Rational Fractal Interpolation Curves And Surfaces With Function Scaling Factors And Their Applications

The curve and surface modeling is everlasting demand and one of the major research areas of Computer-Aided Geometry Design(CAGD).Fractal interpolation has become a powerful tool for dealing with highly irregular data because it can provide a good deterministic expression for complex natural phenomena.Most of the existing fractal interpolation function are generated based on the polynomial iterated function systems.It is well known,rational functions are more effective to describe complex phenomena than the polynomial functions.In this paper,on the basis of the existing references,we have studied a new type of fractal interpolation functions,that is,rational fractal interpolation functions with function scaling factors.The primary strategies axe as following:In Section one,some preliminaries from the fractal interpolation are briefly introduced,including the iterated function systems(IFSs)of fractal curves and surfaces,the dimension of the fractal set.In Section two,a method of constructing fractal curves by using univariate rational fractal interpolation is proposed.Firstly,a rational IFS with function scaling factors is developed based on classical rational spline interpolation functions with shape parameters.The IFS is hyperbolic and its attractor is a rational fractal curve.Secondly,some properties of rational fractal interpolation functions(RFIFs)are investigated,including smoothness,convergence and stability.And then,the box-dimension of rational fractal interpolation curves is obtained.In Section three,we extend one-dimensional rational fractal curves to two-dimensional surfaces,and present a method of constructing rational fractal interpolation surfaces with function scaling factors.Firstly,a novel type of C1-continuous piecewise rational spline interpolation surfaces with shape parameters is developed on rectangular grids.Further,treating fractal surfaces as the fractal perturbation of the bivariate rational interpolation functions,a kind of bivariate rational IFS is constructed.Secondly,some analysis properties of rational fractal interpolating surfaces are studied.Finally,we estimate the box-counting dimension of the rational fractal surfaces.In Section four,we provide some examples of practical applications for the developed rational fractal interpolation curves and surfaces with function scaling factors.Mainly including:the applications of univariate rational fractal interpolation in curve modeling and stock price fitting,the applications of bivariate rational fractal interpolation in natural object modeling and image interpolation.The experimental results demonstrate the effectiveness of the rational fractal interpolating function with the function scaling factor in dealing with the practical problems.