Topic

# Iterated function system

About: Iterated function system is a research topic. Over the lifetime, 1842 publications have been published within this topic receiving 32687 citations. The topic is also known as: IFS Iterated Function System.

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16 Mar 1990

TL;DR: In this article, a mathematical background of Hausdorff measure and dimension alternative definitions of dimension techniques for calculating dimensions local structure of fractals projections of fractality products of fractal intersections of fractalities.

Abstract: Part I Foundations: mathematical background Hausdorff measure and dimension alternative definitions of dimension techniques for calculating dimensions local structure of fractals projections of fractals products of fractals intersections of fractals. Part II Applications and examples: fractals defined by transformations examples from number theory graphs of functions examples from pure mathematics dynamical systems iteration of complex functions-Julia sets random fractals Brownian motion and Brownian surfaces multifractal measures physical applications.

6,325 citations

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TL;DR: Focusing on how fractal geometry can be used to model real objects in the physical world, this up-to-date edition featurestwo 16-page full-color inserts, problems and tools emphasizing fractal applications, and an answers section.

Abstract: Focusing on how fractal geometry can be used to model real objects in the physical world, this up-to-date edition featurestwo 16-page full-color inserts, problems and tools emphasizing fractal applications, and an answers section. A bonus CD of an IFS Generator provides an excellent software tool for designing iterated function systems codes and fractal images.

4,361 citations

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TL;DR: In this paper, a general framework for the exactly computable moment theory of p -balanced measures for hyperbolic i.f.ss and of probabilistic mixtures of iterated Riemann surfaces is presented.

Abstract: Iterated function systems (i. f. ss) are introduced as a unified way of generating a broad class of fractals. These fractals are often attractors for i. f. ss and occur as the supports of probability measures associated with functional equations. The existence of certain ‘ p -balanced’ measures for i. f. ss is established, and these measures are uniquely characterized for hyperbolic i. f. ss. The Hausdorff—Besicovitch dimension for some attractors of hyperbolic i. f. ss is estimated with the aid of p -balanced measures. What appears to be the broadest framework for the exactly computable moment theory of p -balanced measures — that of linear i. f. ss and of probabilistic mixtures of iterated Riemann surfaces — is presented. This extensively generalizes earlier work on orthogonal polynomials on Julia sets. An example is given of fractal reconstruction with the use of linear i. f. ss and moment theory.

931 citations

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01 Jan 1993

TL;DR: Data compression with fractals is an approach to reach high compression ratios for large data streams related to images, at a cost of large amounts of computation.

Abstract: The top-selling multimedia encyclopedia Encarta, published by Microsoft Corporation, includes on one CD-ROM seven thousand color photographs which may be viewed interactively on a computer screen. The images are diverse; they are of buildings, musical instruments, people’s faces, baseball bats, ferns, etc. What most users do not know is that all of these photographs are based on fractals and that they represent a (seemingly magical) practical success of mathematics. Research on fractal image compression evolved from the mathematical ferment on chaos and fractals in the years 1978–1985 and in particular on the resurgence of interest in Julia sets and dynamical systems. Here I describe briefly some of the underlying ideas. Following Hutchinson [7], see also [5], consider first a finite set of contraction mappings wi, each with contractivity factor s < 1, taking a compact metric space X into itself, i = 1,2, . . .N. Such a setup is called an iterated function system (IFS), [1]. Use this IFS to construct a mapping W from the space H of nonempty compact subsets of X into itself by defining, in the self-explanatory notation, W (B) = N ⋃

867 citations

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TL;DR: In this article, the authors introduce iterated function systems whose attractorsG are graphs of continuous functionsf∶I→R, which interpolate the data according tof(x��i)=y fixmei fori e {0,1,⋯,N}.

Abstract: Let a data set {(x
i,y
i) ∈I×R;i=0,1,⋯,N} be given, whereI=[x
0,x
N]⊂R. We introduce iterated function systems whose attractorsG are graphs of continuous functionsf∶I→R, which interpolate the data according tof(x
i)=y
i fori e {0,1,⋯,N}. Results are presented on the existence, coding theory, functional equations and moment theory for such fractal interpolation functions. Applications to the approximation of naturally wiggly functions, which may show some kind of geometrical self-similarity under magnification, such as profiles of cloud tops and mountain ranges, are envisaged.

736 citations