scholarly journals Box dimension for graphs of fractal functions

1997 ◽  
Vol 40 (2) ◽  
pp. 331-344
Author(s):  
Gavin Brown ◽  
Qinghe Yin
Keyword(s):  
Iterated Function Systems ◽  
Box Dimension ◽  
Affine Maps ◽  
Iterated Function ◽  
Fractal Functions

We calculate the box-dimension for a class of nowhere differentiable curves defined by Markov attractors of certain iterated function systems of affine maps.

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.

BOX DIMENSIONS OF α-FRACTAL FUNCTIONS

Fractals ◽  
2016 ◽  
Vol 24 (03) ◽  
pp. 1650037 ◽  
Author(s):  
MD. NASIM AKHTAR ◽  
M. GURU PREM PRASAD ◽  
M. A. NAVASCUÉS
Keyword(s):  
Continuous Function ◽  
Differentiable Function ◽  
Iterated Function System ◽  
Box Dimension ◽  
Data Sets ◽  
Fractal Function ◽  
Iterated Function ◽  
Nowhere Differentiable Function ◽  
Fractal Functions ◽  
Box Dimensions

The box dimension of the graph of non-affine, continuous, nowhere differentiable function [Formula: see text] which is a fractal analogue of a continuous function [Formula: see text] corresponding to a certain iterated function system (IFS), is investigated in the present paper. The estimates for box dimension of the graph of [Formula: see text]-fractal function [Formula: see text] for equally spaced as well as arbitrary data sets are found.

A new class of markov processes for image encoding

1988 ◽  
Vol 20 (01) ◽  
pp. 14-32 ◽  
Author(s):  
Michael F. Barnsley ◽  
John H. Elton
Keyword(s):  
Invariant Measures ◽  
Metric Spaces ◽  
Iterated Function Systems ◽  
Interesting Case ◽  
Image Encoding ◽  
New Class ◽  
Affine Maps ◽  
Iterated Maps ◽  
Iterated Function ◽  
Infinite Extent

A new class of iterated function systems is introduced, which allows for the computation of non-compactly supported invariant measures, which may represent, for example, greytone images of infinite extent. Conditions for the existence and attractiveness of invariant measures for this new class of randomly iterated maps, which are not necessarily contractions, in metric spaces such as , are established. Estimates for moments of these measures are obtained. Special conditions are given for existence of the invariant measure in the interesting case of affine maps on . For non-singular affine maps on , the support of the measure is shown to be an infinite interval, but Fourier transform analysis shows that the measure can be purely singular even though its distribution function is strictly increasing.

scholarly journals Dimensions of Fractals Generated by Bi-Lipschitz Maps

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Qi-Rong Deng ◽  
Sze-Man Ngai
Keyword(s):  
Iterated Function Systems ◽  
Topological Pressure ◽  
Separation Condition ◽  
Lipschitz Mappings ◽  
Affine Maps ◽  
Lipschitz Maps ◽  
Iterated Function ◽  
Bounded Distortion ◽  
Box Dimensions

On the class of iterated function systems of bi-Lipschitz mappings that are contractions with respect to some metrics, we introduce a logarithmic distortion property, which is weaker than the well-known bounded distortion property. By assuming this property, we prove the equality of the Hausdorff and box dimensions of the attractor. We also obtain a formula for the dimension of the attractor in terms of certain modified topological pressure functions, without imposing any separation condition. As an application, we prove the equality of Hausdorff and box dimensions for certain iterated function systems consisting of affine maps and nonsmooth maps.

Dimension of the generalized 4-corner set and its projections

2011 ◽  
Vol 32 (4) ◽  
pp. 1190-1215 ◽  
Author(s):  
BALÁZS BÁRÁNY
Keyword(s):  
Hausdorff Dimension ◽  
Fixed Points ◽  
Iterated Function Systems ◽  
Common Fixed Points ◽  
New Method ◽  
Dimension Theory ◽  
Box Dimension ◽  
Extended Version ◽  
Iterated Function ◽  
Self Similar

AbstractIn the last two decades, considerable attention has been paid to the dimension theory of self-affine sets. In the case of generalized 4-corner sets (see Figure 1), the iterated function systems obtained as the projections of self-affine systems have maps of common fixed points. In this paper, we extend our result [B. Bárány. On the Hausdorff dimension of a family of self-similar sets with complicated overlaps. Fund. Math. 206 (2009), 49–59], which introduced a new method of computation of the box and Hausdorff dimensions of self-similar families where some of the maps have common fixed points. The extended version of our method presented in this paper makes it possible to determine the box dimension of the generalized 4-corner set for Lebesgue-typical contracting parameters.

A new class of markov processes for image encoding

1988 ◽  
Vol 20 (1) ◽  
pp. 14-32 ◽  
Author(s):  
Michael F. Barnsley ◽  
John H. Elton
Keyword(s):  
Invariant Measures ◽  
Metric Spaces ◽  
Iterated Function Systems ◽  
Interesting Case ◽  
Image Encoding ◽  
New Class ◽  
Affine Maps ◽  
Iterated Maps ◽  
Iterated Function ◽  
Infinite Extent

A new class of iterated function systems is introduced, which allows for the computation of non-compactly supported invariant measures, which may represent, for example, greytone images of infinite extent. Conditions for the existence and attractiveness of invariant measures for this new class of randomly iterated maps, which are not necessarily contractions, in metric spaces such as , are established. Estimates for moments of these measures are obtained.Special conditions are given for existence of the invariant measure in the interesting case of affine maps on . For non-singular affine maps on , the support of the measure is shown to be an infinite interval, but Fourier transform analysis shows that the measure can be purely singular even though its distribution function is strictly increasing.

FRACTAL BASES FOR BANACH SPACES OF SMOOTH FUNCTIONS

2015 ◽  
Vol 92 (3) ◽  
pp. 405-419 ◽  
Author(s):  
M. A. NAVASCUÉS ◽  
P. VISWANATHAN ◽  
A. K. B. CHAND ◽  
M. V. SEBASTIÁN ◽  
S. K. KATIYAR
Keyword(s):  
Hermite Interpolation ◽  
Iterated Function Systems ◽  
Smooth Functions ◽  
Bounded Linear ◽  
Fractal Interpolation Functions ◽  
Iterated Function ◽  
Scaling Parameters ◽  
Fractal Functions ◽  
Present Setting ◽  
Interpolation Functions

This article explores the properties of fractal interpolation functions with variable scaling parameters, in the context of smooth fractal functions. The first part extends the Barnsley–Harrington theorem for differentiability of fractal functions and the fractal analogue of Hermite interpolation to the present setting. The general result is applied on a special class of iterated function systems in order to develop differentiability of the so-called $\boldsymbol{{\it\alpha}}$-fractal functions. This leads to a bounded linear map on the space ${\mathcal{C}}^{k}(I)$ which is exploited to prove the existence of a Schauder basis for ${\mathcal{C}}^{k}(I)$ consisting of smooth fractal functions.

SOLVING THE INVERSE PROBLEM FOR FUNCTION/IMAGE APPROXIMATION USING ITERATED FUNCTION SYSTEMS II: ALGORITHM AND COMPUTATIONS

Fractals ◽  
1994 ◽  
Vol 02 (03) ◽  
pp. 335-346 ◽  
Author(s):  
BRUNO FORTE ◽  
EDWARD R. VRSCAY
Keyword(s):  
Inverse Problem ◽  
Quadratic Programming ◽  
Iterated Function Systems ◽  
Grey Level ◽  
Image Approximation ◽  
Affine Maps ◽  
Iterated Function ◽  
Contractive Maps ◽  
Target Set ◽  
Infinite Set

In this paper, we provide an algorithm for the construction of IFSM approximations to a target set [Formula: see text], where X ⊂ RD and µ = m(D) (Lebesgue measure). The algorithm minimizes the squared "collage distance" [Formula: see text]. We work with an infinite set of fixed affine IFS maps wi: X → X satisfying a certain density and nonoverlapping condition. As such, only an optimization over the grey level maps ϕi: R+ → R+ is required. If affine maps are assumed, i.e. ϕi = αit + βi, then the algorithm becomes a quadratic programming (QP) problem in the αi and βi. We can also define a "local IFSM" (LIFSM) which considers the actions of contractive maps wi on subsets of X to produce smaller subsets. Again, affine ϕi maps are used, resulting in a QP problem. Some approximations of functions on [0,1] and images in [0, 1]2 are presented.

A REVISIT TO α-FRACTAL FUNCTION AND BOX DIMENSION OF ITS GRAPH

Fractals ◽  
2019 ◽  
Vol 27 (06) ◽  
pp. 1950090
Author(s):  
S. VERMA ◽  
P. VISWANATHAN
Keyword(s):  
Fractal Geometry ◽  
Iterated Function System ◽  
Wide Spectrum ◽  
Box Dimension ◽  
Fractal Interpolation ◽  
Fractal Function ◽  
Fractal Interpolation Function ◽  
Iterated Function ◽  
Fractal Functions ◽  
Constructive Approximation

One of the tools offered by fractal geometry is fractal interpolation, which forms a basis for the constructive approximation theory for nondifferentiable functions. The notion of fractal interpolation function can be used to obtain a wide spectrum of self-referential functions associated to a prescribed continuous function on a compact interval in [Formula: see text]. These fractal maps, the so-called [Formula: see text]-fractal functions, are defined by means of suitable iterated function system which involves some parameters. Building on the literature related to the notion of [Formula: see text]-fractal functions, the current study targets to record the continuous dependence of the [Formula: see text]-fractal function on parameters involved in its definition. Furthermore, the paper attempts to study the box dimension of the graph of the [Formula: see text]-fractal function.

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