Solving Partial Differential Equations on Point Clouds and Geometric Understanding of Point Clouds.

摘要

Point cloud is defined simply as a set of unstructured points with no specific ordering and connection. Point cloud is the most basic and intrinsic way for sampling and representation of geometric objects or information in high dimensions. There are several basic problems associated to point clouds including the likes of segmentation, visualization, surface reconstruction and geometric understanding. Point cloud processing is becoming more and more popular, and has many applications in computer vision, data science, manifold learning, etc. In this dissertation, several basic point cloud processing problems will be studied. First, we develop a constrained nonlinear least squares approach for point cloud normal estimate, and we extend this strategy to point cloud denoising and segmentation. Second, we propose novel ways to utilize convexified image segmentation models and fast computational algorithms to achieve implicit surface reconstruction directly from point cloud. Third, we develop a general framework for solving partial differential equations on manifold represented by point cloud, without parametrization or connection information, only based on a local approximation of manifold. Finally, we use the framework for geometric understanding on point clouds, including computation of Laplace-Beltrami eigenvalues and eigenfunctions, extraction of skeletons and extraction of conformal structures. Various examples in each chapter show that our approaches are accurate, robust and efficient.