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基于曲面约束的自适应B样条曲线拟合

Constrained Adaptive B-spline Curve Fitting on Surface

作者: 专业:计算数学 导师:李崇君 年度:2010 学位:硕士  院校: 大连理工大学

Keywords

Constrained Optimization, B-spline Curve Fitting, Dominant Points Selection, Parameter Modification, Least Square Method

        用一条光滑曲线来拟合给定的散乱数据是计算机辅助几何、计算机图像、计算机视觉等很多领域的常见问题。B样条方法是计算机辅助几何设计(CAGD)的一类重要方法,拥有着很多良好的性质。本文对约束在曲面上的B样条曲线拟合方法进行了一些讨论和研究。第一章引入了散乱数据的曲线拟合问题,并介绍了该领域内几种重要的研究方法。第二章简单介绍了B样条曲线及其基本性质。第三章讨论了一种基于曲面约束的自适应B样条曲线拟合方法。该方法利用离散曲率来选取初始的主导点,然后用约束在曲面上的最小二乘方法来进行拟合。在计算数据点和曲线误差的过程中还提出了参数修订的方法。参数修订在拟合过程中起到了至关重要的作用,它使得计算出的误差更加精确,以便于在最适当的地方插入新的主导点。这样就可以使拟合的误差迅速下降。数值试验说明这种方法对于解决约束在曲面上的曲线拟合是高效可行的。第四章给出了用权值的方法解决带有误差的曲线拟合,并给出了数值算例以及误差分析。最后总结全文并提出有待于进一步研究的问题。
    Curve fitting of scattered data is always an important part of approximation theory. It is of great significance and extensive applications in many fields. B-spline method play an important part in Computer Aided Geometry Design (CAGD). Some analysis and discussion on the method of B-spline curve fitting of scattered data are presented in this thesis.Chapter 1 introduces the method to approximate a set of points by a smooth curve when designing curves on surface, and several methods are presented.B-spline curves and their main properties are described in chapter 2.In chapter 3, constrained curve fitting method on surface is discussed, which is motivated by an insight that properly selected points using discrete curvature called dominant points. A smooth curve constrained on surface can be fitted with this dominant points. A method called parameter modification which can play an important role in producing better curve approxima-tion is discussed. Through this method, dominant points can be inserted in the properly position to make a better approximation. Numerical experiments illustrate the effectiveness of our algo-rithm.A weighted least squares method for scattered data fitting is described in chapter 4. Nu-merical experiments illustrate the effectiveness of this algorithm.Finally, a summary of this thesis is given and several problems which need to be solved are proposed.
        

基于曲面约束的自适应B样条曲线拟合

摘要4-5
Abstract5
1 绪论8-11
    1.1 问题引入8-11
        1.1.1 问题的来源和应用背景8
        1.1.2 问题的描述8-9
        1.1.3 插值问题的Haar条件9-10
        1.1.4 多元散乱数据的多项式插值10-11
2 B样条曲线简介11-15
    2.1 一元B样条基函数11-13
    2.2 B样条曲线13-15
        2.2.1 B样条曲线及基本性质13
        2.2.2 B样条曲线的节点插入算法13-15
3 一类基于曲面约束的自适应B样条曲线拟合15-30
    3.1 曲线拟合的算法介绍15-16
    3.2 基于曲面约束的自适应B样条曲线拟合16-30
        3.2.1 相关工作16-19
        3.2.2 拟合算法19-20
        3.2.3 选取主导点并计算节点向量20-22
        3.2.4 最小二乘拟合,重新修订参数值22-23
        3.2.5 计算误差,添加新的主导点23-24
        3.2.6 数值试验24-30
4 自适应的带权最小二乘曲线拟合30-34
    4.1 问题介绍30
    4.2 拟合算法及数值算例30-34
结论34-36
参考文献36-38
攻读硕士学位期间发表学术论文情况38-40
致谢40-43
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