Monotonicity Preserving Rational Quadratic Fractal Interpolation Functions

A K B Chand,N Vijender

doi:10.1155/2014/504825

A K B Chand, N Vijender

Open Access

https://doi.org/10.1155/2014/504825

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Abstract

Fractal interpolation is an advanced technique for analysis and synthesis of scientific and engineering data. We introduce the 𝒞1-rational quadratic fractal interpolation functions (FIFs) through a suitable rational quadratic iterated function system (IFS). The novel notion of shape preserving fractal interpolation without any shape parameter is introduced through the rational fractal interpolation model in the literature for the first time. For a prescribed set of monotonic data, we derive the sufficient conditions by restricting the scaling factors for shape preserving 𝒞1-rational quadratic FIFs. A local modification pertaining to any subinterval is possible in this model if the scaling factors are chosen appropriately. We establish the convergence results of a monotonic rational quadratic FIF to the original function in 𝒞4. For given data with derivatives at grids, our approach generates several monotonicity preserving rational quadratic FIFs, whereas this flexibility is not available in the classical approach. Finally, numerical experiments support the importance of the developed rational quadratic IFS scheme through construction of visually pleasing monotonic rational fractal curves including the classical one.

Highlights

The interpolation of smooth curve shape constitutes a major research area for reconstruction and representation problems in medical imaging, computer aided geometric design, robotics, automobile engineering, architecture, and multimedia data representation
A rational quadratic fractal interpolation functions (FIFs) need not be monotonic for arbitrary choice of scaling factors for given monotonic data in general
A uniform error bound is deduced between the monotonic rational quadratic FIF and an original function

Summary

Introduction

The interpolation of smooth curve shape constitutes a major research area for reconstruction and representation problems in medical imaging, computer aided geometric design, robotics, automobile engineering, architecture, and multimedia data representation. The fractal interpolation is an advance technique in fitting of nonsmooth and smooth data from a physical or experimental set-up. To approximate data that follows some kind of self-similarity under magnification, Barnsley [1] introduced fractal interpolation functions (FIFs) defined on a compact interval in R based on the concept of an IFS [2]. These FIFs are not necessarily differentiable, and they differ from classical interpolants in the sense that (i) FIFs obey an implicit functional relation and (ii) FIFs have noninteger fractal dimensions in general. Strahle [12] found a method to determine

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