Research Of Mesh Parameterization

Surface parameterization is used widely in computer graphics, CAGD ,such as texture mapping, surface fitting, remeshing. There are still many applications in digital geometry processing, like drawing in 3D interactively, 3D Mesh editing, mesh morphing. They all need the 3D mesh parameterized .So the surface parameterization has great importance.Basing on the different parameter domain, the 3D mesh parameterization can be divided into two categories: planar parameterization and spherical parameterization.Then we will introduce the main jobs that I do in this field.Intuitively parameterization in planar domain is to transfer the 3D mesh to planar mesh with the guaranty that the planar mesh is valid and the distortion is minimized. The target of planar parameterization is focused on 2D manifold mesh with single boundary, and any closed mesh can be changed into mesh with boundary in the way of dividing and conquering.Zhonggui Chen proposed a new parameterization method based on aligning the optimal local flattening. Focusing on the disadvantage in this article, we proposed two new flattening ways, the first preserves angle and area, the other preserves angle and boundary length. We have found that surface constructed with the parameterization that the improved methods got is more close to the source surface, and the distortion is less. At the same time the distortion of angle and area has declined.The first thought of parameterizing closed mesh is doing this in the sphere domain. These methods can be divided into three: (1) method based on progressive mesh (2) method based on sphere relaxation (3)conformal method. The three methods have their own advantages and shortcomings. But the method based on progressive mesh is much faster.In this article we propose a new spherical parameterization method. And this method is just based on progressive mesh. The objective function is the generalization of which is used by Jerome Maillot. The algorithm first construct a progressive mesh based on the source mesh and change the mesh into the simplest form, tetrahedron. During the parameterization, we first map the tetrahedron to a unit sphere. Then we insert the deleted vertex to the sphere based on the progressive mesh. And the objective function is the generalized function of Jerome Maillot.However, there are still some problems in my method, such as the method of handling the progressive mesh doesn't preserve the features of the source mesh, and the result hasn't been compared with other algorithms. These problems are the direction of our later investigation.