Applications Of Low Rank And Compact Representations In Geometric Modeling

Constructing smooth surface representations from point clouds is a fundamental problem in geometric modeling and computer graphics,and a wealthy of literature has focused on this problem.Among the many approaches,implicit surface reconstruction has been a central topic in the past two decades,as implicit representation is very suitable for CSG operations,and it is able to represent objects with complicated geometry and topology.However,the storage requirement,e.g.memory or disk usage,for implicit representations of complex models is relatively large.The first part of the current thesis focuses on how to reduce the storage require-ment for implicit representations.In chapter 2,we propose a compact representation for multilevel rational algebraic spline(MRAS)surfaces using low-rank tensor approxima-tion technique,and exploit its applications in surface reconstruction.The reconstruction procedure is recursively performed until a certain accuracy is achieved.Numerous ex-periments show that our approach can greatly reduce the storage of the reconstructed implicit surface while preserve the fitting accuracy compared with the state-of-the-art methods.Furthermore,our method has good adaptability and is able to produce recon-struction results with high quality.However,the MRAS surface is only C0 continuous.In order to construct a surface with higher smoothness and further reduce the storage re-quirement for the implicit representation,in chapter 3,we propose a phase-field guided implicit surface reconstruction method to tackle this problem.Given an unorganized point cloud,we present a method to construct a phase-field function represented by a hierarchical B-spline whose zero level set approximates the point cloud as much as pos-sible.Unlike previous approaches,our mathematical model avoids the use of the normal information of the point cloud.Furthermore,as demonstrated by experimental results,our method can achieve very compact representation since we mainly need to save the coefficients of the hierarchical B-spline function within a narrow band near the point cloud.The ability of our method to produce reconstruction results with high quality is also validated by experiments.Domain parameterization and matrix assembly are two essential steps in isogeo-metric analysis(IGA),the quality of the parametrization has a great effect on the subse-quent analysis accuracy and efficiency in IGA.In addition,an analysis and subsequent optimization of the rank of a parametrization can lead to substantial improvements of the overall efficiency of the numerical simulation in IGA.The focus of chapter 4 is on constructing a low-rank tensor-product spline sur-faces from given boundary curves using low-rank tensor approximation technique.Un-der given correspondence of four boundary curves,a quasi-conformal map with low distortion between a unit square and the computational domain can be obtained by solving an optimization problem.We propose an efficient algorithm to compute the quasi-conformal map.The basic idea is to solve two convex optimization problems alternatively.Numerous experiments show that our approach can produce a low-rank parametric spline representation.Furthermore,our new parametrization method is su-perior to other state-of-the-arts in terms of distortion.