Table Of ContentThe IMA Volumes
in Mathematics
and its Applications
Volume 132
Series Editors
Douglas N. Arnold Fadil Santosa
Springer
New York
Berlin
Heidelberg
Hong Kong
London
Milan
Paris
Tokyo
Institute for Mathematics and
its Applications (IMA)
The Institute for Mathematics and its Applications was estab
lished by a grant from the National Science Foundation to the University
of Minnesota in 1982. The primary mission of the IMA is to foster research
of a truly interdisciplinary nature, establishing links between mathematics
of the highest caliber and important scientific and technological problems
from other disciplines and industry. To this end, the IMA organizes a wide
variety of programs, ranging from short intense workshops in areas of ex
ceptional interest and opportunity to extensive thematic programs lasting
a year. IMA Volumes are used to communicate results of these programs
that we believe are of particular value to the broader scientific community.
Douglas N. Arnold, Director of the IMA
* * * * * * * * * *
IMA ANNUAL PROGRAMS
1982-1983 Statistical and Continuum Approaches to Phase Transition
1983-1984 Mathematical Models for the Economics of Decentralized
Resource Allocation
1984-1985 Continuum Physics and Partial Differential Equations
1985-1986 Stochastic Differential Equations and Their Applications
1986-1987 Scientific Computation
1987-1988 Applied Combinatorics
1988-1989 Nonlinear Waves
1989-1990 Dynamical Systems and Their Applications
1990-1991 Phase Transitions and Free Boundaries
1991-1992 Applied Linear Algebra
1992-1993 Control Theory and its Applications
1993-1994 Emerging Applications of Probability
1994-1995 Waves and Scattering
1995-1996 Mathematical Methods in Material Science
1996-1997 Mathematics of High Performance Computing
1997-1998 Emerging Applications of Dynamical Systems
1998-1999 Mathematics in Biology
1999-2000 Reactive Flows and Transport Phenomena
2000-2001 Mathematics in Multimedia
2001-2002 Mathematics in the Geosciences
2002-2003 Optimization
2003-2004 Probability and Statistics in Complex Systems: Genomics,
Networks, and Financial Engineering
2004-2005 Mathematics of Materials and Macromolecules: Multiple Scales,
Disorder, and Singularities
Continued at the back
Michael F. Barnsley Dietmar Saupe
Edward R. Vrscay
Editors
Fractals in Multimedia
With 82 Figures
Springer
Michael F. Barnsley Dietmar Saupe
Department of Mathematics and Statistics Institut flir Informatik
University of Melbourne Universitat Leipzig
Parkville, Victoria 3052, Australia Augustusplatz 10-11
[email protected] Leipzig, Germany
[email protected]
Edward R. Vrscay Series Editors;
Department of Applied Mathematics Douglas N. Arnold
Faculty of Mathematics Fadil Santosa
University of Waterloo Institute for Mathematics and its
Waterloo, Ontario N2L 3GI, Canada Applications
[email protected] University of Minnesota
Minneapolis, MN 55455, USA
http://www .ima. umn.edu
Mathematics Subject Classification (2000): 60-xx, 68-xx, 51-xx, 51NlO, 97-xx, 97D40, 28-xx,
28D05, 37-xx, 37-06, 37-04
Library of Congress Cataloging-in-Publication Data
Fractals in multimedia /editors, Michael F. Barnsley, Dietmar Saupe, Edward R. Vrscay.
p. cm. - (IMA volumes in mathematics and its application; 132)
Includes bibliographical references.
Based on a meeting held at the IMA in Jan. 2001.
I. Fractals-Congresses. I. Barnsley, M.F. (Michael Fielding), 1946- II. Saupe,
Dietmar, 1954- III. Vrscay, Edward R. IV. IMA volumes in mathematics and its
applications; v. 132.
QA614.86 .F7277 2002
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Softcover reprint of the hardcover 1s t edition 1989
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FOREWORD
This IMA Volume in Mathematics and its Applications
FRACTALS IN MULTIMEDIA
is a result of a very successful three-day minisymposium on the same title.
The event was an integral part of the IMA annual program on Mathemat
ics in Multimedia, 2000-2001. We would like to thank Michael F. Barnsley
(Department of Mathematics and Statistics, University of Melbourne), Di
etmar Saupe (Institut fUr Informatik, UniversiUit Leipzig), and Edward
R. Vrscay (Department of Applied Mathematics, University of Waterloo)
for their excellent work as organizers of the meeting and for editing the
proceedings.
We take this opportunity to thank the National Science Foundation
for their support of the IMA.
Series Editors
Douglas N. Arnold, Director of the IMA
Fadil Santosa, Deputy Director of the IMA
v
PREFACE
This volume grew out of a meeting on Fractals in Multimedia held
at the IMA in January 2001. The meeting was an exciting and intense
one, focused on fractal image compression, analysis, and synthesis, iterated
function systems and fractals in education. The central concerns of the
meeting were to establish within these areas where we are now and to
develop a vision for the future.
In this book we have tried to capture not only the material but also
the excitement of the meeting. What we do not capture is the considerable
effort by Willard Miller in organizing the meeting, raising the funds to
pay for it, giving good advice many times, and providing the wonderful
resources of the IMA to host the conference. Neither elsewhere do we
offer even a nod to his warm and efficient staff, nor to the patient efforts
of Patricia V. Brick in preparing this volume, nor do we note the initial
impetus and effort of Av ner Friedman. Thank you.
Michael F. Barnsley
Department of Mathematics and Statistics
University of Melbourne
Australia
E-mail: [email protected]
Dietmar Saupe
Institut fUr Informatik
UniversiUit Leipzig
Germany
E-mail: [email protected]
Edward R. Vrscay
Department of Applied Mathematics
Faculty of Mathematics
University of Waterloo
Ontario, Canada N2L 3GI
E-mail: [email protected]
vii
CONTENTS
Foreword .............................................................. v
Preface .............................................................. vii
Introduction to IMA fractal proceedings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1
Michael F. Barnsley
Uniqueness of invariant measures for place-dependent
random iterations of functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 13
Orjan Stenjlo
Iterated function systems for lossless data
compression. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 33
Michael F. Barnsley
From fractal image compression to fractal-based
methods in mathematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 65
Edward R. Vrscay
Fractal image compression with fast local search. . . . . . . . . . . . . . . . . . . . . . . .. 107
Raoul Hamzaoui and Dietmar Saupe
Wavelets are piecewise fractal interpolation
functions ............................................................ 121
Douglas P. Hardin
Self-affine vector measures and vector calculus
on fractals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 137
F. Mendivil and E.R. Vrscay
Using the Picard contraction mapping to solve
inverse problems in ordinary differential equations ....................... 157
H.E. Kunze and E.R. Vrscay
Fractal modulation and other applications from
a theory of the statistics of dimension ................................... 175
1.M. Blackledge, S. Mikhailov, and M.J. Turner
ix
x CONTENTS
Signal enhancement based on Holder regularity
analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 197
1. Levy Vehel
Iterated data mining techniques on embedded
vector modeling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 211
Ning Lu
A web-based fractal geometry course for
non-science students .................................................. 233
Michael Frame
List of mini symposium participants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 259
INTRODUCTION TO IMA FRACTAL PROCEEDINGS
MICHAEL F. BARNSLEY*
This volume describes the status of fractal imaging research and looks
to future directions. It is to be useful to researchers in the areas of fractal
image compression, analysis, and synthesis, iterated function systems, and
fractals in education. In particular it includes a vision for the future of
these areas.
It is intended to provide an efficient means by which researchers can
look back over the last decade at what has been achieved, and look forward
towards second-generation fractal imaging. The articles in themselves are
not supposed to be detailed reviews or expositions, but to serve as signposts
to the state-of-the art in their areas. What is important is what they
mention and what tools and ideas are seen now to be relevant to the future.
The contributors, a number of whom have been involved since the
start, are active in fractal imaging, and provide a well-informed viewpoint
on both the status and the future. Most were invited participants at a
meeting on Fractals in Multimedia held at the IMA in January 2001. Some
goals of the mini-symposium, shared with this volume, were to demonstrate
that the fractal viewpoint leads to a broad collection of useful mathemat
ical tools, common themes, new ways of looking at and thinking about
existing algorithms and applications in multimedia; and to consider future
developments.
The fractal viewpoint has developed out of the observation that in the
real world and in the scientific measurement of it, there can occur patterns
that repeat at different scales. It upholds the intuition that the mathe
matical world of geometry and the infinitely divisible Euclidean plane are
relevant to the understanding of the physical world; and in particular that
geometrical entities such as lines, ferns, and other fractal attractors are
related to actual pictures. This viewpoint is captured in fractal mathe
matics, which consists of some basic tools and theorems, such as iterated
function systems (IFS) theory and Hutchinson's theorem, and is centered in
real analysis, geometry, measure theory, dynamical systems, and stochas
tic processes. Its application to multimedia lies principally in the attempt
to bridge the divide between the discrete world of digital representation
and the natural continuum world in which we seem to live. It has served
as inspiration for algorithms that try to recreate sounds, pictures, motion
video and textures, and to organize databases in computer environments.
In the papers in this volume we outline ways in which this bridge, between
intuition and reality, has been built, mainly in the area of imaging. We
*Department of Mathematics and Statistics, University of Melbourne, Parkville,
Victoria 3052, Australia. Currently at 335 Pennbrooke Trace, Duluth, GA 30097
(Mbarnsley@aol. com).
1
M. F. Barnsley et al. (eds.), Fractals in Multimedia
© Springer-Verlag New York, Inc. 2002
2 MICHAEL F. BARNSLEY
try to further define the set of those intuitions and insights that constitute
the fractal viewpoint, the mathematics that sustains it, and to identify
areas where it has potential to increase understanding and lead to new
discoveries.
For completeness we include mention here of material related to the
Fractals in Multimedia meeting that is not in the contributed papers. This
volume does not contain papers corresponding to: a review of block-based
fractal image compression by Dietmar Saupe (Leipzig); a presentation by
Jean-Luc Dugelay (Institut EUROCOM) on fractal image indexing, wa
termarking and recognition; and a discussion of a Lagrangian approach
to fractal video coding by Lyman Hurd (Mediabin Inc., formerly Iterated
Systems Inc.). Also, the meeting would have liked to hear from Geoffrey
Davis (Microsoft) with regard to how he now sees the relationship between
fractals and wavelets in compression, see [1], and to have had a discus
sion of space-filling curves, see [2]. Another unfulfilled hope was that there
would be a presentation or paper by Stephen Demko, who has worked for
a number of years on research and product development at Mediabin Inc.
based on fractal image recognition, see [25]. On the subject of fractals in
education, Vicki Fegers and Mary Beth Johnson of the Broward County
School Board in Florida, who have been working with Heinz-Otto Peitgen
and Richard Voss (Florida Atlantic U.) over a number of years to develop
curriculum content to enrich pre-university education, treated the confer
ence to a presentation of their approach, which we mention further below.
This volume also does not have a paper corresponding to Ken Musgrave's
presentation on fractal graphics.
1. Iterated function systems. An IFS is a mathematical means for
producing, analyzing and describing complex geometrical sets and pictures.
The fractals illustrated in Figures 1 and 2 are examples of sets made by
simple IFS's. The theory of IFS is used to design algorithms for computer
graphics, digital image compression and data compression; to construct
synthetic data, for example for stock price simulation in economics and
oil well modeling in geophysics; to analyze strings of symbols occurring in
data analysis, from DNA [3] to Markov chains [4]; and in education, to
introduce in a unified, intuitive, and visually appealing way, probability,
geometry, iteration, and chaos [3,5].
John Hutchinson introduced the basic mathematical theory of IFS [6]
in 1981 to describe deterministic fractals [7]. The term "Iterated Function
System" was introduced in [8]. The subject was developed and popular
ized during the 1980's by the author and co-workers [9]. The study of
associated invariant measures and stochastic processes predates the recent
fractal literature, having begun in 1937 with [10], see [4]. The field con
tinues to be developed both mathematically, see for example [11-16] and
practically, see for example [5,16]. The exact phrase "Iterated Function
System" in the search engine at www.hotbot. com produced more than 200
links in November 2000.
