基于特征多边形的非均匀细分
发布时间：2024-01-31 06:52
非均匀有理B样条(NURBS)和细分是两种主要的自由曲面表示方法。NURBS是工业的标准,细分是动画的标准表示。为了将NURBS推广到任意拓扑,引入了非均匀细分。本文主要研究了非均匀细分,包括以下三个方面的内容。第一部分,我们通过特征多边形构造了一个新的非均匀Doo-Sabin细分格式,证明了在一个较小的假设下(当λ是细分矩阵的第二和第三特征值),对于任意价奇异面和任意给定的正的节点距,极限曲面总是收敛的,并且是G1连续的。然后,对于随机选取节点距,我们进行了一百万个随机数值实验。所有的实验结果都验证了新的细分方案满足上面的假设条件。然而,对于[1-2]中的另外两个现有的非均Doo-Sabin方案,这是不正确的。而且,数值实验表明,新的极限曲面的质量可以得到改善。第二部分定义了一种基于特征多边形的非均匀插值细分格式。我们首先将边点和面点规则分割为相邻顶点贡献的平均值。然后定义了非均匀奇异点的特征多边形,并求解特征多边形引导下的奇异点对相邻面点和边点的贡献。这种构造可以保证细分矩阵具有两个相同的特征值1/2,这是细分曲面为G1的必要条件。数值实验结果表明,我们的算法可以改进细分曲面,使之...
【文章页数】：131 页
【学位级别】：博士
【文章目录】：
摘要
ABSTRACT
Chapter 1 Introduction
1.1 Background and Motivation
1.2 Main Task
1.3 The organization of this Thesis
Chapter 2 NURBS and Uniform Subdivision
2.1 B-spline
2.1.1 B-spline basis functions
2.1.2 B-spline curves
2.1.3 B-spline surfaces
2.2 NURBS
2.2.1 NURBS curves
2.2.2 NURBS surfaces
2.3 Uniform Subdivision
2.3.1 Chaikin's method
2.3.2 Four point interpolatory curve subdivision scheme
2.3.3 The tensor-product four point subdivision scheme
2.3.4 Butterfly interpolatory subdivision scheme
2.3.5 DS subdivision surfaces
2.3.6 CC subdivision surfaces
2.4 Chapter Summary
Chapter 3 Non-Uniform Subdivision
3.1 Non-Uniform Approximation Subdivision
3.1.1 Non-uniform quadratic B-spline curves and surfaces
3.1.2 Non-uniform cubic B-spline curves and surfaces
3.1.3 NURDS scheme
3.1.4 NURSS scheme
3.2 Non-Uniform Interpolatory Subdivision
3.2.1 Non-uniform four point interpolatory curve subdivision
3.2.2 The non-uniform tensor-product surface subdivision
3.2.3 NUISS scheme
3.3 Chapter Summary
Chapter 4 Eigen Polyhedron
4.1 Introduction
4.2 Eigen Polygon
4.2.1 The basic idea of eigen polygon
4.2.2 The basic idea of eigen polyhedron for non-uniform subdivision
4.2.3 Examples for eigen polyhedron
4.3 Uniform Subdivision through Eigen Polyhedron
4.4 Non-Uniform Subdivision through Eigen Polyhedron
4.4.1 Creating an eigen polyhedron
4.4.2 Solving the subdivision rules from an eigen polyhedron
4.5 Chapter Summary
Chapter 5 Non-Uniform Doo-Sabin Subdivision Surface via EigenPolygon
5.1 Introduction
5.2 Problem Statement
5.2.1 Non-uniform bi-quadratic B-spline surface
5.2.2 Doo-Sabin subdivision scheme
5.2.3 Non-uniform Doo-Sabin subdivision scheme
5.3 Non-Uniform Doo-Sabin Subdivision via Eigen Polygon
5.3.1 The basic idea of eigen polygon
5.3.2 Eigen polygon for CCVDS and non-uniform biquadratic B-spline refinementrule
5.3.3 Define the subdivision rule from eigen polygon
5.4 Result
5.4.1 Convergence and continuity analysis
5.4.2 Limit surfaces
5.5 Chapter Summary
Chapter 6 Non-Uniform Interpolatory Subdivision Surface via EigenPolyhedron
6.1 Introduction
6.2 Problem Statement and Main Challenge
6.3 Non-Uniform Interpolatory Subdivision Scheme
6.3.1 The basic framework of the scheme
6.4 Construction of our New Non-Uniform Interpolatory Subdivision Scheme
6.4.1 The limit point rule
6.4.2 The eigen-polyhedron
6.4.3 Define unknown weighted parameters for the points FV
i and EV
i via eigen poly-hedron
6.5 Numerical Examples
6.6 Chapter Summary
Chapter 7 Exact Evaluation of Non-Uniform Subdivision SurfaceScheme via Eigen Polyhedron
7.1 Introduction
7.2 Evaluation of Uniform Catmull-Clark Subdivision Scheme
7.2.1 Parameterization for uniform Catmull-Clark subdivision surfaces
7.2.2 Calculate control vertices through subdivision
7.2.3 Eigenanalysis, convergence analysis and evaluation of uniform Catmull-Clarksurfaces
7.3 Evaluation of Non-Uniform Subdivision Scheme via Eigen Polyhedron
7.3.1 Non-uniform subdivision basis functions
7.4 Numerical Examples
7.5 Chapter Summary
Chapter 8 Conclusion and Future Directions
Bibliography
Acknowledgements
Publications
本文编号：3891161
【文章页数】：131 页
【学位级别】：博士
【文章目录】：
摘要
ABSTRACT
Chapter 1 Introduction
1.1 Background and Motivation
1.2 Main Task
1.3 The organization of this Thesis
Chapter 2 NURBS and Uniform Subdivision
2.1 B-spline
2.1.1 B-spline basis functions
2.1.2 B-spline curves
2.1.3 B-spline surfaces
2.2 NURBS
2.2.1 NURBS curves
2.2.2 NURBS surfaces
2.3 Uniform Subdivision
2.3.1 Chaikin's method
2.3.2 Four point interpolatory curve subdivision scheme
2.3.3 The tensor-product four point subdivision scheme
2.3.4 Butterfly interpolatory subdivision scheme
2.3.5 DS subdivision surfaces
2.3.6 CC subdivision surfaces
2.4 Chapter Summary
Chapter 3 Non-Uniform Subdivision
3.1 Non-Uniform Approximation Subdivision
3.1.1 Non-uniform quadratic B-spline curves and surfaces
3.1.2 Non-uniform cubic B-spline curves and surfaces
3.1.3 NURDS scheme
3.1.4 NURSS scheme
3.2 Non-Uniform Interpolatory Subdivision
3.2.1 Non-uniform four point interpolatory curve subdivision
3.2.2 The non-uniform tensor-product surface subdivision
3.2.3 NUISS scheme
3.3 Chapter Summary
Chapter 4 Eigen Polyhedron
4.1 Introduction
4.2 Eigen Polygon
4.2.1 The basic idea of eigen polygon
4.2.2 The basic idea of eigen polyhedron for non-uniform subdivision
4.2.3 Examples for eigen polyhedron
4.3 Uniform Subdivision through Eigen Polyhedron
4.4 Non-Uniform Subdivision through Eigen Polyhedron
4.4.1 Creating an eigen polyhedron
4.4.2 Solving the subdivision rules from an eigen polyhedron
4.5 Chapter Summary
Chapter 5 Non-Uniform Doo-Sabin Subdivision Surface via EigenPolygon
5.1 Introduction
5.2 Problem Statement
5.2.1 Non-uniform bi-quadratic B-spline surface
5.2.2 Doo-Sabin subdivision scheme
5.2.3 Non-uniform Doo-Sabin subdivision scheme
5.3 Non-Uniform Doo-Sabin Subdivision via Eigen Polygon
5.3.1 The basic idea of eigen polygon
5.3.2 Eigen polygon for CCVDS and non-uniform biquadratic B-spline refinementrule
5.3.3 Define the subdivision rule from eigen polygon
5.4 Result
5.4.1 Convergence and continuity analysis
5.4.2 Limit surfaces
5.5 Chapter Summary
Chapter 6 Non-Uniform Interpolatory Subdivision Surface via EigenPolyhedron
6.1 Introduction
6.2 Problem Statement and Main Challenge
6.3 Non-Uniform Interpolatory Subdivision Scheme
6.3.1 The basic framework of the scheme
6.4 Construction of our New Non-Uniform Interpolatory Subdivision Scheme
6.4.1 The limit point rule
6.4.2 The eigen-polyhedron
6.4.3 Define unknown weighted parameters for the points FV
i and EV
i via eigen poly-hedron
6.5 Numerical Examples
6.6 Chapter Summary
Chapter 7 Exact Evaluation of Non-Uniform Subdivision SurfaceScheme via Eigen Polyhedron
7.1 Introduction
7.2 Evaluation of Uniform Catmull-Clark Subdivision Scheme
7.2.1 Parameterization for uniform Catmull-Clark subdivision surfaces
7.2.2 Calculate control vertices through subdivision
7.2.3 Eigenanalysis, convergence analysis and evaluation of uniform Catmull-Clarksurfaces
7.3 Evaluation of Non-Uniform Subdivision Scheme via Eigen Polyhedron
7.3.1 Non-uniform subdivision basis functions
7.4 Numerical Examples
7.5 Chapter Summary
Chapter 8 Conclusion and Future Directions
Bibliography
Acknowledgements
Publications
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