Spherical Parameterization And Remeshing In Digital Geometry Processing

With the development of modern science techniques. 3D geometrical models are recently becoming a new type of medium after sound, images and video, which are applied in industry widely. Consequently, a new discipline processing 3D geometrical models is born, which is called digital geometry processing. It has been a focus of research activity in digital signal processing. Unfortunately, 3D geometrical models are significantly different from other traditional mediums. The topology of 3D geometrical models are very complicated, that is to say, they have any genus and boundary. So traditional signal processing techniques can not be extended to process 3D geometrical models. We must develop some more intricate digital geometry processing algorithms. In this dissertation. we take the triangular mesh models as our research object, and do some research on spherical parameterization and remeshing in digital geometry processing. The main work can be summarized as follows:1. We present a uniform quasi-conformal spherical parameterization method (UQCSP method) based on barycentric coordinates and M(o|¨)bius transformation for closed genus-zero triangular meshes. An initial spherical parameterization is generated by lifting fixed circle boundary planar parameterization methods to the sphere through the stereographic projection. Then we develop a new approach to improve the initial spherical parameterization by barycentric coordinates. At last, a uniform spherical parameterization is generated by a M(o|¨)bius transformation. This method can reduce the area distortion of the spherical parameterization whose angle distortion is low. Furthermore, it can avoid fold-overs which may exist for other angle preserving spherical parameterization approaches. We also extend this method to the singular boundary genus-zero meshes. Experiments reveal that it has obvious improvement in uniform texture mapping for complicated meshes than the available discrete conformal mapping and the discrete authalic mapping.2. We present a new idea based on symmetry analysis for the construction of spherical parameterization for closed genus-zero triangular meshes. It reduces the intricate spherical parameterization problem to a planar one by splitting a closed genus-zero mesh into two pieces with the help of its geometry symmetry traits. According to this simplification we can get two spherical parameterization methods balancing angle and area distortion: the spherical parameterization method based on the stretch metric (BSMSP method) and the spherical parameterization method based on the quasi-harmonic maps (BQHSP method). Both of them can solve the problem that spherical conformal mapping is often having no control in area distortion. They can produce almost isometric parameterizations for the objects close to the sphere. Furthermore, their computation time is dominated by solving only linear systems. The BSMSP method can guarantee the validity of the spherical parameterization. Although the BQHSP method can not guarantee it in theory, it can reduce the angle distortion of the spherical parameterizations for the complicated objects. Besides, we also apply the spherical parameterization generated by the BQHSP method to the subdivision connectivity remeshing. Experiments reveal that this method can obtain high quality subdivision connectivity mesh.3. We present an adaptive isotropic remeshing method based on distance field for arbitrary genus and boundary 3D meshes. Many approaches generate the isotropic remeshing by global parameterization. However, as for complicated meshes with high genus and arbitrary boundary, the construction of the global parameterization is a very complicated problem. Consequently, we process the mesh directly and do not use global parameterization. Firstly, a mesh with the required number of vertices can be obtained according to edge-collapse or edge-split. Then the vertex sampling and the triangle quality can be improved by geometry optimization and connectivity optimization. In order to reduce the accumulation of the error, we introduce a novel method based on mesh distance field to remain the optimized vertices on the original mesh. Experiments and comparisons are taken with some nontrivial 3D models, which reveals that our approach is effective, fast and robust.