Shape and data analysis using computational differential geometry

由于现代技术、实验和计算机模拟的快速和大量的数据采集，计算和数据分析方法在科学和工程中的许多重要应用中变得越来越重要。数据表示、分析和理解的一个重要而又具有挑战性的任务是从大数据集中提取感兴趣的信息，这些信息包括内在的几何特征和全局结构。本项目旨在引入微分几何中先进的数学理论和分析工具，并将其转化为有效的数据分析计算算法。该项目的主要工作是为一些重要但具有挑战性的任务开发新的数学模型和计算工具，用于三维建模和高维数据分析。这些模型和工具将用于在各种应用中提取和表征数据分析的几何结构。特别地，拟共形映射和共形结构将用于表面映射和形状建模的表示和分析。本课程将探讨点云分析和高维流形学习的内在几何微分算子，如拉普拉斯-贝尔特拉米算子及其特征系统。本项目开发的模型、方法和计算工具将在基准数据集和实际应用中进行测试。跨学科的应用和与计算机科学家和统计学家的合作也将继续进行。

Due to rapid and vast data acquisition by modern technology, experiments and computer simulations, computational and data analytical approaches have become increasingly important for many important applications in science and engineering. An important but challenging task in data representation, analysis and understanding is to extract information of interest, which includes intrinsic geometric features and global structures from large data set. The aim of this project is to introduce advanced mathematical theories and analytical tools in differential geometry and translate them into efficient computation algorithms for data analysis.The main effort of this project is to develop new mathematical models and computational tools for a few important but challenging tasks in 3D modeling and data analysis in higher dimensions. These models and tools will be utilized to extract and characterize geometric structures for data analysis in various applications. In particular, quasi-conformal map and conformal structure will be used for representation and analysis for surface maps and shape modeling. Intrinsic geometric differential operators, such as Laplace-Beltrami operator and its eigen-system, will be explored for point cloud analysis and manifold learning in high dimensions. Models, methods and computational tools developed in this project will be tested on benchmark data sets as well as real applications. Interdisciplinary applications and collaborations with computer scientists and statisticians will also be pursued.