Fractal Geometry: Mathematical Methods, Algorithms, Application

245 Pages·2004·37.16 MB·English

by  Jonathan M Blackledge

FRACTAL GEOMETRY:
Mathematical Methods, Algorithms, Applications
Based on the proceedings of the First IMA Conference on 'Fractal Geometry: Mathematical
Methods Algorithms and Applications'. Organised by the Institute of Mathematics and its
Applications and held at De Montfort University, 20-22 September 2000
Co-sponsored by the Institute of Physics and the Institute of Electrical Engineers
The Institute of The Institute of Physics, The Institution of
Mathematics and its 76 Portland Place, Electrical Engineers,
Applications, London Savoy Place,
Catherine Richards House, WIN 3DH London
16 Nelson Street, (http://www.iop.org/) WC2ROBL
Southend-on-Sca, {http://www.iec.org/}
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{http://www.ima.org.uk/}
Horwood Publishing Series: Mathematics and Applications
Cultural Diversity in Mathematics (Education)
Eds: A. Ahmed el al, University College, Chichester
Geometry with Trigonometry
P.D. Barry, National University of Ireland, Cork
Fractal Geometry: Mathematical Methods, Algorithms, Applications
J.M. Blackledge, A.K. Evans & M.J.Tumer, De Montfort University, Leicester
Advanced Imaging: Theory, Computing, Practice
J.M. Blackledge & M.J.Rycroft, Dc Montfort University, Leicester
Digital Signal Processing: Software Solutions and Applications
J.M. Blackledge & M.J. Turner, De Montfort University
Image Processing III: Mathematical Methods, Algorithms & Applications
J.M. Blackledge el al, De Montfort University
Image Processing II: Mathematical Methods, Algorithms & Applications
J.M. Blackledge & M.J. Turner, De Montfort University
Electrical Engineering Mathematics
R. Clarke, Imperial College, London
Mathematical Modelling (ICTMA 8)
Eds: P. Galbraith et al. University of Queensland, Brisbane, Australia
Experimental Design Techniques in Statistical Practice
W.P. Gardiner & Gettinby, Glasgow Caledonian University
Delta Functions: Fundamental Introduction to Generalised Functions
R.F. Hoskins, De Montfort University
Teaching & Learning Mathematical Modelling (ICTMA 7)
Eds: S.K. Houston et al. University of Ulster, Northern Ireland
Calculus: Introduction to Theory and Applications in Physical and Life Sciences
R.M. Johnson, University of Paisley
Linear Differential and Difference Equations
R.M. Johnson, University of Paisley
Game Theory: Mathematical Models of Conflict
A.J. Jones, University College of Wales, Cardiff
Stochastic Differential Equations and Applications
X. Mao, University of Strathclyde
Teaching Practice in Mathematics
J.H. Mason, The Open University, Milton Keynes
Modelling and Mathematics Education (ICTMA 9)
J.F. Matos et al, University of Lisbon, Portugal
Manifold Theory
D. Martin, University of Glasgow
Fundamentals of University Mathematics, edition
C. McGregor et al, University of Glasgow
Mathematics for Earth Sciences
P. Sharkey, University of Portsmouth
Decision and Discrete Mathematics
The Spode Group, Truro School, Cornwall
Mathematical Analysis and Proof
D.S.G. Stirling, University of Reading
Statistical Mechanics: An Introduction
D.H. Trevena, University of Aberystwyth
Ordinary Differential Equations and Applications
W.S. Weiglhofer & K.A. Lindsay, University of Glasgow
Geometry of Navigation
R. Williams, Master Mariner
F r a c t al G e o m e t r y:
Mathematical Methods, Algorithms, Applications
Editors:
J.M. Blackledge, institution of Simulation Sciences
A.K. Evans, Modern Optics Centre
M. J. Turner, Institution of Simulation Sciences
DeMontfort University, Leicester
Horwood Publishing far The Institute of Mathematics
Chichester and its Applications
Published in 2002 by
HORWOOD PUBLISHING LIMITED
Coll House, Westergate, Chichester, West Sussex, PO20 6QL England
COPYRIGHT NOTICE
All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval
system, or transmitted, in any form or by any means, electronic, mechanical, photocopying,
recording, or otherwise, without the permission of Horwood Publishing, International
Publishers, Coll House, Westergate, Chichester, West Sussex, England
Ο J.M. Blackledge, A.K. Evans, M.J. Turner, 2002
British Library Cataloguing in Publication Data
A catalogue record of this book is available from the British Library
ISBN 1-904275-00-1
Printed in Great Britain by Martins Printing Group, Bodmin, Cornwall
Contents
Chaotic Dynamics in a Simple Aeromechanical System:
M.G. Goman, A.N. Khrabrov and A.V. Khramtsovsky . .. 1
Random walks with fluctuating step number, scale invariant be­
haviour and self-organised-criticality:
R.M.J. Tanner, K.I. Hopcraft and E. Jakeman 16
Fractional Integrals, Singular Measures and Epsilon Functions:
R.F. Hoskins 39
Diffusion on Fractals: Efficient Algorithms to Compute the Ran­
dom Walk Dimension:
A. Franz, C. Schuizky, S. Seeger and K.H. Hoffmann . . .. 52
Why study financial time series?:
M.D. London, A.K. Evans and M.J. Turner 68
Analysis of the Limitations of Fractal Dimension Texture Segmen­
tation for Image Characterisation:
M.J. Turner and J.M. Blackledge 114
Fractals Basins of Attraction in the Inversion of Gravity and Mag­
netic Data:
G.R.J. Cooper 138
Properties of Fractal Compression and their use within Texture
Mapping:
M.J. Turner 149
Fractal Time and Nested Detectors:
S. Vrobel 173
Deterministic Chaos in Digital Cryptography:
N.V. Ptitsyn, J.M. Blackledge and V.M. Chernenky . . .. 189
The Making of "Fractal Geometry in Digital Imaging":
M.J. Turner and J.M. Blackledge 223
ν
Index of Authors
Bak, Ρ iv
Niels Bohr Institute for Astronomy, Physics and Geophysics,
Blegdamsvej 17, 2100 Copenhagen 0, Denmark
Blackledge, J.M 114, 189, 223
Institute of Simulation Sciences, SERC, Hawthorn Building, De
Montfort University, Leicester LEI 9BH, UK
Chernenky, V.M 189
Department of Information Processing and Management Systems,
Faculty of Information and Control Systems, Moscow State Technical
University (NA Bauman), Moscow, Russia
Cooper, G.R.J 138
Department of Geophysics, University of the Witwatersrand,
Johannesburg, South Africa
Evans, Α.Κ 68
Centre for Modern Optics, SERC, Hawthorn Bldg., De Montfort
University, Leicester LEI 9BH, UK
Franz, A 52
Institut fur Physik, Technische Universitat, D-09107 Chemnitz,
Germany
Goman, M.G 1
Institute of Simulation Sciences, SERC, Hawthorn Building, De
Montfort University, Leicester LEI 9BH, UK
Hoffmann, K.H 52
Institut fur Physik, Technische Universitat, D-09107 Chemnitz,
Germany
Hopcraft, K.I 16
Theoretical Mechanics Division, School of Mathematical Sciences,
University of Nottingham, Nottingham, NG7 2RD, UK
Hoskins, R 39
Institute of Simulation Sciences, SERC, Hawthorn Building, De
Montfort University, Leicester LEI 9BH, UK
vi
Index of Authors vii
Jakeman, Ε 16
Theoretical Mechanics Division, School of Mathematical Sciences,
University of Nottingham, Nottingham, NG7 2RD, UK
Khrabrov, A.N 1
Central Aerohydrodynamic Institute (TsAGI), Zhukovsky, Russia
Khramtsovsky, A.V 1
Central Aerohydrodynamic Institute (TsAGI), Zhukovsky, Russia
Schulzky, C 52
Department of Applied Mathematics, The University of Western
Ontario, London N6A 5B, Canada
Seeger, S 52
Institut fur Physik, Technische Universitat, D-09107 Chemnitz,
Germany
London, M.D 68
Institute of Simulation Sciences, SERC, Hawthorn Building, De
Montfort University, Leicester LEI 9BH, UK
Ptitsyn, N.V 189
Institute of Simulation Sciences, SERC, Hawthorn Building, De
Montfort University, Leicester LEI 9BH, UK
Tanner, R.M.J 16
Theoretical Mechanics Division, School of Mathematical Sciences,
University of Nottingham, Nottingham, NG7 2RD, UK
Turner, M.J 68, 114, 149, 223
Institute of Simulation Sciences, SERC, Hawthorn Building, De
Montfort University, Leicester LEI 9BH, UK
Vrobel, S 173
The Institute for Fractal Research Schachtenstr. 5 34130 Kassel
Germany
Preface
This volume contains the proceedings of the First IMA Conference on Frac­
tal Geometry: Mathematical Methods, Algorithms and Applications, that
was held at De Montfort University in September 2000. The emphasis was
given to work that related the mathematical exposure given to a problem
and the practicalities required to create and implement an algorithm.
Eleven papers were presented, and finally accepted and published here,
by researchers from many areas of Europe, Russia and America giving a
multi-national emphasis to the fractal geometry problems described and
the conference as a whole. Prof. Per Bak from the Neils Bohr Institute for
Astronomy, Physics and Geophysics, while on sabbatical with the Depart­
ment of Mathematics and Theoretical Physics at Imperial College, London,
gave an entertaining keynote talk on being able to observe the global struc­
ture from local fractal geometric properties, under the title Forest Fires,
Measles, and the Structure of the Universe.
Due to unforeseen circumstances a full transcript is unavailable, but a
copy of the abstract is presented in full, "Forest fire models can be used to
describe turbulent phenomena, where energy is injected at a large length
scale and dissipated at a small scale. The fractal dimension of fires varies
gradually from zero to three as the length scale is increased. A study of
galaxy catalogues indicates that the distribution of luminous matter in the
universe follows a similar pattern. At small distances, the universe is zero-
dimensional and point-like; at distances of the order of 1 Mpc the dimension
is unity, indicating a filamentary, string-like structure. When viewed at
larger scales it gradually becomes two dimensional and pancake-like; finally,
at the correlation length, 300 Mpc, it becomes uniform. The dissipative
field in turbulence may follow a similar pattern. Moreover, the distribution
of real forest fires in the US and Australia, and the distribution of measles
epidemics mimics the distribution of fires in a modified version of the model
due to Drossel and Schwabl." Further details are available in an extensive
publication list of Per Bak's from the famous popular science book How
Nature Works (5 June, 1997) Oxford University Press; ISBN: 0198501641,
and Self-similarity of extinction statistics in the Fossil Record Ricard V.
Sole, Susanna C. Manrubia, Michael Benton, and Per Bak. Nature 388,
p.764, 21 August 1997 (Featured in New Scientist, 8 November 1997, and
in New York Times, 2 September 1997), to the more directly relevant and
recent works including Scale Independent Dimension of Luminous Matter
in the Universe Per Bak and Ken Chen, Phys. Rev. Lett. 86 4215 (2001)
and Solitons in the one-dimensional forest fire-model Per Bak and Ken
Chen, Phys. Rev. Lett. 86 2475 (2001).
Preface ix
The proceedings presented here contain eleven of the presentations as
full published papers. Some of the papers were not published, due to patent
issues and it is hoped and planned to publish them in a following conference
and proceedings when the IPR issues are fully resolved. The IMA is already
planning to host the next conference on fractal geometry in September 2003
(details can found on the IMA web site http://www.ima.org.uk/). As is
healthy, in any field, there is a good mixture of theoretical and applied
papers.
The variation of applications of fractal geometry is immense, as the
study has been embraced by many different scientific fields, and this is
reflected here in their use from finance to geology and from aeromechanical
design to encryption. To quote John Archibald Wheeler, protege of Neils
Bohr "No one will be considered scientifically literate tomorrow who is not
familiar with fractals".
The editors would like to thank all those who contributed to the pro­
ceedings and those involved in organising the conference. We wish to show
our gratitude to the continual support of the main sponsors; the Institute
of Mathematics and its Applications and the co-sponsors; the Institute of
Physics and the Institute of Electrical Engineers.
Jonathan M. Blackledge, Allan K. Evans, Martin J. Turner
Front Cover Image
The Institute of Simulation Sciences (ISS) currently consists of twelve re­
search professors, five researchers and over 25 PhD students looking at a
range of diverse areas from fractal coding techniques and noise modelling
to shock wave implosion simulations and laser scattering.
The front cover shows a stylised version in the blue spectrum of an
electrographic photograph of a leaf. This represents one of many projects
being carried out by members of the Institute. The image was captured by
discharging a 50KV Tesla coil charge and photographing the result using
a Canon EOS10 with a non-coated lens captured on ISO 800 film, with an
exposure time of about 5 seconds.
Electrography was developed mainly by the Armenian Semyon Davi-
dovich Kirlian from 1939, although it had been known about before: Georg
Lichtenberg in 1777 recorded electrical discharges in resin and after pho­
tography was developed, Batholomew Navratil created many examples and
first used the term electrography in 1888. It is often known by the name Kir­
lian photography, and the corona has unfortunately been associated with
unfounded 'parapsychological' causes.