Fractals: Theory and Applications in Engineering PDF
Preview Fractals: Theory and Applications in Engineering
Fractals: Theory and Applications in Engineering Springer London Berlin Heidelberg New York Barcelona HongKong Milan Paris Santa Clara Singapore Tokyo Michel Dekking, Jacques Levy Vehel, Evelyne Lutton and Claude Tricot (Eds.) Fractals: Theory and Applications in Engineering i Springer MichelDekking,Professor FacultyofTechnicalMathematicsausSC,DelftUniversityofTechnology, Melalweg4,2628CDDelft,TheNetherlands JacquesLevyWhel,Doctor INRIA,Rocquencourt,B.P. 105,78153LeChesnayCedex,France EvelyneLutton,Doctor INRIA,Rocquencourt,B.P. 105,78153LeChesnayCedex,France ClaudeTricot,Professor UniversiteBlaisePascal,DepartementMathematiques,63177AubiereCedex,France ISBN-13:978-1-4471-1225-9 Springer-VerlagLondonBerlinHeidelberg BritishLibraryCataloguinginPublicationData Fractals:theoryandapplicationsinengineering l.Fractals2.Engineeringmathematics I.Dekking,Michel 620'.0015'14742 ISBN-13:978-1-4471-1225-9 e-ISBN-13:978-1-4471-0873-3 DOl:10.1007/978-1-4471-0873-3 LibraryofCongressCataloging-in-PublicationData AcatalogrecordforthisbookisavailablefromtheLibraryofCongress Apartfromanyfairdealingforthepurposesofresearchorprivatestudy,orcriticismorreview,aspermitted under the Copyright, Designs and Patents Act 1988, this publication may onlybe reproduced, storedor transmitted,inanyformorbyanymeans,withthepriorpermissioninwritingofthepublishers,orinthe caseofreprographicreproductioninaccordancewiththetermsoflicencesissuedbytheCopyrightLicensing Agency. Enquiriesconcerningreproductionoutsidethosetermsshouldbesenttothepublishers. ©Springer-VerlagLondonLimited1999 Softcoverreprintofthehardcover 1stedition 1999 The useofregisterednames, trademarks,etc.in thispublicationdoesnot imply,evenin the absence ofa specificstatement,thatsuchnamesareexemptfromtherelevantlawsandregulationsandthereforefreefor generaluse. Thepublishermakesnorepresentation,expressorimplied,withregardto theaccuracyoftheinformation containedinthisbookandcannotacceptanylegalresponsibilityorliabilityforanyerrorsoromissionsthat maybemade. Typesetting:Camerareadybyeditors 69/3830-543210Printedonacid-freepaper Foreword The activity around the fractal models develops itselfso rapidly that it is neces sary, at regulartimes, tosketcha surveyofthe new applicationsand discoveries. Someoldtopicsseemto cometoa kind ofmaturity, likea localmaximum; whilst othertopics springon a positiveslope, bringing discussions and dynamics to the subject. The readers of the former book Fractals in Engineering; From Theory to Industrial Applications (Springer-Verlag 1997, ed. J. Levy-Vehel, E. Lutton and C. Tricot) may find after two yearssome major differences with the present collection ofworks. At first, a general impression is the following: Mathematics are more and more involved in the definition and use of fractal models. Let us mention two particularly active areas which both are strongly based upon theoretical argu ments. Firstly, stochastic processes defined as extensions offractional Brownian Motion functions become more and more sophisticated and adaptable to many experimental situations. Secondly, multifractal analysis has proved itselfa pow erful and versatile technique in Signal Processing. In both directions, new tools and new theorems are found regularly and the field ofapplications continues to grow. The readers will find consequently several chapters on fractal stochastic pro cesses in this book. Multifractal spectraare used in texture analysis and several new models are proposed. Some new achievements on IFS theory and image compression are given. Wavelets occur in different places (processes, fractal lat tices,compression)and continuetoplaytheirfundamental roleinsignalanalysis. Some intriguing vector calculus on fractal curves is sketched: This may well be a major topic in the near future. The chapters on fractal pores, fractal tunnels, chaoticflows and monolayerstructures are good representativesofthe wealth of activity occuring in physics and chemistry in connection with fractals. Note in particular a paper involving conformal invariance, quantum gravity and multi fractal analysisthat bringsnew insightsinto percolationphenomenaand, among other results, proves the famous 4/3 Mandelbrot's conjecture on the boundary ofa Brownian trajectory. Finally, it seems that the time when the field was mainly concerned with a qualitative observation of fractal phenomena has definitely gone. In order to VI get operational and closer to the real world, the models are now deep-rooted in mathematics, with new theory behind. We wish to thank all the authors who have generously contributed through recent and original work to the composition ofthis book. We also thank Mitzi Adams, PierreAdler, AntoineAyache,ChristopheCanus,MarcChassery, Nathan Cohen, Kenneth Falconer, Bertrand Guiheneuf, Stephane Jaffard, Georges Op penheim, JacquesPeyriere,PeterPfeifer, MichelRosso,for theirhelp andcareful reviews, and Nathalie Gaudechoux, for her Latexskill. Oncemore, the efficiency ofour publisher Springer-Verlag is warmly thanked. We are also deeply grateful to INRIA and the University ofTechnology ofDelft for their support. Michel DEKKING, Jacques LEVY VEHEL, Evelyne LUTTON, Claude TRICOT. Table of Contents LOCALLY SELF SIMILAR PROCESSES From Self-Similarity to Local Self-Similarity: the Estimation Problem. .... 3 Serge Cohen Generalized Multifractional Brownian Motion: Definition and Preliminary Results.. .. .. .... .. .. .. .. ....... .. .. ..... .... 17 Antoine Ayache, Jacques Levy Vehel Elliptic SelfSimilar Stochastic Processes. ............................. 33 Albert Benassi, Daniel Roux Wavelets for Scaling Processes. ...................................... 47 Patrick Flandrin, Patrice Abry MULTIFRACTAL ANALYSIS Classification of Natural Texture Images from Shape Analysis ofthe Leg endre Multifractal Spectrum. ........................................ 67 Piotr Stanczyk, Peter Sharpe AGeneralizationofMultifractalAnalysisBasedonPolynomialExpansions ofthe Generating Function. ......................................... 81 Antoine Saucier, Jiri Muller Local Effective Holder Exponent Estimation on the Wavelet Transform Maxima Tree ...................................................... 93 Zbigniew R. Struzik Easy and Natural Generation of Multifractals: Multiplying Harmonics of Periodic Functions 113 Marc-Olivier Coppens, Benoit B. Mandelbrot IFS IFS-Type Operators on Integral Transforms 125 Bruno Forte, Franklin Mendivil, Edward R. Vrscay Comparison ofDimensions ofa Self-Similar Attractor 139 Serge Dubuc, Jun Li VIII FRACTIONAL CALCULUS Vector Analysis on Fractal Curves 155 Massimiliano Giona Local Fractional Calculus: a Calculus for Fractal Space-Time 171 Kiran M. Kolwankar, Anil D. Gangal PHYSICAL SCIENCES Conformal Multifractality of Random Walks, Polymers, and Percolation in Two Dimensions 185 Bertrand Duplantier FractalPoresand FractalTunnels: Trapsfor "Particles" or "SoundParticles"207 Jerome Dorignac, Bernard Sapoval Fractal Pores and the Degradation ofShales 229 Luis E. Vallejo, Ann Stewart Murphy Continuous Wavelet Transform Analysis ofFractal Superlattices 245 Herve Aubert, Dwight L. Jaggard CHEMICAL ENGINEERING MixinginLaminarChaoticFlows: DifferentiableStructuresand Multifrac- tal Features 263 Massimiliano Giona Adhesion AFM Applied to Lipid Monolayers. A Fractal Analysis. . 277 Gianina Dobrescu, Camelia Obreja, Mircea Rusu IMAGE COMPRESSION Faster Fractal Image Coding Using Similarity Search in a KL-transformed Feature Space 293 Jean Cardinal Can One Break the "Collage Barrier" in Fractal Image Coding? 307 Edward R. Vrscay, Dietmar Saupe Two Algorithms for Non-Separable Wavelet Transforms and Applications to Image Compression 325 Franklin Mendivil, Daniel Piche LOCALLY SELF SIMILAR PROCESSES From Self-Similarity to Local Self-Similarity: the Estimation Problem Serge Cohen Universite de Versailles-St Quentin en Yvelines 45, avenue des Etats-Unis, 78035 Versailles FRANCE or CERMICS-ENPC cohen~ath.uvsq.fr Abstract. In this article we review some methods used to identify the orderH ofafractional Brownianmotion.Thisdiscussion isintroducedto seehowsuchtechniquescan beextendedtolocallyself-similar processes. Moreover the model ofthe multifractional Brownian motion which is a locally self-similar process is further studied. In particular it is shown that its presentation given by J. Levy Vehel and R. Peltier and the one given by A. Benassi, S. Jaffard and D. Roux are in some sense Fourier transformed of each other. Last some results for the estimation of the multifractionnal Brownian motion are recalled. Introduction It is now a classical technique to estimate the scalar index that is relevant to describe the self-similarity property of a Gaussian process. Let us recall that in the Gaussian framework there is only one self-similar Gaussian process with stationary increments (see also [8] for a complete discussion of stationary self similar Gaussian random fields defined in a generalized sense): the fractional Brownian motion (fBm) of index H (0 < H < 1) which can be defined by Xo=0 a.s. and lE(Xt - Xs)2 =It- SI2H. By computing the covariance of the fBm one can see easily that: VA> 0 (1) where (!j means that for every (t1,... ,tn) the distribution of (X-xtll'" ,X-XtJ is thesameas thoseofAH(Xtl,... ,XtJ. This means that thefBm is self-similar with index H. Estimating the parameter H by experimental data is a classical but major issue for applications, and various methods has been proposed. Actually the parameter H governs at least two other properties of the fBm beside the self-similarity and I will classify these techniques by stressing which property is used for the estimation. Moreover the aim ofthis classification is to discuss whether these techniques can also be applied to locally self-similar pro cesses. Theneedfor localizingtheself-similarity (1) is motivated by applications.
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