Papers by Michael Barnsley
Springer eBooks, 1988
Fractals are geometric or data structures which do not simplify under magnification. Fractal Imag... more Fractals are geometric or data structures which do not simplify under magnification. Fractal Image Compression is a technique which associates a fractal to an image. On the one hand, the fractal can be described in terms of a few succinct rules, while on the other, the fractal contains much or all of the image information. Since the rules are described with less bits of data than the image, compression results. Data compression with fractals is an approach to reach high compression ratios for large data streams related to images. The high compression ratios are attained at a cost of large amounts of computation. Both lossless and lossy modes are supported by the technique. The technique is stable in that small errors in codes lead to small errors in image data. Applications to the NASA mission are discussed. OVERVIEW Fractals are geometric or data structures which do not simplify under magnification. Fractal Image Compression is a technique which associates a fractal to an image. On the one hand, the fractal can be described in terms of a few succinct rules, while on the other, the fractal contains much or all of the image information. Since the rules are described with less bits of data than the image, compression results. Fractal image compression is a computationally intensive technique.
Proceedings of SPIE, Oct 1, 1998
ABSTRACT High quality video compression is necessary for reduction of transmission bandwidth and ... more ABSTRACT High quality video compression is necessary for reduction of transmission bandwidth and in archiving applications. We propose a compression scheme which, depending on the available bandwidth, can vary from lossless compression to lossy compression, but always with guaranteed quality. In the case of lossless compression, the customer receives the original content without any loss. Even the lower compression ratios obtained with lossless compression can represent significant savings in the communication bandwidth. In the case of lossy compression, the maximum error between recovered and the original video is mathematically bounded. The amount of compression achieved is a function of the error bounds. Furthermore, errors are statistically independent from the video content, and thus guaranteed not to create any type of artifacts. So the recovered video has the same quality, visually indistinguishable from the original, at all times and all motion conditions.
Annals of the New York Academy of Sciences, Jul 1, 1987
Siam Journal on Imaging Sciences, 2009
Advances in Mathematics, Aug 1, 2008
We establish properties of a new type of fractal which has partial self similarity at all scales.... more We establish properties of a new type of fractal which has partial self similarity at all scales. For any collection of iterated functions systems with an associated probability distribution and any positive integer V there is a corresponding class of V-variable fractal sets or measures. These V-variable fractals can also be obtained from the points on the attractor of a single deterministic iterated function system. Existence, uniqueness and approximation results are established under average contractive assumptions. We also obtain extensions of some basic results concerning iterated function systems.
International Journal of Bifurcation and Chaos, Mar 1, 1997
arXiv (Cornell University), Aug 10, 2009
The two main theorems of this paper provide a characterization of hyperbolic affine iterated func... more The two main theorems of this paper provide a characterization of hyperbolic affine iterated function systems defined on R m. Atsushi Kameyama (Distances on Topological Self-Similar Sets, Proceedings of Symposia in Pure Mathematics, Volume 72.1, 2004) asked the following fundamental question: given a topological self-similar set, does there exist an associated system of contraction mappings? Our theorems imply an affirmative answer to Kameyama's question for selfsimilar sets derived from affine transformations on R m .
arXiv (Cornell University), Dec 16, 2003
Deterministic and random fractals, within the framework of Iterated Function Systems, have been u... more Deterministic and random fractals, within the framework of Iterated Function Systems, have been used to model and study a wide range of phenomena across many areas of science and technology. However, for many applications deterministic fractals are locally too similar near distinct points while standard random fractals have too little local correlation. Random fractals are also slow and difficult to compute. We construct V-variable fractals, which have the property that at each level of decomposition there are at most V distinct components up to a natural equivalence relation. We show that V-variable fractals are elements of a superfractal, for which there is a fast forward algorithm. Finally we show that V-variable fractals approximate standard random fractals for large V and thereby obtain a fast forward algorithm for obtaining standard random fractals and their natural probability distribution to within any prescribed degree of approximation. The main ideas are developed by means of examples with the intention of being accessible to a wide readership.
Fractals, Jun 1, 2005
We describe new families of random fractals, referred to as "Vvariable", which are intermediate b... more We describe new families of random fractals, referred to as "Vvariable", which are intermediate between the notions of deterministic and of standard random fractals. The parameter V describes the degree of "variability": at each magnification level any V-variable fractals has at most V key "forms" or "shapes". V-variable random fractals have the surprising property that they can be computed using a forward process. More precisely, a version of the usual Random Iteration Algorithm, operating on sets (or measures) rather than points, can be used to sample each family. To present this theory, we review relevant results on fractals (and fractal measures), both deterministic and random. Then our new results are obtained by constructing an iterated function system (a super IFS) from a collection of standard IFSs together with a corresponding set of probabilities. The attractor of the super IFS is called a superfractal; it is a collection of V-variable random fractals (sets or measures) together with an associated probability distribution on this collection. When the underlying space is for example R 2 , and the transformations are computationally straightforward (such as affine transformations), the superfractal can be sampled by means of the algorithm, which is highly efficient in terms of memory usage. The algorithm is illustrated by some computed examples. Some variants, special cases, generalizations of the framework, and potential applications are mentioned.
Fractal image coding generally seeks to express an image as a union of spatially contracted and g... more Fractal image coding generally seeks to express an image as a union of spatially contracted and greyscale modified copies of subsets of itself. Generally, images are represented as functions u(x) and the fractal coding method is conducted in the framework of L2 or L1. Here we formulate a method of fractal image coding on measure-valued images: At each point x, u(x) is a probability measure over the range of allowed greyscale values. It is natural to consider the possibility of more than one domain subblock mapping onto a given range block Rj. We consider in particular the case in which all subblocks from a common domain pool D can contribute to the image supported on each range block. Using an appropriate weighting of these subblocks, in invariant measure that reflects the local self-similarity properties of the image can be constructed.
Horizons of Fractal Geometry and Complex Dimensions, 2019
New tilings of certain subsets of R M are studied, tilings associated with fractal blow-ups of ce... more New tilings of certain subsets of R M are studied, tilings associated with fractal blow-ups of certain similitude iterated function systems (IFS). For each such IFS with attractor satisfying the open set condition, our construction produces a usually infinite family of tilings that satisfy the following properties: (1) the prototile set is finite; (2) the tilings are repetitive (quasiperiodic); (3) each family contains self-similar tilings, usually infinitely many; and (4) when the IFS is rigid in an appropriate sense, the tiling has no non-trivial symmetry; in particular the tiling is non-periodic.
The IMA Volumes in Mathematics and its Application, 2002
Probability in the Engineering and Informational Sciences, 1988
Recently, orbits of two-dimensional Markov chains have been used to generate computer images. The... more Recently, orbits of two-dimensional Markov chains have been used to generate computer images. These chains evolve according to products of i.i.d. affine maps. We deal with mixing models, whereby one mixes together several of these Markov chains, so as to create a mixed image. These mixtures involve starting one Markov chain off at the stationary distribution of another, and then running it for a geometrically distributed number of steps. We use this to analyze various mixing scenarios.
Journal of Statistical Physics, 1985
Beginning with an LRC network with impedance function Z(cn), a sequence of iterated networks N~ w... more Beginning with an LRC network with impedance function Z(cn), a sequence of iterated networks N~ with impedance functions Z~ k=l, 2, 3 ..... is introduced. The asymptotic comportment of Z~ and the spectra of Nk are analyzed in terms of the Julia set of Z. An example is given of an iterated network associated with a cascade of period-doubling bifurcations.
It is proved that homeomorphic images of certain two-dimensional aperiodic tilings, such as Amman... more It is proved that homeomorphic images of certain two-dimensional aperiodic tilings, such as Ammann-A2 tilings, are recognizable, in both mathematical and practical senses. One implication of the results is that it is possible to search for distorted aperiodic structures in nature, where they may be hiding in plain sight.
We present a general theory of fractal transformations and show how it leads to a new type of met... more We present a general theory of fractal transformations and show how it leads to a new type of method for filtering and transforming digital images. This work substantially generalizes earlier work on fractal tops. The approach involves fractal geometry, chaotic dynamics, and an interplay between discrete and continuous representations. The underlying mathematics is established and applications to digital imaging are described and exemplified.
The fast basin of an attractor of an iterated function system (IFS) is the set of points in the d... more The fast basin of an attractor of an iterated function system (IFS) is the set of points in the domain of the IFS whose orbits under the associated semigroup intersect the attractor. Fast basins can have non-integer dimension and comprise a class of deterministic fractal sets. The relationship between the basin and the fast basin of a point-fibred attractor is analyzed. To better understand the topology and geometry of fast basins, and because of analogies with analytic continuation, branched fractal manifolds are introduced. A branched fractal manifold is a metric space constructed from the extended code space of a point-fibred attractor, by identifying some addresses. Typically, a branched fractal manifold is a union of a nondenumerable collection of nonhomeomorphic objects, isometric copies of generalized fractal blowups of the attractor.
Abstract. A fractal function is a function whose graph is the attractor of an iterated function s... more Abstract. A fractal function is a function whose graph is the attractor of an iterated function system. This paper generalizes analytic continuation of an analytic function to continuation of a fractal function. 1.
Abstract. Theorems and explicit examples are used to show how transformations between self-simila... more Abstract. Theorems and explicit examples are used to show how transformations between self-similar sets (general sense) may be continuous almost everywhere with respect to station-ary measures on the sets and may be used to carry well known flows and spectral analysis over from familiar settings to new ones. The focus of this work is on a number of surprising applications including (i) what we call fractal Fourier analysis, in which the graphs of the basis functions are Cantor sets, being discontinuous at a countable dense set of points, yet have very good approximation properties; (ii) Lebesgue measure-preserving flows, on polygonal laminas, whose wave-fronts are fractals. The key idea is to exploit fractal transformations to provide unitary transformations between Hilbert spaces defined on attractors of iterated function sys-tems. Some of the examples relate to work of Oxtoby and Ulam concerning ergodic flows on regions bounded by polygons. 1.
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Papers by Michael Barnsley