Rational iterated function system for positive/monotonic shape preservation

Akb Chand,N Vijender,Rp Agarwal

doi:10.1186/1687-1847-2014-30

Akb Chand, N Vijender + Show 1 more

Open Access

https://doi.org/10.1186/1687-1847-2014-30

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Abstract

In this paper we consider the (inverse) problem of determining the iterated function system (IFS) which produces a shaped fractal interpolant. We develop a new type of rational IFS by using functions of the form E i F i , where E i are cubics and F i are preassigned quadratics having 3-shape parameters. The fixed point of the developed rational cubic IFS is in C 1 , but its derivative varies from a piecewise differentiable function to a continuous nowhere differentiable function. An upper bound of the uniform error between the fixed point of a rational IFS and an original function Φ∈ C 4 is deduced for the convergence results. The automatic generations of the scaling factors and shape parameters in the rational IFS are formulated so that its fixed point preserves the positive/monotonic features of prescribed data. The presence of scaling factors provides additional freedom to the shape of the fractal interpolant over its classical counterpart in the modeling of discrete data.

Highlights

Setting a novel platform for the approximation of natural objects such as trees, clouds, feathers, leaves, flowers, landscapes, glaciers, galaxies, and torrents of water, Mandelbrot [ ] introduced the term fractal in the literature
(ii) Again from the error estimation ( ), O(hp) (p =, ) convergence can be obtained if the derivative values are available such that ζi = O(hpi – ) (p =, ), and the scaling factors are chosen as |ξi| ≤ κapi – (p =, ) for i =, . . . , n
6 Conclusion A new type of rational cubic iterated function system (IFS) with -shape parameters is introduced in this work such that its fixed point can be used for shaped data

Summary

Introduction

Setting a novel platform for the approximation of natural objects such as trees, clouds, feathers, leaves, flowers, landscapes, glaciers, galaxies, and torrents of water, Mandelbrot [ ] introduced the term fractal in the literature. We introduce the rational cubic IFS with -shape parameters in each subinterval of the interpolation domain such that its fixed point generalizes the corresponding classical rational cubic spline functions [ ]. The developed rational cubic spline FIF is bounded, and is unique by fixed point theory for a given set of scaling factors and shape parameters. By varying the scaling factors (within the shape-preserving interval) and shape parameters (according to the conditions derived in our theory), we can make the fixed point of a rational cubic IFS more pleasant and suitable for aesthetic requirements in a modeling problem. By taking different sets of scaling and shape parameters, we can generate an infinite number of fixed points for the above rational cubic IFS. The rational cubic FIF reduces to the rational quadratic function as in [ ]

Error analysis of fixed point of rational cubic IFS

Conclusion