Differential Methods For Mesh Surface Analysis And Processing

With the development of social science and technology, digital geometry processing has gained unprecedented attention, and a lot of classic algorithms and new technologies have been proposed. The analysis and processing for3D model, such as feature extraction, cluster segmen-tation and deformation, are the most fundamental and important subjects in computer graphics, and they also have a wide range of applications in the production and living. This article fo-cuses on the mesh surface and the techniques based on differential methods. The main work issummarized as follows:(1) For the problem of heat diffusion on mesh surface, we propose an anisotropic heat kernel to better control the heat conduction according to the geometry. We also propose a differ-ential representation, called normal controlled coordinates (NCC), and rigorously prove a proposition that the NCC is always parallel with the corresponding normal. Then, we use the NCC to assign the initial field, and combine with the anisotropic heat diffusion to ana-lyze and describe the geometric properties of the mesh surface. We use a concept of local convolution in heat region to transfer the heat diffusion problem into an efficient product problem between sparse matrixes and vectors. Further, we apply the anisotropic heat dif-fusion to some applications, including the scalar field and mesh smoothing, multi-scale feature extraction, signal decomposition on3D models and so on. Finally, a large number of experiments are conducted to verify the validity and robustness of our approach.(2) We propose two scalar field driven mesh cutting the clustering methods. One is arbitrary genus mesh cutting that based on Poisson scalar field. The Poisson scalar field is used to select the critical points, and find the fastest downward path. Then we use the Morse theory to cut the arbitrary genus mesh into a single boundary mesh that is homeomor-phic to a unit disk mesh. The other segmentation method is based on the quasi-harmonic field, which is the steady state of heat diffusion. By constructing a high-dimensional quasi-harmonic field, we can easily cluster the similar points, and finally obtain the mesh segmentation. Since our method inherits the advantages of heat diffusion, it is robust to the noise and the models with holes, and is applicable to the general models.(3) We propose two types of feature-preserving deformation methods:structure-preserving deformation based on local tensor, and feature-preserving deformation based on differ- ential coordinates. The former first analyze the feature structure of a given mesh by lo-cal tensors, then use tensors, areas, angles and other geometric quantities to construct a nonlinear energy function by treating different types of points differently. In order to sim-plify the optimization problem, we transfer the problem to a hierarchical feature subspace, which can accelerate the calculation a lot. Our method can achieve the desired deforma-tion with very little control constraints. The latter characterizes the local geometry by the differential coordinates, then the only purpose is to preserve the magnitudes of differential coordinates while updating the directions of them. Such as normal control coordinates based deformation, in which we only need to calculate the normal during the iterative pro-cess, and all the other information can be obtained by back substitution, is very efficient. We conduct experiments to analyze and validate the effectiveness and stability of these methods.