| 英文摘要 | The multimedia revolution has encompassed three types of data so far: sound, image, and video. Digital Geometry Processing (DGP) is an entirely new and much needed research area. Its goal is the research of mathematical and computational foundations needed for the next wave of multimedia data: geometry. Unlike previous data types, we cannot rely on existing theoretical and algorithmic tools since geometric data has intrinsic properties, such as topology, curvature, and non-uniform sampling that render most traditional tools useless. We develop new methods for efficient, reliable and scalable tools for cross-parameterization (consistent correspondence), mesh deformation and editing, mesh segmentation, mesh faring and many other types of DGP operations. This provides a complete mesh processing pipeline to allow for easy and efficient manipulation of 3D mesh models. The main contributions of this thesis include following issues: We proposes a novel retargetting method, called model transduction, to directly transfer pose between different meshes, without the need of building the skeleton configurations for meshes. Different from previous retargetting methods, such as deformation transfer, model transduction does not require a reference source mesh to obtain source deformation, thus effectively avoids unsatisfying results. Moreover, we show that the transduction method also can be used for pose correction after various mesh editing operations. We propose a novel framework for consistent correspondence between arbitrary manifold meshes. Different from most existing methods, our approach directly maps the connectivity of the source mesh onto the target mesh without needing to segment input meshes, thus effectively avoids dealing with unstable extreme conditions (e.g. complex boundaries or high genus). We propose a sketch-based interactive framework for real-time mesh segmentation. With an easy-to-use tool, the user can freely segment a 3D mesh while needing little effort or skill. In order to meaningfully segment the mesh, two dimensionless feature sensitive metrics are proposed, which are independent of the model and part size. We show that these metrics give the clear physical meaning to illustrate discrete differential geometric features, such as the curvature tensor and the curve length of gaussian image. We propose a novel scheme for shape-preserving "divide and rule" cross-parameterization between 3D meshes. Our scheme exploits the excellent properties of convex hull, e.g. good approximating ability and linear convex representation for interior vertices. Based on the novel mean-value manifold operator, we present a general framework for performing mesh processing tasks. The mean-value manifold operator is a shape-preserving operator defined on manifold meshes, and furnishes a variety of processing applications, such as mesh editing, mesh fairing, cross-parameterization and model transduction. |
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