CGV: Small: Scalable high-quality surface parameterization and resampling

有效地处理几何图形对于许多领域至关重要，包括计算机辅助设计、科学计算、城市规划和文化遗产保护，仅举几例。因此，生成和存储的几何数据量正在迅速增加。原始几何数据有多种形式(例如，距离图像、激光雷达数据和体数据)，最常见的是转换为非结构化网格，这些网格具有许多吸引人的特征，但固有地限制了几何处理、曲面数据处理和物理模拟算法的效率和精度。在这些情况下，将几何体转换为类似图像的规则采样形式已被证明具有显著的优势。网格参数化是用于对结构化网格上的曲面进行重采样和将数据映射到曲面的基本技术。全局参数化算法可以生成具有四边形区域的任意曲面的无缝拼接，从而可以在不包括孤立点的任何地方进行规则的重采样。这一领域的不断进步导致了越来越可靠和高质量的算法，但还没有完全健壮的自动方法可用。PI在这个项目中的目标是解决这个问题，并开发由严格的分析支持的全局参数化的基本算法，以及这些算法的健壮和可扩展的实现。具体地说，他的目标是解决以下密切相关的问题：质量(优化适当选择的失真度量，并明确保证局部失真)；稳健性(直接参数化和重采样可能有噪声的几何数据，避免中间网格表示，并保证得到的参数化结构)；以及可伸缩性和效率(使具有数亿个采样点的网格能够在台式计算机上处理)。尽管和平倡议将在该领域最近取得的重大进展的基础上再接再厉，但要实现所述目标，将需要重新思考问题的一些核心方面。更广泛的影响：这项研究不仅将导致实用和高效的算法的开发，而且将促进我们对相关几何问题(例如，对失真和几何重采样的内在限制)的基本理解。PI计划开发的软件有可能实现几何处理中各种任务的新方法，如果可以假定有强大的参数化工具，则可以在结构网格和参数域中执行这些任务。人们对稳健和可伸缩的参数化技术的兴趣远远超出了几何处理领域，扩展到使用复杂几何模型的许多科学和工程领域(例如，复杂几何中的大规模流体流动模拟，这需要对从不同来源提取的曲面进行高阶近似)。

Processing geometry efficiently is crucial for many domains including computer-aided design, scientific computing, urban planning, and cultural heritage preservation, to name just a few. As a consequence, the amount of geometric data generated and stored is rapidly increasing. Raw geometric data comes in a variety of forms (e.g., range images, LIDAR data, and volumetric data), and most commonly is converted to unstructured meshes which enjoy many attractive features but inherently limit efficiency and accuracy of algorithms for geometric processing, processing data on surfaces and physical simulation. Conversion of geometry to an image-like, regularly sampled form has been demonstrated to have significant advantages in these contexts. Mesh parameterization is the fundamental technique used both for resampling surfaces on structured grids and for mapping data to surfaces. Global parameterization algorithms can generate seamless tilings of arbitrary surfaces with quadrilateral domains enabling regular resampling everywhere excluding isolated points. The continuing progress in this area has led to increasingly reliable and high-quality algorithms, yet no fully robust automatic method is available yet. The PI's goal in this project is to tackle this problem and to develop fundamental algorithms for global parameterization supported by rigorous analysis, as well as robust and scalable implementations of these algorithms. Specifically, he aims to address the following closely interconnected problems: Quality (optimization of suitably chosen distortion measures, and explicit guarantees on local distortion); Robustness (direct parameterization and resampling of possibly noisy geometric data avoiding intermediate mesh representations, with guarantees on the resulting parameterization structure); and Scalability and Efficiency (to enable meshes with hundreds of millions of sample points to be processed on a desktop computer). Although the PI will build on the substantial recent advances in the area, achieving the stated goals will require rethinking some of the core aspects of the problem. Broader Impacts: This research will lead not only to the development of practical and efficient algorithms, but also to advances in our fundamental understanding of related geometric problems (for example, intrinsic limitations on distortion and geometric resampling). The software the PI plans to develop has the potential to enable new approaches to a variety of tasks in geometric processing, which can be performed on structured grids and in the parametric domain if a robust tool for parameterization can be assumed to be available. The interest in robust and scalable parameterization techniques extends far beyond the domain of geometric processing, to many domains of science and engineering where complex geometric models are used (e.g., large-scale fluid flow simulation in complex geometry, which requires high-order approximation of surfaces extracted from different sources).