Surface Discretization Mapping Method And Application Based On Harmonic Mapping

Searching for the mapping between surfaces has always been the focus and founda-tion in the fields such as computational geometry,computer graphics,computer vision,computer-aided design/manufacturing and other fields.For the mapping between any two surfaces,it has to be guaranteed that the mappings are corresponding to each other and smooth enough,besides,the mapping must be diffeomorphism.During the mapping pro-cess,the distortion caused by the mapping should be minimized.The quality of mapping is also mainly dependent on the distortion,scientists in related fields have been looking for ways to reduce the distortion of mapping.Mapping transformation has a wide variety of application in academic and industrial fields.In this dissertation,based on the theory and algorithm of harmonic mapping,we conducted research on surfaces conformal mapping,area-preserving mapping,measure-driven mapping and high-genus surface mapping.The main work and conclusions are as follows:A nonlinear diffusion algorithm for harmonic energy degradation is proposed to cal-culate the conformal mapping of topological disk surfaces.Then the algorithm is extend-ed to compute the conformal mapping of high-genus surfaces.On the basis of harmonic mapping,the Laplace tangential direction of the mapping can be adjusted through a non-linear diffusion process,during which the energy decreases constantly.And finally,the harmonic energy is minimized,so as to obtain the conformal mapping of the surface.The experimental results demonstrate that the algorithm is accurate and the mapping result can guarantee the angle relation of mesh.The algorithm does not require high-quality mesh and has better stability,compared with the three classical conformal mapping methods,the results obtained by this dissertation are more uniform and achieves better conformal effect.The proposed method can be applied to the fields such as surface parameterization,texture mapping and surface registration.Improve the existing area-preserving mapping algorithm that is based on the optimal mass transport.Conformal mapping will cause area distortion on the parameterization domain,while the area-preserving mapping can reduce the area distortion of the map.Some scholars proposed an optimal mass transport mapping algorithm that is based on Monge-Brenier theory to realize the area-preserving mapping.In this dissertation,an improved algorithm is proposed based on this algorithm,the adjustive rule of height vector in the original mapping algorithm is improved to adapt to the situation of zero area cell in the calculation process.Thus the stability of the algorithm is improved.The improved area-preserving mapping algorithm is applied to normal mapping and normal map design.By experimental data,the method in this dissertation can effectively improve the ability of normal mapping in presenting geometric details of low-resolution models.A method of normal map design based on measure-driven parameterization is proposed:on the basis of area-preserving mapping method,different target area measure-ment are set to obtain different mapping results,thus different parameterization results are obtained.By taking advantage of this feature,the surface area where the designer is interested in can be enlarged proportionally in the parameterization domain.This method can be used to control the two-dimensional parameterization domain interactively and provide a powerful tool for normal map design.The experimental results show that the method is convenient and effective.Improve the existing hyperbolic harmonic mapping algorithm.In the initial mapping stage,the existing hyperbolic harmonic mapping algorithm needs to calculate the pants decomposition.The improved algorithm simplifies the calculation of the initial mapping,transforms the original pants decomposition procedures into the calculating canonical basis of the fundamental group of the high-genus surface.Then,in Euclidean space,utilizes the Euclidean harmonic mapping of boundary constraint to calculate the initial mapping.This method is simpler and more intuitive.Meanwhile,the improved algorithm is applied to smooth distortion between high-genus surfaces,provides a tool that can automatically realize deformation between the surfaces with a handle.