A rational iterated function system for resolution of univariate constrained interpolation

Abstract

Iterated Function Systems (IFSs) provide a standard framework for generating Fractal Interpolation Functions (FIFs) that yield smooth or non-smooth approximants. Nevertheless, the most widely studied FIFs so far in the literature that are obtained through polynomial IFSs are, in general, incapable of reproducing important shape properties inherent in a given data set. Abandoning the polynomiality of the functions defining the IFS, we introduce a new class of rational IFS that generates fractal functions (self-referential functions) for solving constrained interpolation problems. Suitable values of the rational IFS parameters are identified so that: (i) the corresponding FIF inherits positivity and/or monotonicity properties present in the data set, and (ii) the attractor of the IFS lies within an axis-aligned rectangle. The proposed IFS schemes for the shape preserving interpolation generalize some of the classical non-recursive interpolation methods, and expand the interpolation/approximation, including approximants for which functions themselves or the first derivatives can even be non-differentiable in a dense set of points of the domain. For appropriate values of the IFS parameters, the resulting rational quadratic FIF converges uniformly to the original function Φ ∈ C 3 [x 1 , x n ] with h 3 order of convergence, where h denotes the norm of the partition. We also provide a number of examples intended to demonstrate the proposed schemes and to suggest how these schemes outperform their classical counterparts.

Adaptive Interpolation Scheme for Image Magnification Based on Local Fractal Analysis

“…The more suitable the value of s i,j is, the more accurate the fractal functions are. Scaling factor is usually given a range of values or as a free parameter [33], [34]. In this paper, the scaling factor is calculated by fractal dimension, since the fractal dimension has a strong correlation with the scaling factor.…”