Geometric PDEs Based Methods for Analyzing Point Clouds in 3D and Higher

基于几何偏微分方程的 3D 及更高维点云分析方法

• 批准号: 1522645
• 负责人: Rongjie Lai
• 金额: $ 15.79万
• 依托单位: Rensselaer Polytechnic Institute
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-08-01 至 2019-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1522645&HistoricalAwards=false
• 关键词: Geometric; PDEs; Based; Methods; Analyzing

Recent advances of sensor and data collection techniques have led to "data deluge" from wide applications in science and engineering. In order for data scientists to make sense of this massive amount of data, it becomes increasingly important to develop new tools in both theoretical and computational point of views for data representation, analysis and mining. In practice, it is natural to represent data as a collection of points sampled from a low dimensional manifold in a high dimension space. In this scenario, intrinsic geometric information extraction is a crucial step to conduct high level structure understanding of data.The PI will investigate geometry and partial differential equation (PDE) based methods for analyzing point clouds data in 3D and higher dimension. One objective is to develop and improve numerical methods for solving PDEs on point clouds sampled from Riemannian manifolds. Another one is to utilize solutions of specially designed geometric PDEs on point clouds to conduct intrinsic and global understanding of data sets, such as topological information extraction and point clouds intrinsic comparisons. This investigation will lead to different viewpoints for tackling problems in intrinsic data analysis. The research also contributes to related areas in numerical methods for solving PDEs, computational differential geometry and non-convex optimization. By collaborating with computer scientists and biomedical engineers, applications to real problems will also be explored by the PI.

传感器和数据采集技术的最新进展导致了科学和工程中广泛应用的“数据泛滥”。为了让数据科学家能够理解如此大量的数据，从理论和计算的角度开发用于数据表示、分析和挖掘的新工具变得越来越重要。在实践中，将数据表示为从高维空间中的低维流形采样的点的集合是很自然的。在这种情况下，内在几何信息提取是对数据进行高层次结构理解的关键步骤。PI将研究基于几何和偏微分方程（PDE）的方法，用于分析3D和更高维度的点云数据。一个目标是发展和改进的数值方法求解点云采样黎曼流形上的偏微分方程。另一种是利用专门设计的点云上的几何偏微分方程的解决方案进行内在的和全局的数据集的理解，如拓扑信息提取和点云内在的比较。这一研究将导致不同的观点来解决问题的内在数据分析。该研究也有助于相关领域的数值方法求解偏微分方程，计算微分几何和非凸优化。通过与计算机科学家和生物医学工程师合作，PI还将探索真实的问题的应用。