Michael F. Barnsley

Iterated Function Systems and the Global Construction of Fractals

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.

Fractal Functions and Interpolation

Let a data set {(xi,yi) ∈I×R;i=0,1,⋯,N} be given, whereI=[x0,xN]⊂R. We introduce iterated function systems whose attractorsG are graphs of continuous functionsf∶I→R, which interpolate the data according tof(xi)=yi fori ε {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.

Recurrent iterated function systems

Recurrent iterated function systems generalize iterated function systems as introduced by Barnsley and Demko [BD] in that a Markov chain (typically with some zeros in the transition probability matrix) is used to drive a system of mapswj:K →K,j=1, 2,⋯,N, whereK is a complete metric space. It is proved that under “average contractivity,” a convergence and ergodic theorem obtains, which extends the results of Barnsley and Elton [BE]. It is also proved that a Collage Theorem is true, which generalizes the main result of Barnsleyet al. [BEHL] and which broadens the class of images which can be encoded using iterated map techniques. The theory of fractal interpolation functions [B] is extended, and the fractal dimensions for certain attractors is derived, extending the technique of Hardin and Massopust [HM]. Applications to Julia set theory and to the study of the boundary of IFS attractors are presented.

The calculus of fractal interpolation functions

Abstract The calculus of deterministic fractal functions is introduced. Fractal interpolation functions can be explicitly indefinitely integrated any number of times, yielding a hierarchy of successively smoother interpolation functions which generalize splines and which, just as in the case for the original fractal functions, are attractors for iterated function systems. The fractal dimension for a class of fractal interpolation functions is explicitly computed.

Application Of Recurrent Iterated Function Systems To Images

A new fractal technique for the analysis and compression of digital images is presented. It is shown that a family of contours extracted from an image can be modelled geometrically as a single entity, based on the theory of recurrent iterated function systems (RIFS). RIFS structures are a rich source for deterministic images, including curves which cannot be generated by standard techniques. Control and stability properties are investigated. We state a control theorem - the recurrent collage theorem - and show how to use it to constrain a recurrent IFS structure so that its attractor is close to a given family of contours. This closeness is not only perceptual; it is measured by means of a min-max distance, for which shape and geometry is important but slight shifts are not. It is therefore the right distance to use for geometric modeling. We show how a very intricate geometric structure, at all scales, is inherently encoded in a few parameters that describe entirely the recurrent structures. Very high data compression ratios can be obtained. The decoding phase is achieved through a simple and fast reconstruction algorithm. Finally, we suggest how higher dimensional structures could be designed to model a wide range of digital textures, thus leading our research towards a complete image compression system that will take its input from some low-level image segmenter.

Harnessing chaos for image synthesis

Chaotic dynamics can be used to model shapes and render textures in digital images. This paper addresses the problem of how to model geometrically shapes and textures of two dimensional images using iterated function systems. The successful solution to this problem is demonstrated by the production and processing of synthetic images encoded from color photographs. The solution is achieved using two algorithms: (1) an interactive geometric modeling algorithm for finding iterated function system codes; and (2) a random iteration algorithm for computing the geometry and texture of images defined by iterated function system codes. Also, the underlying mathematical framework, where these two algorithms have their roots, is outlined. The algorithms are illustrated by showing how they can be used to produce images of clouds, mist and surf, seascapes and landscapes and even faces, all modeled from original photographs. The reasons for developing iterated function systems algorithms include their ability to produce complicated images and textures from small databases, and their potential for highly parallel implementation.

Fractal modelling of real world images

Mankind seems to be obsessed with straight lines. I am not sure where it all began, but I like to imagine two Druids overseeing the construction of Stonehenge, and using a piece of stretched string to check the straightness of the edges of those huge stone rectangular blocks, or the path that the light beam from the equinox sun would follow. It was at least an efficient form of quality control; for very little effort by those two Druids it kept the laborers with their eyes sharply on the job.

The chaos game on a general iterated function system

The main theorem of this paper establishes conditions under which the ‘chaos game’ algorithm almost surely yields the attractor of an iterated function system. The theorem holds in a very general setting, even for non-contractive iterated function systems, and under weaker conditions on the random orbit of the chaos game than obtained previously.

On the Invariant Sets of a Family of Quadratic Maps

The Julia setBλ for the mappingz → (z−λ)2 is considered, where λ is a complex parameter. For λ≧2 a new upper bound for the Hausdorff dimension is given, and the monic polynomials orthogonal with respect to the equilibrium measure onBλ are introduced. A method for calculating all of the polynomials is provided, and certain identities which obtain among coefficients of the three-term recurrence relations are given. A unifying theme is the relationship betweenBλ and λ-chains λ±√ (λ±√ (λ± ...), which is explored for −1/4≦λ≦2 and for λ∈ℂ with |λ|≦1/4, with the aid of the Böttcher equation. ThenBλ is shown to be a Hölder continuous curve for |λ|<1/4.

A Mandelbrot set for pairs of linear maps

The set of points A ( s )= {± 1 ± s ± s 2 ± s 3 ± … for all sequences of + and −}, generically a fractal. For example A (1/3) is the classical Cantor set and A (1/2 + i/2) is a dragon curve. This set of fractals can be classified in terms of an associated Mandelbrot set D = s ∈ : ¦s¦ A(s) is disconnected. The structure of D and its boundary are investigated.