Hidden variable bivariate fractal interpolation functions and errors on perturbations of function vertical scaling factors

2019 ◽  
Vol 12 (02) ◽  
pp. 1950021 ◽  
Author(s):  
Chol-Hui Yun ◽  
Mi-Kyong Ri
Keyword(s):  
Iterated Function System ◽  
Scaling Factor ◽  
Function System ◽  
Vertical Scaling ◽  
Fractal Interpolation ◽  
Scaling Factors ◽  
Hidden Variable ◽  
Fractal Interpolation Functions ◽  
Iterated Function ◽  
Interpolation Functions

In this paper, we present a construction of hidden variable bivariate fractal interpolation functions (HVBFIFs) with function vertical scaling factors and estimate errors of HVBFIFs on perturbation of the function vertical scaling factor. We construct HVBFIFs on the basis of the iterated function system (IFS) with function vertical scaling factors. The perturbation of the function vertical scaling factors in the IFS causes a change in the HVBFIF. An upper estimation of the errors between the original HVBFIF and the perturbed HVBFIF is given.

scholarly journals On the Bernstein Affine Fractal Interpolation Curved Lines and Surfaces

Axioms ◽  
2020 ◽  
Vol 9 (4) ◽  
pp. 119
Author(s):  
Nallapu Vijender ◽  
Vasileios Drakopoulos
Keyword(s):  
Generating Function ◽  
Iterated Function System ◽  
Three Dimensions ◽  
Function System ◽  
Fractal Interpolation ◽  
Scaling Factors ◽  
Fractal Interpolation Functions ◽  
Iterated Function ◽  
Interpolation Functions

In this article, firstly, an overview of affine fractal interpolation functions using a suitable iterated function system is presented and, secondly, the construction of Bernstein affine fractal interpolation functions in two and three dimensions is introduced. Moreover, the convergence of the proposed Bernstein affine fractal interpolation functions towards the data generating function does not require any condition on the scaling factors. Consequently, the proposed Bernstein affine fractal interpolation functions possess irregularity at any stage of convergence towards the data generating function.

HIDDEN VARIABLE RECURRENT FRACTAL INTERPOLATION FUNCTIONS WITH FUNCTION CONTRACTIVITY FACTORS

Fractals ◽  
2019 ◽  
Vol 27 (07) ◽  
pp. 1950113
Author(s):  
CHOL-HUI YUN
Keyword(s):  
Iterated Function System ◽  
Function System ◽  
Fractal Interpolation ◽  
Interpolation Function ◽  
Hidden Variable ◽  
Fractal Interpolation Function ◽  
Fractal Interpolation Functions ◽  
Iterated Function ◽  
Better Than ◽  
Interpolation Functions

In this paper, we introduce a construction of hidden variable recurrent fractal interpolation functions (HVRFIFs) with four function contractivity factors. The HVRFIF is a hidden variable fractal interpolation function (HVFIF) constructed using a recurrent iterated function system (RIFS). In the fractal interpolation theory, it is very important to ensure flexibility and diversity of the construction of interpolation functions. RIFSs produce fractal sets with local self-similarity structure. Therefore, the RIFS can describe the irregular and complicated objects in nature better than the iterated function system (IFS). The HVFIF is neither self-similar nor self-affine one. Hence, the HVFIF is more complicated, diverse and irregular than the fractal interpolation function (FIF). The contractivity factors of IFS are very important one that determines characteristics of FIFs. The IFS and RIFS with function contractivity factors can describe the fractal objects in nature better than one with constant contractivity factors. To ensure higher flexibility and diversity of the construction of the FIFs, we present constructions of one variable HVRFIFs and bivariable HVRFIFs using RIFS with four function contractivity factors.

SENSITIVITY ANALYSIS FOR HIDDEN VARIABLE FRACTAL INTERPOLATION FUNCTIONS AND THEIR MOMENTS

Fractals ◽  
2009 ◽  
Vol 17 (02) ◽  
pp. 161-170
Author(s):  
HONG-YONG WANG
Keyword(s):  
Sensitivity Analysis ◽  
Iterated Function System ◽  
Fractal Interpolation ◽  
Hidden Variable ◽  
Fractal Interpolation Functions ◽  
Iterated Function ◽  
The Difference ◽  
The Moment ◽  
Upper Estimate ◽  
Interpolation Functions

The sensitivity analysis for a class of hidden variable fractal interpolation functions (HVFIFs) and their moments is made in the work. Based on a vector valued iterated function system (IFS) determined, we introduce a perturbed IFS and investigate the relations between the two HVFIFs generated by the IFS determined and its perturbed IFS, respectively. An explicit expression for the difference between the two HVFIFs is presented, from which, we show that the HVFIFs are not sensitive to a small perturbation in IFSs. Furthermore, we compute the moment integrals of the HVFIFs and discuss the error of moments of the two HVFIFs. An upper estimate for the error is obtained.

scholarly journals Fractal Interpolation Using Harmonic Functions on the Koch Curve

2021 ◽  
Vol 5 (2) ◽  
pp. 28
Author(s):  
Song-Il Ri ◽  
Vasileios Drakopoulos ◽  
Song-Min Nam
Keyword(s):  
Harmonic Functions ◽  
Iterated Function System ◽  
Function System ◽  
Fractal Interpolation ◽  
Koch Curve ◽  
Fractal Sets ◽  
Fractal Interpolation Function ◽  
Fractal Interpolation Functions ◽  
Iterated Function ◽  
Interpolation Functions

The Koch curve was first described by the Swedish mathematician Helge von Koch in 1904 as an example of a continuous but nowhere differentiable curve. Such functions are now characterised as fractal since their graphs are in general fractal sets. Furthermore, it can be obtained as the graph of an appropriately chosen iterated function system. On the other hand, a fractal interpolation function can be seen as a special case of an iterated function system thus maintaining all of its characteristics. Fractal interpolation functions are continuous functions that can be used to model continuous signals. An in-depth discussion on the theory of affine fractal interpolation functions generating the Koch Curve by using fractal analysis as well as its recent development including some of the research made by the authors is provided. We ensure that the graph of fractal interpolation functions on the Koch Curve are attractors of an iterated function system constructed by non-constant harmonic functions.

scholarly journals Spline coalescence hidden variable fractal interpolation functions

2006 ◽  
Vol 2006 ◽  
pp. 1-17 ◽  
Author(s):  
A. K. B. Chand ◽  
G. P. Kapoor
Keyword(s):  
Iterated Function System ◽  
Hidden Variables ◽  
Fractal Interpolation ◽  
Fractal Function ◽  
Hidden Variable ◽  
Fractal Interpolation Function ◽  
Fractal Interpolation Functions ◽  
Iterated Function ◽  
New Construction ◽  
Interpolation Functions

This paper generalizes the classical spline using a new construction of spline coalescence hidden variable fractal interpolation function (CHFIF). The derivative of a spline CHFIF is a typical fractal function that is self-affine or non-self-affine depending on the parameters of a nondiagonal iterated function system. Our construction generalizes the construction of Barnsley and Harrington (1989), when the construction is not restricted to a particular type of boundary conditions. Spline CHFIFs are likely to be potentially useful in approximation theory due to effects of the hidden variables and these effects are demonstrated through suitable examples in the present work.

The Study on Bivariate Fractal Interpolation Functions and Creation of Fractal Interpolated Surfaces

Fractals ◽  
1997 ◽  
Vol 05 (04) ◽  
pp. 625-634 ◽  
Author(s):  
Heping Xie ◽  
Hongquan Sun
Keyword(s):  
Fractal Dimension ◽  
Iterated Function System ◽  
Function System ◽  
Fractal Interpolation ◽  
Fractal Surface ◽  
Fractal Interpolation Functions ◽  
Iterated Function ◽  
Interpolation Functions

In this paper, the methods of construction of a fractal surface are introduced, the principle of bivariate fractal interpolation functions is discussed. The theorem of the uniqueness of an iterated function system of bivariate fractal interpolation functions is proved. Moreover, the theorem of fractal dimension of fractal interpolated surface is derived. Based on these theorems, the fractal interpolated surfaces are created by using practical data.

scholarly journals ANALYTIC PROPERTIES OF HIDDEN VARIABLE BIVARIABLE FRACTAL INTERPOLATION FUNCTIONS WITH FOUR FUNCTION CONTRACTIVITY FACTORS

Fractals ◽  
2019 ◽  
Vol 27 (02) ◽  
pp. 1950018 ◽  
Author(s):  
CHOL-HUI YUN ◽  
MI-KYONG LI
Keyword(s):  
Experimental Data ◽  
Data Fitting ◽  
Vertical Scaling ◽  
Fractal Interpolation ◽  
Self Similarity ◽  
Scaling Factors ◽  
Small Perturbations ◽  
Hidden Variable ◽  
Fractal Interpolation Functions ◽  
Interpolation Functions

In this paper, we present some analytic properties of hidden variable bivariable fractal interpolation functions (HVBFIFs) with four function contractivity factors presented in [C. H. Yun and M. K. Li, Hidden variable bivariate fractal interpolation functions and errors on perturbations of function vertical scaling factors, Asian-Eur. J. Math. (2017), doi:10.1142/s1793557119500219]. Since four contractivity factors of these HVBFIFs are all functions, the construction of these HVBFIFs has more flexibility and diversity in fitting and approximation of complicated surfaces in nature and irregular experimental data with less self-similarity than one whose four contractivity factors are all constants or only one factor is function. The smoothness and stability of HVBFIFs are needed to ensure the applicability of the HVBFIFs in many practical problems such as the simulation of the objects of the nature, data fitting, etc. We first obtain the results related to their smoothness in nine different cases and then prove that the HVBFIFs are stable to the small perturbations of the interpolation points.

NONLINEAR RECURRENT HIDDEN VARIABLE FRACTAL INTERPOLATION CURVES WITH FUNCTION VERTICAL SCALING FACTORS

Fractals ◽  
2020 ◽  
Vol 28 (06) ◽  
pp. 2050096
Author(s):  
JINMYONG KIM ◽  
HAKMYONG MUN
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Iterated Function Systems ◽  
Vertical Scaling ◽  
Fractal Interpolation ◽  
Scaling Factors ◽  
Hidden Variable ◽  
Iterated Function

In this paper, we present a construction of new nonlinear recurrent hidden variable fractal interpolation curves. In order to get new fractal curves, we use Rakotch’s fixed point theorem. We construct recurrent hidden variable iterated function systems with function vertical scaling factors to generate more flexible fractal interpolation curves. We also give an illustrative example to demonstrate the effectiveness of our results.

Hidden Variable Bivariate Fractal Interpolation Surfaces

Fractals ◽  
2003 ◽  
Vol 11 (03) ◽  
pp. 277-288 ◽  
Author(s):  
A. K. B. Chand ◽  
G. P. Kapoor
Keyword(s):  
Iterated Function System ◽  
Degree Of Freedom ◽  
Function System ◽  
Fractal Interpolation ◽  
Fractal Surface ◽  
Hidden Variable ◽  
Iterated Function ◽  
Self Similar ◽  
Vector Valued

We construct hidden variable bivariate fractal interpolation surfaces (FIS). The vector valued iterated function system (IFS) is constructed in ℝ4 and its projection in ℝ3 is taken. The extra degree of freedom coming from ℝ4 provides hidden variable, which is an important factor for flexibility and diversity in the interpolated surface. In the present paper, we construct an IFS that generates both self-similar and non-self-similar FIS simultaneously and show that the hidden variable fractal surface may be self-similar under certain conditions.

CONSTRUCTION OF NONLINEAR HIDDEN VARIABLE FRACTAL INTERPOLATION FUNCTIONS AND THEIR STABILITY

Fractals ◽  
2019 ◽  
Vol 27 (06) ◽  
pp. 1950103
Author(s):  
JINMYONG KIM ◽  
HYONJIN KIM ◽  
HAKMYONG MUN
Keyword(s):  
Iterated Function Systems ◽  
Continuous Functions ◽  
Vertical Scaling ◽  
Fractal Interpolation ◽  
Hidden Variable ◽  
Fractal Interpolation Functions ◽  
The Stability ◽  
And Function ◽  
Stability Results ◽  
Interpolation Functions

This paper presents a method to construct nonlinear hidden variable fractal interpolation functions (FIFs) and their stability results. We ensure that the projections of attractors of vector-valued nonlinear iterated function systems (IFSs) constructed by Rakotch contractions and function vertical scaling factors are graphs of some continuous functions interpolating the given data. We also give an explicit example illustrating obtained results. Then, we get the stability results of the constructed FIFs in the case of the generalized interpolation data having small perturbations.

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