Rational Fractal Curves with Function Scaling Factors

Fractal interpolation with constant vertical scaling factors is an effective tool for describing things with obvious self-similarity, and yet, which is difficult to accurately characterize irregular data with weak self-similarity. Therefore, an interpolation method of rational spline fractal with function scaling factors is proposed. Firstly, a type of rational fractal interpolation curves with shape parameters is constructed by introducing function scaling factors into the iterated function system. And then, analytical properties of fractal curves are discussed, including smoothness of fractal curves under the appropriate condition of scaling factors, stability of fractal curves to perturbation of interpolation data, and convergence of fractal interpolation functions. Finally, the box-counting dimension of fractal curves is studied, and the upper and lower bounds of box-counting dimension are given. The numerical examples verify the curve controllability and robustness against noise; for interpolating the coastline data, this algorithm performs better in restoring the coastline coarse appearance than B Spline, Bézier curve and cubic spline method; for processing the time series data in the stock market, this algorithm performs better than ARIMA and SVM method under multiple indexes such as RMSE. © 2020, Beijing China Science Journal Publishing Co. Ltd. All right reserved.