几何映射计算的理论与应用研究

Geometric mapping computation is a fundamental problem in computer graphics and digital geometry processing. It is the core of many problems and applications, such as the triangular mesh parameterization, the real walking problem in virtual reality. It has a very broad application prospects. These problems and applications usually require the mappings to be foldover-free and with low distortion. However, these two constraints are highly non-linear and non-convex, thus it is difficult to efficiently and effectively solve. To this end, this project will first conduct an in-depth study of the theory and algorithms of geometric mapping computations, including theoretical conditions for ensuring no foldovers and more efficient practical algorithms. Then, the project will apply the geometric mapping in various applications, such as mesh parameterizations, shape correspondence, domain parametrization in iso-geometric analysis, and virtual scene mapping. It would stimulate development in various 3D modeling and processing algorithms. Finally, the applicant will also actively cooperate with domestic and foreign experts to explore the broader application prospect of geometric mapping and promote the intersection and development of computer graphics, computer vision, virtual / augmented reality and artificial intelligence.

几何映射计算是计算机图形学和数字几何处理中的一个基础问题。它是很多问题与应用的核心，比如三角网格参数化、虚拟现实中的实际行走问题，具有非常广阔的应用前景。这些问题与应用通常要求映射满足无翻转、低形变的约束。然而这两个约束是高度非线性的且非凸的，于是增加了求解的难度和降低了求解的效率。为此，本项目首先将深入研究几何映射计算的理论和算法，包括保证无翻转的理论条件、更高效的映射求解算法。然后，项目将发展在网格参数化、三维形状对应、等几何分析中的区域参数化、虚拟场景映射等问题的应用，从而进一步探索和发展几何映射在三维形状建模和处理中的各种算法。最后，申请人也将积极同国内外专家开展合作，探索几何映射更广阔的应用前景，促进计算机图形学、计算机视觉、虚拟/增强现实和人工智能的交叉和发展。

几何映射计算是计算机图形学和数字几何处理中的一个基础问题。它是很多问题与应用的核心，比如三角网格参数化、虚拟现实中的实际行走问题，具有非常广阔的应用前景。本项目主要研究几何映射计算的理论基础、高效数值优化算法，以及在几个典型问题上的应用。项目组取得发明专利一项，在学术期刊发表论文 20 篇，其中JCR一区期刊8篇，二区期刊4篇。提出了目前最快的双射参数化算法；提出了目前唯一能保证装配效率有界的算法；在科普、艺术方面，申请人提出了橘皮成形技术在多个科技活动周进行科普宣传；在动画制作方面，提出了高质量相容性网格生成算法；将几何映射理论应用到网格生成、虚拟现实中的基本问题中。研究成果及相关论文引起了学界和业界的较大关注，部分成果已经进入实际工业应用，产生巨大经济价值。