【作者】 陈龙；

【导师】 冯结青；

【作者基本信息】 浙江大学 ， 电子信息（专业学位）， 2025， 硕士

【摘要】 在B-样条曲线曲面拟合中,基于几何距离的拟合能够保证所得结果对数据点实现最佳逼近,但由于需要考虑控制顶点、节点向量以及数据点参数等多个自由变量,且目标函数具有高度非线性特征,导致优化过程极为复杂。因此,在基于几何距离的B-样条曲线曲面拟合中,如何有效地优化这些相互关联的变量,已成为计算机辅助几何设计领域内亟待突破的难题。首先,针对平面数据点序列,提出了基于几何距离、全变量优化的B-样条曲线拟合方法。针对该非线性优化问题,设计并实现了两种求解策略:一是基于L-BFGS-B算法的整体求解法,在迭代中同时优化控制顶点、节点向量和数据点参数,具有全局收敛性;二是基于Gauss-Newton算法的交替迭代法,通过交替地优化各组变量,显著提高了计算效率。实验结果表明,与已有的线性最小二乘和基于几何距离的B-样条曲线拟合方法相比,该方法在逼近精度方面具有显著优势。特别地,在处理具有尖锐特征的数据点时,生成的B-样条曲线包含重节点,从而精确地复现这些特征。其次,针对空间散乱点集,提出了基于几何距离、全变量优化的B-样条曲面拟合方法。首先,采用基于主成分分析的分治策略,对数据点集进行初始参数化;其次,提出一阶与二阶逼近混合的投影策略,加速参数更新并保持数值稳定性;最后,设计了基于L-BFGS-B算法的整体求解法和基于Gauss-Newton算法的交替迭代法。针对周期曲面,在上述优化框架中引入了周期性约束,确保曲面的整体光滑性。实验结果表明,与传统的线性最小二乘和基于几何距离的B-样条曲面拟合方法相比,新方法在逼近精度上具有显著优势。通过系统地研究基于几何距离、全变量优化的B-样条曲线曲面拟合问题,并设计相应的数值优化算法,丰富了B-样条曲线曲面造型的理论和方法,为曲线曲面建模提供了逼近精度更高的解决方案。更多还原

【Abstract】 In B-spline curve and surface fitting,geometric distance-based fitting ensures optimal ap-proximation to the data points but involves multiple interdependent free variables—control points,knot vectors,and data parameters.This interdependency coupled with the high non-linearity of the objective function complicates the optimization process.Therefore,effectively optimizing these variables in terms of geometric distance remains a challenging problem in computer-aided geometric design.First,for planar data point sequences,a geometric distance-based B-spline curve fitting method with all-variable optimization is proposed.To solve the nonlinear optimization prob-lem,two solution strategies are designed and implemented:(1)a global method based on the L-BFGS-B algorithm,which simultaneously optimizes control points,knot vectors,and data point parameters during iteration and guarantees global convergence;(2)an alternating itera-tive method based on the Gauss-Newton algorithm,which alternately optimizes different sets of variables,significantly improving computational efficiency.Experimental results show that the proposed method achieves superior approximation accuracy compared to existing least-squares and geometric distance-based B-spline curve fitting methods.Notably,for data sets with sharp features,the generated B-spline curves include repeated knots,accurately reproducing these features.Second,for scattered spatial point sets,a geometric distance-based B-spline surface fitting method with all-variable optimization is proposed.First,an initial parameterization of the data points is performed using a divide-and-conquer strategy based on principal component analysis(PCA).Then,a mixed projection strategy combining first-order and second-order approxima-tions is proposed to accelerate parameter updates and maintain numerical stability.Finally,both a global solution method based on the L-BFGS-B algorithm and an alternating iterative method based on the Gauss-Newton algorithm are designed.For periodic surfaces,periodic constraints are incorporated into the optimization framework to ensure overall smoothness.Experimental results demonstrate that the proposed method achieves significantly higher fitting accuracy than traditional least-squares and geometric distance-based B-spline surface fitting methods.By systematically investigating geometric distance-based B-spline fitting with all variables optimized and developing corresponding numerical algorithms,this work advances the theory and methodology of B-spline modeling and provides higher-accuracy solutions for curve and surface fitting.更多还原