Several Key Techniques In Digital Geometry Processing

Following sound, image and video, digital geometry is considered as the fourth wave of digital multi-media. Recently, with the development of 3D scanning technology, digital geometry has been widely used in many areas such as digital entertainment, manufacture and biology information. Digital geometry processing technology is playing a more important role in computer graphics and computer aided design. This thesis addresses several important problems in digital geometry processing.In chapter 1, We briefly recall the history of digital geometry processing at first. The several important problems are reviewed in digital geometry processing.In chapter 2, we propose a novel approach for computing the oriented normal field on a point cloud. We integrate the unsigned normal estimation and consistent normal orientation into one variational model. By spectral relaxation, it results in solving a standard eigenvalue problem. Experiments demonstrate that our approach is more robust than other normal estimation methods. It can not only estimate more accurate normal for clean data, but also can provide a good normal estimation for data in some difficult scenarios(e.g. close-by surface, sharp feature and thick structure).In chapter 3, we propose a new shape representation—implicit PHT-spline, which allows us to efficiently reconstruct surface models from vary large sets of points. A PHT-spline is a piecewise tricubic polynomial over a 3D hierarchical T-mesh. For surface reconstruction, it has some good properties including natural hierarchical structure, simple local refinement, great geometry description power and hermite interpolation. Given a point cloud, an implicit PHT-spline surface is reconstructed by interpolating the Hermitian information at basis vertices of the T-mesh, and the Hermitian information is obtained by locally estimating the geometric quantities on the underlying surface of the point cloud. We use the natural hierarchical structure of PHT-splines to reconstruct surface adaptively, with simple error-guided local refinements that adapt to the regional geometric details of the target object. Unlike some previous methods that heavily depend on the normal information of the point cloud, our approach only uses it for orientation and is less sensitive to the noise of normals. Our approach also has the data parallelism property and is very suitable for multi-core system. Experimental results show that our approach can produce high quality reconstruction surface very efficiently.In chapter 4, we study the deformation technology for complex mesh models, which often consist of multiple components connected by various type of joints. We propose a joint-aware deformation framework that supports the direct manip-ulation of an arbitrary mix of rigid and deformable components. We apply slip-pable motion analysis to automatically detect multiple types of joint constraints that are implicit in model geometry. For single-component geometry or models with disconnected components, our approach supports user-defined virtual joints. We integrate manipulation handle constraints, multiple components, joint con-straints, joint limits, and deformation energy into a single volumetric-cell-based space deformation problem. An iterative, parallelized Gauss-Newton solver is used to solve the resulting non-linear optimization. Interactive deformable ma-nipulation is demonstrated on a variety of geometric models while automatically respecting their multi-component nature and the natural behavior of their joints.Finally, in chapter 5 we focus on a special quandrangulation—planar quan-drangulation. We propose a novel conjugate direction field CDF design scheme to control the layout of a planar quad mesh. By introducing a novel signed-permutation based smoothness measure for CDF, our method can model±κ/4(κ∈Z) singularities and this is the first to model±κ/4 singularities in CDF design. The vector association in CDF is treated as a signed-permutation operation. It can avoid the additional integer variables used in the period jump technique for the N-Rotation symmetry field, and results in an efficient direction field optimiza- tion algorithm. After obtaining the CDF, we apply a global parameterization to generate the quad mesh, whose edges align with the CDF. Finally we adopt a pla-nar ity optimization for the quad mesh, resulting in a planar quad mesh which is also close to the original shape. Experimental results demonstrate that our CDF design algorithm is efficient in producing smooth planar quad mesh on a variety of models.