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.

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 Attractors of directed graph IFSs that are not standard IFS attractors and their Hausdorff measure

2012 ◽  
Vol 154 (2) ◽  
pp. 325-349 ◽  
Author(s):  
G. C. BOORE ◽  
K. J. FALCONER
Keyword(s):  
Directed Graph ◽  
Hausdorff Measure ◽  
Iterated Function Systems ◽  
New Class ◽  
Iterated Function ◽  
Separation Conditions

AbstractFor directed graph iterated function systems (IFSs) defined on ℝ, we prove that a class of 2-vertex directed graph IFSs have attractors that cannot be the attractors of standard (1-vertex directed graph) IFSs, with or without separation conditions. We also calculate their exact Hausdorff measure. Thus we are able to identify a new class of attractors for which the exact Hausdorff measure is known.

scholarly journals Ledrappier–Young formulae for a family of nonlinear attractors

2021 ◽  
Author(s):  
Natalia Jurga ◽  
Lawrence D. Lee
Keyword(s):  
Invariant Measures ◽  
Gibbs Measures ◽  
Iterated Function Systems ◽  
Hölder Continuous ◽  
Natural Class ◽  
Iterated Function ◽  
Bernoulli Measures

AbstractWe study a natural class of invariant measures supported on the attractors of a family of nonlinear, non-conformal iterated function systems introduced by Falconer, Fraser and Lee. These are pushforward quasi-Bernoulli measures, a class which includes the well-known class of Gibbs measures for Hölder continuous potentials. We show that these measures are exact dimensional and that their exact dimensions satisfy a Ledrappier–Young formula.

scholarly journals Attractors of iterated function systems and Markov operators

2003 ◽  
Vol 2003 (8) ◽  
pp. 479-502 ◽  
Author(s):  
Józef Myjak ◽  
Tomasz Szarek
Keyword(s):  
Iterated Function Systems ◽  
Markov Operators ◽  
New Class ◽  
Borel Measures ◽  
Transition Functions ◽  
Iterated Function ◽  
Banach Contraction ◽  
Complete Separable Metric Space ◽  
The Banach Contraction Principle ◽  
Separable Metric Space

This paper contains a review of results concerning “generalized” attractors for a large class of iterated function systems{wi:i∈I}acting on a complete separable metric space. This generalization, which originates in the Banach contraction principle, allows us to consider a new class of sets, which we call semi-attractors (or semifractals). These sets have many interesting properties. Moreover, we give some fixed-point results for Markov operators acting on the space of Borel measures, and we show some relations between semi-attractors and supports of invariant measures for such Markov operators. Finally, we also show some relations between multifunctions and transition functions appearing in the theory of Markov operators.

scholarly journals Approximating distribution functions by iterated function systems

2005 ◽  
Vol 2005 (1) ◽  
pp. 33-46 ◽  
Author(s):  
Stefano Maria Iacus ◽  
Davide La Torre
Keyword(s):  
Monte Carlo ◽  
Distribution Function ◽  
Iterated Function System ◽  
Asymptotic Properties ◽  
Iterated Function Systems ◽  
Monte Carlo Analysis ◽  
Distribution Functions ◽  
Density Estimator ◽  
New Class ◽  
Iterated Function

An iterated function system (IFS) on the space of distribution functions is built with the aim of proposing a new class of distribution function estimators. One IFS estimator and its asymptotic properties are studied in detail. We also propose a density estimator derived from the IFS distribution function estimator by using Fourier analysis. Relative efficiencies of both estimators, for small and moderate sample sizes, are presented via Monte Carlo analysis.

scholarly journals Duality results for iterated function systems with a general family of branches

2017 ◽  
Vol 17 (03) ◽  
pp. 1750021 ◽  
Author(s):  
Jairo K. Mengue ◽  
Elismar R. Oliveira
Keyword(s):  
Cost Function ◽  
Relative Entropy ◽  
Metric Spaces ◽  
Iterated Function Systems ◽  
Compact Spaces ◽  
Compact Metric Spaces ◽  
Duality Results ◽  
General Family ◽  
Iterated Function ◽  
Kantorovich Duality

Given [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] compact metric spaces, we consider two iterated function systems [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text] are contractions. Let [Formula: see text] be the set of probabilities [Formula: see text] with [Formula: see text]-marginal being holonomic with respect to [Formula: see text] and [Formula: see text]-marginal being holonomic with respect to [Formula: see text]. Given [Formula: see text] and [Formula: see text], let [Formula: see text] be the set of probabilities in [Formula: see text] having [Formula: see text]-marginal [Formula: see text] and [Formula: see text]-marginal [Formula: see text]. Let [Formula: see text] be the relative entropy of [Formula: see text] with respect to [Formula: see text] and [Formula: see text] be the relative entropy of [Formula: see text] with respect to [Formula: see text]. Given a cost function [Formula: see text], let [Formula: see text]. We will prove the duality equation: [Formula: see text] In particular, if [Formula: see text] and [Formula: see text] are single points and we drop the entropy, the equation above can be rewritten as the Kantorovich duality for the compact spaces [Formula: see text] and a continuous cost function [Formula: see text].

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.

A NEW CLASS OF FRACTAL INTERPOLATION SURFACES BASED ON FUNCTIONAL VALUES

Fractals ◽  
2016 ◽  
Vol 24 (01) ◽  
pp. 1650007 ◽  
Author(s):  
A. K. B. CHAND ◽  
N. VIJENDER
Keyword(s):  
Iterated Function Systems ◽  
Fractal Interpolation ◽  
Shape Parameters ◽  
Positive Real ◽  
Scaling Factors ◽  
Numerical Illustration ◽  
New Class ◽  
Convergence Results ◽  
Iterated Function

Fractal interpolation is a modern technique for fitting of smooth/non-smooth data. Based on only functional values, we develop two types of [Formula: see text]-rational fractal interpolation surfaces (FISs) on a rectangular grid in the present paper that contain scaling factors in both directions and two types of positive real parameters which are referred as shape parameters. The graphs of these [Formula: see text]-rational FISs are the attractors of suitable rational iterated function systems (IFSs) in [Formula: see text] which use a collection of rational IFSs in the [Formula: see text]-direction and [Formula: see text]-direction and hence these FISs are self-referential in nature. Using upper bounds of the interpolation error of the [Formula: see text]-direction and [Formula: see text]-direction fractal interpolants along the grid lines, we study the convergence results of [Formula: see text]-rational FISs toward the original function. A numerical illustration is provided to explain the visual quality of our rational FISs. An extra feature of these fractal surface schemes is that it allows subsequent interactive alteration of the shape of the surfaces by changing the scaling factors and shape parameters.

scholarly journals Applications of Multivalued Contractions on Graphs to Graph-Directed Iterated Function Systems

2015 ◽  
Vol 2015 ◽  
pp. 1-16
Author(s):  
T. Dinevari ◽  
M. Frigon
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Metric Space ◽  
Metric Spaces ◽  
Iterated Function System ◽  
Iterated Function Systems ◽  
Complete Metric Spaces ◽  
Fixed Point Result ◽  
Iterated Function ◽  
Multivalued Contractions

We apply a fixed point result for multivalued contractions on complete metric spaces endowed with a graph to graph-directed iterated function systems. More precisely, we construct a suitable metric space endowed with a graphGand a suitableG-contraction such that its fixed points permit us to obtain more information on the attractor of a graph-directed iterated function system.

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