A fractal interpolation function (FIF) f is a special type of continuous function that interpolates a given set of data {(xi, f(xi)) ∈ ℝ2: i=0,1,...,N}. The attractor of a hyperbolic iterated function system (IFS) associated with the set of data {(xi, yi): i=0,1,...,N} is the graph of the FIFs f with f(xi)=yifor i=0,1,...,N.
The free parameter in the IFS, called the vertical scaling factor, plays an important role in the shape and properties of the corresponding FIFs. By taking constant free parameters in an IFS and making the same vertical compression ratios on an identical subinterval, one can get the obvious self-similarity for the corresponding FIFs.
In this paper, the authors used variable free parameters instead of constant free parameters, which in the more general sense allow more flexibility and diversity in the nature of the corresponding FIFs. They are more suitable for fitting and approximating many complicated curves and for data that display less self-similarity. The smoothness of the FIFs discussed here and the technique used are different from those in other papers in the literature. The stability and sensitivity analysis of the FIFs are also discussed. The authors investigate errors caused by the perturbations of the IFSs (in the scaling parameter) generating these FIFs, and obtain an upper bound for the errors. From the stability result, one can see that the FIFs with variable parameters introduced in this paper remain stable in response to small perturbations of the interpolation points. Also, the sensitivity result shows that slight perturbations in the iterated mappings lead to small changes in the corresponding FIFs. The results on stability and sensitivity of the FIFs with variable parameters establish a theoretical foundation for their practical application in areas such as data fitting, approximation of functions, signal processing, and computer graphics.
I found this paper well written and very interesting.
