids, as well as the approach for generating measure preserving parameterization sequence clusters. Furthermore, basing on our research, we will establish the significant theoretical and algorithmic foundation of measure preserving parameterization methods for meshes, and open up some new research directions with wide application prospects.

网格参数化是数字几何处理的基本工具，具有广泛的应用背景和重要的理论研究价值。现有的网格参数化主要关注于如何减少参数化的几何畸变，而忽略更一般意义上的测度畸变。本课题将研究保测度的面网格和体网格的全局参数化方法，在保测参数化的理论、关键技术以及应用等方面展开深入、系统的研究，建立整套覆盖全部拓扑系列的保测度的全局参数化的理论和算法。保测参数化的总体研究思路是先用Ricci曲率流将测度前推到某个由网格拓扑性质决定的常曲率参数域上，然后在该常曲率参数域上通过离散最优传输对测度进行修正，使其和目标测度完全一致。其研究重点是各种常曲率空间的离散Ricci曲率流和最优传输算法，具体研究内容包括：常曲率空间上的最优传输理论、面网格的保测参数化方法、体网格的保测参数化方法、测度控制的参数化序列簇的生成方法。同时，在网格的保测参数化的理论和算法基础上开辟一些全新的有广泛应用前景的研究方向。

基于最优传输，我们对网格的保测参数化展开深入、系统的研究。在最优传输的理论方面，我们构建了最优传输的凸几何理论。在最优传输的算法方面，我们提出了离散最优传输的几何变分算法和基于GPU渲染的离散最优传输算法。在网格的保测参数化方面，我们构建网格保测参数化的通用模型，提出了多种网格参数化方法，具体包括基于卡拉比流、李导数、法向流和单位法丛的参数化方法。在测度控制的参数化序列簇方面，我们提出了通过操控测度建立了测度可控的参数化序列簇的算法框架，通过操控三角形网格的面元、体元、曲率来实现对参数化结果的自由控制来构建参数化序列簇。另外，我们还基于最优传输对深度学习给出了理论解释，提出了生成对抗网络(GAN)的最优传输(OT)观点。研究取得了具有一定影响力的研究成果，发表了15篇论文，获批3项国家发明专利，培养了一批优秀的研究生。在成果应用方面，我们与武汉市三医院合作在智能烧伤智能诊断中得到应用，与华为黎曼实验室合作在三维场景纹理优化中得到应用。