scholarly journals On Non-Tensor Product Bivariate Fractal Interpolation Surfaces on Rectangular Grids

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 525 ◽  
Author(s):  
Vasileios Drakopoulos ◽  
Polychronis Manousopoulos
Keyword(s):  
Comparative Study ◽  
Tensor Product ◽  
Necessary Conditions ◽  
Iterated Function System ◽  
Fractal Interpolation ◽  
Fractal Surfaces ◽  
Iterated Function ◽  
Data Points ◽  
Rectangular Grids ◽  
Pass Through

Some years ago, several authors tried to construct fractal surfaces which pass through a given set of data points. They used bivariable functions on rectangular grids, but the resulting surfaces failed to be continuous. A method based on their work for generating fractal interpolation surfaces is presented. Necessary conditions for the attractor of an iterated function system to be the graph of a continuous bivariable function which interpolates a given set of data are also presented here. Moreover, a comparative study for four of the most important constructions and attempts on rectangular grids is considered which points out some of their limitations and restrictions.

scholarly journals CONSTRUCTION OF RECURRENT FRACTAL INTERPOLATION SURFACES WITH FUNCTION SCALING FACTORS AND ESTIMATION OF BOX-COUNTING DIMENSION ON RECTANGULAR GRIDS

Fractals ◽  
2015 ◽  
Vol 23 (04) ◽  
pp. 1550030 ◽  
Author(s):  
CHOL-HUI YUN ◽  
HUI-CHOL CHOI ◽  
HYONG-CHOL O
Keyword(s):  
Iterated Function System ◽  
Vertical Scaling ◽  
Fractal Interpolation ◽  
Scaling Factors ◽  
Bivariate Interpolation ◽  
Box Counting ◽  
Box Counting Dimension ◽  
Iterated Function ◽  
Rectangular Grids ◽  
Bivariate Functions

We consider a construction of recurrent fractal interpolation surfaces (RFISs) with function vertical scaling factors and estimation of their box-counting dimension. A RFIS is an attractor of a recurrent iterated function system (RIFS) which is a graph of bivariate interpolation function. For any given dataset on rectangular grids, we construct general RIFSs with function vertical scaling factors and prove the existence of bivariate functions whose graph are attractors of the above-constructed RIFSs. Finally, we estimate lower and upper bounds for the box-counting dimension of the constructed RFISs.

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.

MULTI-DIMENSIONAL PIECE-WISE SELF-AFFINE FRACTAL INTERPOLATION MODEL IN TENSOR FORM

2006 ◽  
Vol 09 (03) ◽  
pp. 287-293 ◽  
Author(s):  
TONG ZHANG ◽  
JIANLIN LIU ◽  
ZHUO ZHUANG
Keyword(s):  
Partial Derivative ◽  
Iterated Function System ◽  
Model Parameters ◽  
Fractal Interpolation ◽  
Interpolation Model ◽  
Tensor Form ◽  
Real Numbers ◽  
Inverse Algorithm ◽  
Iterated Function ◽  
Discrete Sequences

Iterated Function System (IFS) models have been used to represent discrete sequences where the attractor of the IFS is piece-wise self-affine in R2 or R3 (R is the set of real numbers). In this paper, the piece-wise self-affine IFS model is extended from R3 to Rn (n is an integer greater than 3), which is called the multi-dimensional piece-wise self-affine fractal interpolation model. This model uses a "mapping partial derivative" and a constrained inverse algorithm to identify the model parameters. The model values depend continuously on all the model parameters, and represent most data which are not multi-dimensional self-affine in Rn. Therefore, the result is very general. Moreover, the multi-dimensional piece-wise self-affine fractal interpolation model in tensor form is more terse than in the usual matrix form.

SHAPE PRESERVING RATIONAL QUARTIC FRACTAL FUNCTIONS

Fractals ◽  
2019 ◽  
Vol 27 (08) ◽  
pp. 1950141 ◽  
Author(s):  
S. K. KATIYAR ◽  
A. K. B. CHAND
Keyword(s):  
Iterated Function System ◽  
Linear Equations ◽  
Fractal Interpolation ◽  
Fractal Function ◽  
Shape Preserving ◽  
Fractal Interpolation Function ◽  
Fractal Interpolation Functions ◽  
Iterated Function ◽  
Fractal Functions ◽  
Interpolation Functions

The appearance of fractal interpolation function represents a revival of experimental mathematics, raised by computers and intensified by powerful evidence of its applications. This paper is devoted to establish a method to construct [Formula: see text]-fractal rational quartic spline, which eventually provides a unified approach for the generalization of various traditional nonrecursive rational splines involving shape parameters. We deduce the uniform error bound for the [Formula: see text]-fractal rational quartic spline when the original function is in [Formula: see text]. By solving a system of linear equations, appropriate values of the derivative parameters are determined so as to enhance the continuity of the [Formula: see text]-fractal rational quartic spline to [Formula: see text]. The elements of the iterated function system are identified befittingly so that the class of [Formula: see text]-fractal function [Formula: see text] incorporates the geometric features such as positivity, monotonicity and convexity in addition to the regularity inherent in the germ [Formula: see text]. This general theory in conjunction with shape preserving aspects of the traditional splines provides algorithms for the construction of shape preserving fractal interpolation functions.

scholarly journals Non-Stationary Fractal Interpolation

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 666 ◽  
Author(s):  
Peter Massopust
Keyword(s):  
Fixed Points ◽  
Iterated Function System ◽  
Function System ◽  
Global Behavior ◽  
Fractal Interpolation ◽  
The Novel ◽  
Countable Sequence ◽  
Iterated Function ◽  
Fractal Functions ◽  
Novel Concept

We introduce the novel concept of a non-stationary iterated function system by considering a countable sequence of distinct set-valued maps { F k } k ∈ N where each F k maps H ( X ) → H ( X ) and arises from an iterated function system. Employing the recently-developed theory of non-stationary versions of fixed points and the concept of forward and backward trajectories, we present new classes of fractal functions exhibiting different local and global behavior and extend fractal interpolation to this new, more flexible setting.

EXISTENCE AND BOX DIMENSION OF GENERAL RECURRENT FRACTAL INTERPOLATION FUNCTIONS

2020 ◽  
pp. 1-13
Author(s):  
HUO-JUN RUAN ◽  
JIAN-CI XIAO ◽  
BING YANG
Keyword(s):  
Iterated Function System ◽  
General Setting ◽  
Iterated Function Systems ◽  
Box Dimension ◽  
Fractal Interpolation ◽  
Fractal Interpolation Functions ◽  
Iterated Function ◽  
Recurrent System ◽  
Definition Of ◽  
Interpolation Functions

Abstract The notion of recurrent fractal interpolation functions (RFIFs) was introduced by Barnsley et al. [‘Recurrent iterated function systems’, Constr. Approx.5 (1989), 362–378]. Roughly speaking, the graph of an RFIF is the invariant set of a recurrent iterated function system on $\mathbb {R}^2$ . We generalise the definition of RFIFs so that iterated functions in the recurrent system need not be contractive with respect to the first variable. We obtain the box dimensions of all self-affine RFIFs in this general setting.

Some Results on Recurrent Fractal Interpolation Function

2020 ◽  
Vol 12 (8) ◽  
pp. 1038-1043
Author(s):  
Wadia Faid Hassan Al-Shameri
Keyword(s):  
Iterated Function System ◽  
Iterated Function Systems ◽  
Function System ◽  
Fractal Interpolation ◽  
Interpolation Function ◽  
Data Set ◽  
Fractal Interpolation Function ◽  
Iterated Function ◽  
Fractal Functions ◽  
Fractal Approximation

Barnsley (Barnsley, M.F., 1986. Fractal functions and interpolation. Constr. Approx., 2, pp.303–329) introduced fractal interpolation function (FIF) whose graph is the attractor of an iterated function system (IFS) for describing the data that have an irregular or self-similar structure. Barnsley et al. (Barnsley, M.F., et al., 1989. Recurrent iterated function systems in fractal approximation. Constr. Approx., 5, pp.3–31) generalized FIF in the form of recurrent fractal interpolation function (RFIF) whose graph is the attractor of a recurrent iterated function system (RIFS) to fit data set which is piece-wise self-affine. The primary aim of the present research is investigating the RFIF approach and using it for fitting the piece-wise self-affine data set in ℜ2.

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 Uniform Continuity of Fractal Interpolation Function

2020 ◽  
Vol 2020 ◽  
pp. 1-5
Author(s):  
Xuezai Pan ◽  
Minggang Wang ◽  
Xudong Shang
Keyword(s):  
Iterated Function System ◽  
Uniform Continuity ◽  
Fractal Interpolation ◽  
Interpolation Function ◽  
Covering Theorem ◽  
Fractal Interpolation Function ◽  
Iterated Function ◽  
Definition Of ◽  
Uniformly Continuous ◽  
Research Analysis

In order to research analysis properties of fractal interpolation function generated by the iterated function system defined by affine transformation, the continuity of fractal interpolation function is proved by the continuous definition of function and the uniform continuity of fractal interpolation function is proved by the definition of uniform continuity and compactness theorem of sequence of numbers or finite covering theorem in this paper. The result shows that the fractal interpolation function is uniformly continuous in a closed interval which is from the abscissa of the first interpolation point to that of the last one.

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