可变参数的有理分形插值曲线建模

Curve modeling using rational fractal interpolation with variable parameters

为了有效地处理复杂真实现象中的不规则数据,提出一种利用有理分形插值进行分形曲线建模的方法。首先,基于传统的具有形状参数的有理样条,构造了一类具有函数尺度因子的有理迭代函数系统,并定义了有理分形插值曲线。然后,研究了有理分形曲线的一些重要性质,包括光滑性、稳定性以及收敛性。最后,估计了有理分形曲线计盒维数的上下界。提出的可变参数的有理分形插值推广了传统的单变量有理样条,适用于拟合不规则数据或逼近具有连续但不规则导数的函数,具有更好的灵活性和多样性。数值实例和曲线建模表明,该方法不仅在视觉效果上明显优于Bézier插值,B样条插值以及基于多项式的分形插值方法,而且在均方根误差的数值对比中也具有显著优势。

In order to deal with irregular data from complex real phenomena,a constructive approach to fractal curves was proposed using the rational fractal interpolation.First,based on the traditional rational spline with shape parameters,we constructed one class of rational Iterated Function Systems(IFS)with function vertical scaling factors.Rational IFSs were hyperbolic and rational fractal interpolation curves were defined.Then,some important properties of rational fractal curves were investigated,including smoothness,stability and convergence.Finally,lower and upper bounds in the box-counting dimension of rational fractal curves were estimated.The presented rational fractal interpolation with variable parameters generalized the traditional univariate rational spline,which is more suitable for fitting irregular data or approximating a function with continuous but irregular derivatives,and is more flexible and diversiform.Numerical examples and curve modeling show that this method can not only significantly outperform the Bézier interpolation,B-spline interpolation,and polynomial-based fractal interpolation methods in terms of visual effects,but also display prominent advantages in numerical comparison of root mean square errors.