【作者】 张婷婷；

【导师】 韩力文；

【作者基本信息】 河北师范大学 ， 计算机应用技术， 2010， 硕士

【摘要】 拟插值作为一种逼近方法在计算机辅助几何设计、数据分析等领域有广泛应用,尤其在逆向工程领域,它能够直接拟合散乱数据点而不需要所有插值点都落在目标曲线或曲面上,在处理坏点、尖锐点方面非常有效。拟插值方法计算效率高、能够很好地局部逼近插值点,受到越来越多的关注。同时,近年来许多国内外学者开始关注参数样条的形状控制,构造出多种带有形状参数的B样条扩展模型,但由于这些模型普遍缺少可加细性质,不能应用细分方法实现曲线曲面的离散生成。本文进一步研究Manni等构造的三次张力参数B样条,在三次张力参数B样条曲线的多尺度细分和基于三次张力参数B样条的拟插值曲线曲面方面开展了大量深入的研究,主要研究成果如下:1、探讨了三次张力参数B样条的构造和性质,研究了此样条曲线与Beta样条曲线、Gamma样条曲线的转换,并就张力参数对曲线的影响进行了详细讨论。2、研究了三次张力参数B样条曲线的细分,总结出C 1连续条件下具有统一形式的M -尺度细分规则(2≤M≤5,M∈Ν),具体给出3-尺度细分面具,重点讨论了Gamma样条和Beta样条曲线细分,并以2、3-尺度细分为例,分析了细分曲线误差。特别地,为使细分曲线达到更好的光滑性,重点研究了样条曲线在C 2连续下的细分条件及2-尺度细分规则。3、研究了基于三次张力参数B样条的拟插值曲线,总结出拟插值曲线的M -尺度细分规则(2≤M≤5,M∈Ν),进一步以2、3-尺度为例给出细分曲线实例。最后,将单变量样条曲线推广至双变量样条曲面(包括三次张力参数B样条曲面及拟插值曲面)。更多还原

【Abstract】 As an approximation method, quasi-interpolation is widely used in computer-aided geometric design (CAGD), data processing, especially in the field of scattered point cloud reconstruction. And quasi-interpolation could fit data points directly, without all the interpolation points on the curve or surface, especially effective in dealing with dead pixels and sharp point. The advantages of Quasi-interpolation have received a considerable attention by many authors, such as low computational cost, efficient approximants to a given set of data. What’s more, many scholars have been beginning to focus on the shape control of splines with parameters in recent years. They have constructed kinds of expansion of B-splines models with shape parameters, however, these models generally lacked of the essence of splines’refinability, and couldn’t express curves and surfaces’discrete generation by subdivision schemes. This paper discusses cubic B-splines with tension parameters constructed by Manni and the feature of cubic B-splines’ refinement, further more, the paper focuses on quasi-interpolation based on cubic B-splines with tension parameters. The main research results in this article are as follows:First, the paper discusses the construction and properties of cubic B-splines curve with tension parameters, transformation theorem between cubic B-splines curve with tension parameters and Beta-spline curves or Gamma-spline curves. What’s more, it gives a detailed discussion about the influences of tension parameters on the curve.Second, the paper makes research on the subdivision schemes of cubic B-splines curve with tension parameters, and then summarizes a unified rules of M-band subdivision in the condition of C 1 continuous, where 2≤M≤5,M∈Ν, especially obtains ternary subdivision mask. Beta-spline curves and Gamma-spline curves subdivision conditions are discussed, the binary and ternary subdivision numerical examples are given to illustrate the advantage of approximating to the limit curve. In particular, it studies subdivision conditions and binary subdivision rules of cubic B-splines curve with tension parameters in the condition of C 2 continuous.Third, the paper studies quasi-interpolation based on cubic B-splines with tension parameters, moreover, summarizes its M-band subdivision rules as well as numerical examples based on binary and ternary subdivision. Finally, it further extends the curve results to surface (including cubic B-splines surface with tension parameters and quasi-interpolation surface based on cubic B-splines with tension parameters).更多还原