Fractal Interpolation Research Articles

On bivariate fractal interpolation for countable data and associated nonlinear fractal operator

Abstract Fractal interpolation has been conventionally treated as a method to construct a univariate continuous function interpolating a given finite data set with the distinguishing property that the graph of the interpolating function is the attractor of a suitable iterated function system. On the one hand, attempts have been made to extend the univariate fractal interpolation from a finite data set to a countably infinite set. On the other hand, fractal interpolation in higher dimensions, particularly the theory of fractal interpolation surfaces (FISs), has received increasing attention for more than a quarter century. This article targets a two-fold extension of the notion of fractal interpolation by providing a general framework to construct FISs for a prescribed set consisting of countably infinite data on a rectangular grid. By using this as a crucial tool, we obtain a parameterized family of bivariate fractal functions simultaneously interpolating and approximating a prescribed bivariate continuous function. Some elementary properties of the associated nonlinear (not necessarily linear) fractal operators are established, thereby allowing the interaction of the notion of fractal interpolation with the theory of nonlinear operators.

Multifractal analysis of fractal interpolation functions

This paper presents a novel algorithm to utilize multifractal spectrum as a quantitative measure for the fractal interpolation functions with respect to scaling factor and fractional order. As of yet, there were no error estimation techniques to interpret the fractal interpolation functions in the literature. To bridge this gap, this paper sketches multifractality as a quantitative measure for inquiring and comparing the effects of different scaling factors. The proposed algorithm for analyzing the multifractal measure depends on the probability measure of data points, which fractal function passes through, enabling to effectively discuss the heterogeneity of fractal interpolation functions. In addition, the impact of fractional orders on the fractional derivative (integral) of fractal interpolation functions is also discussed tailoring the multifractal measure.

Mpox outbreak: Time series analysis with multifractal and deep learning network.

This article presents an overview of an mpox epidemiological situation in the most affected regions-Africa, Americas, and Europe-tailoring fractal interpolation for pre-processing the mpox cases. This keen analysis has highlighted the irregular and fractal patterns in the trend of mpox transmission. During the current scenario of public health emergency of international concern due to an mpox outbreak, an additional significance of this article is the interpretation of mpox spread in light of multifractality. The self-similar measure, namely, the multifractal measure, is utilized to explore the heterogeneity in the mpox cases. Moreover, a bidirectional long-short term memory neural network has been employed to forecast the future mpox spread to alert the outbreak as it seems to be a silent symptom for global epidemic.

Positivity-Preserving Rational Cubic Fractal Interpolation Function Together with Its Zipper Form

In this paper, a novel class of rational cubic fractal interpolation function (RCFIF) has been proposed, which is characterized by one shape parameter and a linear denominator. In interpolation for shape preservation, the proposed rational cubic fractal interpolation function provides a simple but effective approach. The nature of shape preservation of the proposed rational cubic fractal interpolation function makes them valuable in the field of data visualization, as it is crucial to maintain the original data shape in data visualization. Furthermore, we discussed the upper bound of error and explored the mathematical framework to ensure the convergence of RCFIF. Shape parameters and scaling factors are constraints to obtain the desired shape-preserving properties. We further generalized the proposed RCFIF by introducing the concept of signature, giving its construction in the form of a zipper-rational cubic fractal interpolation function (ZRCFIF). The positivity conditions for the rational cubic fractal interpolation function and zipper-rational cubic fractal interpolation function are found, which required a detailed analysis of the conditions where constraints on shape parameters and scaling factor lead to the desired shape-preserving properties. In the field of shape preservation, the proposed rational cubic fractal interpolation function and zipper fractal interpolation function both represent significant advancement by offering a strong tool for data visualization.

Shape preserving aspects of a novel class of bi-cubic partially blended rational zipper fractal interpolation surfaces

Shape preserving aspects of a novel class of bi-cubic partially blended rational zipper fractal interpolation surfaces

On the box dimension of recurrent fractal interpolation functions defined with Matkowski contractions

On the box dimension of recurrent fractal interpolation functions defined with Matkowski contractions

Modeling two-dimensional ice shape based on fractal interpolation

The ice accretion data obtained from ice wind tunnel tests reveal a multiscale structure and rough surface. In the follow-up aerodynamic evaluation of icing airfoils, simplified two-dimensional ice shapes are generally used as substitutes, but this simplification changes the aerodynamic effects of the original ice shape. Therefore, finding a simple and effective two-dimensional ice shape simulation method is urgent. Due to the self-similarity characteristics of ice shape surfaces, fractal interpolation is proposed for ice shape simulation. First, the geometric characteristics of the ice shapes are analyzed to determine interpolation points, and an iterative function system is constructed for interpolation simulation. Considering the influence of various characteristics of ice shapes on aerodynamics, interpolation parameters are limited to simulating more realistic ice shapes. High-order numerical simulation methods were utilized to numerically simulate and analyze the aerodynamic characteristics of icing airfoil while also verifying the feasibility of fractal interpolation for simulating ice shapes. The analysis revealed that this method could effectively simulate ice profiles of various feature scales with minimal ice shape data. These simulated shapes closely resemble real ice formations and maintain the original aerodynamic characteristics of the icing airfoil. This method can be used to improve the computational accuracy of ice accretion codes and provide improvement strategies for complex ice shape prediction; thus, this method has great application prospects in engineering.

On [formula omitted]-fractal functions and their applications to analyzing the S&P BSE Sensex in India

Following the seminal work of Barnsley on fractal interpolation, Navascués (2005) defined a class of parametrized continuous functions called α-fractal functions. In this paper, we construct α-fractal functions in the Lebesgue spaces defined with respect to a fractal measure. We also show the existence of these fractal functions in smooth-dimensional spaces. Further, we define some set-valued maps associated with them and study some properties of these maps. Ultimately, we analyze the S&P BSE Sensex in India with the help of α-fractal functions.

Analytical properties and the box-counting dimension of nonlinear hidden variable recurrent fractal interpolation functions constructed by using Rakotch's fixed point theorem

Rakotch contraction is a generalization of Banach contraction, which implies that in the case of using Rakotch's fixed point theorem, we can model more objects than using Banach's fixed point theorem. Moreover, hidden variable recurrent fractal interpolation functions (HVRFIFs) with Hölder function factors are more general than the fractal interpolation functions (FIFs), recurrent FIFs and hidden variable FIFs with Lipschitz function factors. We demonstrate that HVRFIFs can be constructed using the Rakotch's fixed point theorem, and then investigate the analytical and geometric properties of those HVRFIFs. Firstly, we construct a nonlinear hidden variable recurrent fractal interpolation functions with Hölder function factors on the basis of a given data set using Rakotch contractions. Next, we analyze the smoothness of the HVRFIFs and show that they are stable on the small perturbations of the given data. Finally, we get the lower and upper bounds for their box-counting dimensions.

An approximation theoretic revamping of fractal interpolation surfaces on triangular domains

An approximation theoretic revamping of fractal interpolation surfaces on triangular domains

Zipper rational fractal interpolation functions

Zipper rational fractal interpolation functions

Contractive Multivariate Zipper Fractal Interpolation Functions

In this paper we introduce a new general multivariate fractal interpolation scheme using elements of the zipper methodology. Under the assumption that the corresponding Read-Bajraktarevic operator is well-defined, we enlarge the previous frameworks occurring in the literature, considering the constitutive functions of the iterated function system whose attractor is the graph of the interpolant to be just contractive in the last variable (so, in particular, they can be Banach contractions, Matkowski contractions, or Meir-Keeler contractions in the last variable). The main difficulty that should be overcome in this multivariate framework is the well definedness of the above mentioned operator. We provide three instances when it is guaranteed. We also display some examples that emphasize the generality of our scheme.

A novel fractal interpolation function algorithm for fractal dimension estimation and coastline geometry reconstruction: a case study of the coastline of Kingdom of Saudi Arabia

A novel fractal interpolation function algorithm for fractal dimension estimation and coastline geometry reconstruction: a case study of the coastline of Kingdom of Saudi Arabia

Fractal Interpolation Function based Thin Wire Antennas

This paper presents an approach for the design of wire antennas based on fractal interpolation functions (FIFs). The interpolation points and the contraction factors of the FIFs are chosen as free parameters to modify the antenna geometry. The proposed structures’ gain and radiation pattern can be optimized using FIF parameters. Producible prefractal antennas obtained in the intermediate iterations of fractal generation have compact sizes compared to classical counterparts. The error in prefractal geometry and the original fractal is bounded, and can be determined in terms of the finest producible detail’s dimensions. The emerging structures have multiband behavior due to their self-similar and symmetric nature. To illustrate the approach, we have provided finite element based simulations for several prefractal antennas. |S11|, the gain, the radiation efficiency, the radiation patterns, and feed point impedances for the demonstrated antennas are calculated numerically. The results indicate that produced antennas can be used in applications that require limited mechanical size, multiple operating bands, and controlled radiation patterns.

Some results on the space of rational cubic fractal interpolation functions

Some results on the space of rational cubic fractal interpolation functions

A very general framework for fractal interpolation functions

M. Barnsley introduced the concept of fractal interpolation function as an alternative method for construction of interpolation functions. Such a function is the attractor of an iterated function system comprising Banach contractions (on the product of two real compact intervals) with respect to the second variable. In the present paper we enlarge classical Barnsley's framework by allowing the constitutive functions of the system to be Edelstein contractions in the second variable. As the class of Edelstein contractions contains the class of Matkowski contractions, the class of Meir-Keeler contractions, the class of F-contractions and the class of θ-contractions, we provide a much more flexible framework for the construction of fractal interpolation functions. We also obtain a result concerning the estimation of lower and upper box dimensions of the graph of a fractal interpolation function constructed via our general method.

On the stability of Fractal interpolation functions with variable parameters

<abstract><p>Fractal interpolation function (FIF) is a fixed point of the Read–Bajraktarević operator defined on a suitable function space and is constructed via an iterated function system (IFS). In this paper, we considered the generalized affine FIF generated through the IFS defined by the functions $ W_n(x, y) = \big(a_n(x)+e_n, \alpha_n(x) y +\psi_n(x)\big) $, $ n = 1, \ldots, N $. We studied the shift of the fractal interpolation curve, by computing the error estimate in response to a small perturbation on $ \alpha_n(x) $. In addition, we gave a sufficient condition on the perturbed IFS so that it satisfies the continuity condition. As an application, we computed an upper bound of the maximum range of the perturbed FIF.</p></abstract>

CONSTRUCTION OF MONOTONOUS APPROXIMATION BY FRACTAL INTERPOLATION FUNCTIONS AND FRACTAL DIMENSIONS

In this paper, we research on the dimension preserving monotonous approximation by using fractal interpolation techniques. A constructive result of the approximating sequence of self-affine continuous functions has been given, which can converge to the object continuous function of bounded variation on [Formula: see text] monotonously and unanimously, meanwhile their graphs can be any value of the Hausdorff and the Box dimension between one and two. Further, such approximation for continuous functions of unbounded variation or even general continuous functions with non-integer fractal dimension has also been discussed elementarily.

The uniformly continuous theorem of fractal interpolation surface function and its proof

<abstract> <p>In order to research uniform continuity of fractal interpolation surface function on a closed rectangular area, the accumulation principle was applied to prove uniform continuity of fractal interpolation surface function on a closed rectangular area. First, fractal interpolation surface function was constructed by affine mapping. Second, the continuous concept of fractal interpolation surface function at a planar point in a three-dimensional cartesian coordinate space system and uniform continuity of fractal interpolation surface function on a closed rectangular area were defined in the paper. Finally, the uniformly continuous theorem of fractal interpolation surface function was proven through accumulation principle in the paper. The conclusion showed that fractal interpolation surface was uniformly continuous function on a closed rectangular area.</p> </abstract>

Zipper Quintic Fractal Interpolation Function for Curve Fitting

Zipper Quintic Fractal Interpolation Function for Curve Fitting