Efficient Mesh Parameterizations

Digital geometry processing technology has been developed rapidly in recent years and has been widely used in emerging industries such as 3D printing,autonomous driv-ing,virtual reality,and smart city.The development of these emerging industries re-quires a large amount of 3D data processing technologies to support.One of the impor-tant basic problems is how to compute the low-distortion parameterizations of a mesh.Since mesh parameterizations connects the surface in 3D and the planar parameter do-main,allowing convenient information transfer and computational conversion between them,many applications in computer graphics incorporate the low-distortion parame-terizations as a part of the algorithm.The efficiency and robustness of the parameteri-zations algorithm have a great impact on the performance of related applications.This dissertation focuses on the topic of mesh parameterizations and studies the issues such as computing high-quality parameterizations with high efficiency and practical robust-ness,memory-efficient bijective parameterizations of very-large-scale models,and the efficient optimization for the developability of triangle meshes.To tackle the difficulties emerging from the mesh parameterizations such as high initial distortion and the non-convex nonlinear optimization energy,this dissertation proposes a progressive parameterizations method that can efficiently compute low-distortion and flip-free mesh parameterizations.This method is based on the obser-vation that when the distortion between all the parameterized and reference triangles is less than a certain threshold,only very few iterations are required to reduce the dis-tortion to a level close to convergence.Therefore,unlike previous parameterizations methods that directly use the input mesh as the optimization target,this approach em-ploys interpolation to construct a series of reference triangles,which are then treated as intermediate targets to optimize progressively.When the distortion drops to a certain level,the optimization is finally performed with the input mesh as the target until con-vergence.Besides,the method adopts a hybrid solver,which combines the advantages of two solvers for fast distortion reduction and second-order convergence.We have tested the performance of the method on more than 20,000 models,and the experiments show that this algorithm performs consistently well for models with different triangula-tion,different initialization,different resolution,and different mesh quality.Compared with existing methods,this method has high efficiency and practical robustness.Since computing parameterizations requires solving a linear system with a compa-rable size as the mesh vertices,the existing parameterizations algorithms are only suit-able for processing small and medium-sized meshes.Given a very-large-scale mesh,the algorithm usually crashes due to memory limitations.To address this problem,this dis-sertation proposes a memory-efficient bijective parameterizations method specifically for very-large-scale meshes.The key idea of this method is to convert large-scale prob-lems into small problems with controllable size through mesh partition and reduced-space approximation.This method consists of three parts.In the initialization phase,based on the divide-and-conquer strategy,the mesh is partitioned into multiple patches.The algorithm first maps the dividing line and then the patches one by one,which si-multaneously guarantees the bijection and limits the upper bound of memory usage.In the optimization stage,the parameterized domain is remeshed into more sparse discrete elements,on which the spline functions are defined,thus converting the variables from mesh vertices to spline control vertices with controllable scale.In the post-processing stage,this method employs a combination of pointwise optimization and patchwise op-timization to further reduce distortion.This algorithm integrates these techniques to keep the scale of the linear system to be solved below a given threshold.This method can successfully compute the low-distortion bijective parameterizations of a model con-taining nearly 100 million triangles on a desktop computer equipped with 16G RAM.The isometric distortion is inevitable when computing the parameterization of a general mesh,while the developable surfaces can be isometrically mapped to the plane.Given a general triangle mesh,this dissertation proposes a joint bilateral filtering-based algorithm for optimizing mesh developability.This method consists of two stages.Firstly,apply the joint bilateral filtering to the surface normals to obtain the target nor-mal of each triangle,where the guidance normal of the joint bilateral filter is constructed by using the normal consistency of the vertex half-neighborhood.Then update the ver-tex positions according to the calculated target normals.This method performs the above two operations alternately and iteratively,and the mesh developability is gradu-ally improved by modifying the mesh geometry.The optimized mesh can be parame-terized with extremely low isometric distortion after appropriate cutting.This method is simple to implement.Compared with the previous algorithms,this method can obtain comparable or even better results more efficiently.