Analysis of Global Properties of Shapes

With increasing amounts of data describing 3D geometry at scales small and large, shape analysis is becoming increasingly important in fields ranging from computer graphics to robotics to computational biology. While a great deal of research exists on local shape analysis, less work has been done on global shape analysis. This thesis aims to advance global shape analysis in three directions: symmetry-aware mesh processing, part decomposition of 3D models, and analysis of 3D scenes. First, we propose a pipeline for making mesh processing algorithms “symmetry-aware”, using large-scale symmetries to aid the processing of 3D meshes. Our pipeline can be used to emphasize the symmetries of a mesh, establish correspondences between symmetric features of a mesh, and decompose a mesh into symmetric parts and asymmetric residuals. We make technical contributions towards two of the main steps in this pipeline: a method for symmetrizing the geometry of an object, and a method for remeshing an object to have a symmetric triangulation. We offer several applications of this pipeline: modeling, beautification, attribute transfer, and simplification of approximately symmetric surfaces. Second, we conduct several investigations into part decomposition of 3D meshes. We propose a hierarchical mesh segmentation method as a basis for consistently segmenting a set of meshes. We show how our method of consistent segmentation can be used for the more specific applications of symmetric segmentation and segmentation transfer. Then, we propose a probabilistic version of mesh segmentation, which we call a “partition function”, that aims to estimate the likelihood that a given mesh edge is on a segmentation boundary. We describe several methods of computing this structure, and demonstrate its robustness to noise, tessellation, and pose and intra-class shape variation. We demonstrate the utility of the partition function for mesh visualization, segmentation, deformation, and registration. Third, we develop a system for object recognition in 3D scenes, and test it on a large point cloud representing a city. We make technical contributions towards three key steps of our system: localizing objects, segmenting them from the background, and extracting features that describe them. We conduct an extensive evaluation of the system: we perform quantitative evaluation on a point cloud consisting of about 100 million points, with about 1000 objects of interest belonging to 16 classes. We evaluate our system as a whole, as well as each individual step, trying several alternatives for each component.