关键词: 有理分形插值, 函数尺度因子, 不规则数据, 分析性质, 曲线建模

Abstract: 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. 

Key words: rational fractal interpolation, function scaling factor, irregular data, analytical property, curve modeling