A Type Of Bicubic Fractal Rational Spline Interpolation And Application In Image Processing

Fractal interpolation which is an alternative to traditional interpolation techniques,gives a broader set of interpolants,and provides a good deter?ministic way for the understanding of the real world phenomena.Using this method,we can construct not only interpolants with non-integral dimension but also smooth interpolants.Fractal interpolation techniques are effective and applied widely to deal with a highly irregular data from many real phe-nomena.The most of the current fractal interpolation functions(FIFs)are generated by iterated function system(IFS)based on the polynomial func-tion.In this paper,based on existing research literature,we studied a class of bicubic rational spline fractal interpolation functions(BRFIFs)which have shape constraint parameters.The primary strategies are as following:Firstly,we propose a constructive approach of fractal rational surfaces,and theoretically proved the attractor generated by this construction is con-tinuous or C1-continuous fractal interpolation surface.Secondly,based on the construction method of bivariate rational fractal surface,we constructed a type of bicubic rational spline fractal interpolation functions with the help of classic rational cubic spline interploants.First,x-direction cubic rational function P*i,j(x)with a shape parameter is constructed,and then a bicubic rational spline interpolation functionPi,j(X,y)is defined by using the interpolant Pi,j*(x);Further,the perturbation basis functionBi,j(x,y)is presented based on the idea of constricting Pi,y(x,y).At last,we use the bivariate function Pi,y(x,y)and perturbation function Bi,y(x,y)to develop a bicubic rational spline fractal interpolation system.And also,the fractal interpolation functions can be explicitly expressed by the symmetric bases and the simple matrix form.Thirdly,we discussed some analytic properties of fractal interpolation surfaces.And the convergence property shows that the BRFIFs converge to original functions;stability shows that the BRFIFs have good capacity of stability when there are perturbation at data points.Fourthly,we provide a monotonicity-preserving bicubic rational fractal interpolation system.The monotonicity preserving properties of bicubic ratio-nal fractal surface can realize by selecting suitable scaling factors and shape parameters.The monotonicity properties of the curve and surface are impor-tant research topic in curve and surface modeling,it has a wide applications in practical design.Fifthly,we demonstrate the stability,quasi-locality and shape-preserving of the rational spline fractal interpolants by numerical Experiment.The surfaces generated by BRFIFs not only have good stability,but also shape-preserving.At last,we employ the BRFIFs to process image.The results of experi-ment demonstrate the model based on BRFIFs is superior to other compared algorithm in visual effect and objective data.