Fractal Series by John Driscoll Rough Draft 10/22/2012
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Using Fractal Dimension for Target Detection in Clutter
KIM T. CONSTANTIKES USING FRACTAL DIMENSION FOR TARGET DETECTION IN CLUTTER The detection of targets in natural backgrounds requires that we be able to compute some characteristic of target that is distinct from background clutter. We assume that natural objects are fractals and that the irregularity or roughness of the natural objects can be characterized with fractal dimension estimates. Since man-made objects such as aircraft or ships are comparatively regular and smooth in shape, fractal dimension estimates may be used to distinguish natural from man-made objects. INTRODUCTION Image processing associated with weapons systems is fractal. Falconer1 defines fractals as objects with some or often concerned with methods to distinguish natural ob all of the following properties: fine structure (i.e., detail jects from man-made objects. Infrared seekers in clut on arbitrarily small scales) too irregular to be described tered environments need to distinguish the clutter of with Euclidean geometry; self-similar structure, with clouds or solar sea glint from the signature of the intend fractal dimension greater than its topological dimension; ed target of the weapon. The discrimination of target and recursively defined. This definition extends fractal from clutter falls into a category of methods generally into a more physical and intuitive domain than the orig called segmentation, which derives localized parameters inal Mandelbrot definition whereby a fractal was a set (e.g.,texture) from the observed image intensity in order whose "Hausdorff-Besicovitch dimension strictly exceeds to discriminate objects from background. Essentially, one its topological dimension.,,2 The fine, irregular, and self wants these parameters to be insensitive, or invariant, to similar structure of fractals can be experienced firsthand the kinds of variation that the objects and background by looking at the Mandelbrot set at several locations and might naturally undergo because of changes in how they magnifications. -
Project-Team Virtual Plants Modeling Plant Morphogenesis from Genes To
INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE Project-Team Virtual Plants Modeling plant morphogenesis from genes to phenotype Sophia Antipolis THEME BIO c t i v it y ep o r t 2005 Table of contents 1. Team 1 2. Overall Objectives 2 2.1. Overall Objectives 2 3. Scientific Foundations 2 3.1. Analysis of structures resulting from meristem activity 2 3.2. Meristem functioning and development 3 3.3. A software platform for plant modeling 4 4. New Results 4 4.1. Analysis of structures resulting from meristem activity 4 4.1.1. Analysis of longitudinal count data and underdispersion 4 4.1.2. Growth synchronism between Aerial and Root Systems 4 4.1.3. Changes in branching structures within the whole plants 5 4.1.4. Growth components in trees 5 4.1.5. Markov switching models 5 4.1.6. Diagnostic tools for hidden Markovian models 5 4.1.7. Hidden Markov tree models for investigating physiological age within plants 6 4.1.8. Branching processes for plant development analysis 6 4.1.9. Self-similarity in plants 6 4.1.10. Reconstruction of plant foliage density from photographs 6 4.1.11. Fractal analysis of plant geometry 6 4.1.12. Light interception by canopy 7 4.1.13. Heritability of architectural traits 7 4.2. Meristem functioning and development 8 4.2.1. 3D surface reconstruction and cell lineage detection in shoot meristems 8 4.2.2. Simulation of auxin fluxes in the mersitem 8 4.2.3. Dynamic model of phyllotaxy based on auxin fluxes 8 4.2.4. -
Turbulence, Fractals, and Mixing
Turbulence, fractals, and mixing Paul E. Dimotakis and Haris J. Catrakis GALCIT Report FM97-1 17 January 1997 Firestone Flight Sciences Laboratory Guggenheim Aeronautical Laboratory Karman Laboratory of Fluid Mechanics and Jet Propulsion Pasadena Turbulence, fractals, and mixing* Paul E. Dimotakis and Haris J. Catrakis Graduate Aeronautical Laboratories California Institute of Technology Pasadena, California 91125 Abstract Proposals and experimenta1 evidence, from both numerical simulations and laboratory experiments, regarding the behavior of level sets in turbulent flows are reviewed. Isoscalar surfaces in turbulent flows, at least in liquid-phase turbulent jets, where extensive experiments have been undertaken, appear to have a geom- etry that is more complex than (constant-D) fractal. Their description requires an extension of the original, scale-invariant, fractal framework that can be cast in terms of a variable (scale-dependent) coverage dimension, Dd(X). The extension to a scale-dependent framework allows level-set coverage statistics to be related to other quantities of interest. In addition to the pdf of point-spacings (in 1-D), it can be related to the scale-dependent surface-to-volume (perimeter-to-area in 2-D) ratio, as well as the distribution of distances to the level set. The application of this framework to the study of turbulent -jet mixing indicates that isoscalar geometric measures are both threshold and Reynolds-number dependent. As regards mixing, the analysis facilitated by the new tools, as well as by other criteria, indicates en- hanced mixing with increasing Reynolds number, at least for the range of Reynolds numbers investigated. This results in a progressively less-complex level-set geom- etry, at least in liquid-phase turbulent jets, with increasing Reynolds number. -
Fractal Curves and Complexity
Perception & Psychophysics 1987, 42 (4), 365-370 Fractal curves and complexity JAMES E. CUTI'ING and JEFFREY J. GARVIN Cornell University, Ithaca, New York Fractal curves were generated on square initiators and rated in terms of complexity by eight viewers. The stimuli differed in fractional dimension, recursion, and number of segments in their generators. Across six stimulus sets, recursion accounted for most of the variance in complexity judgments, but among stimuli with the most recursive depth, fractal dimension was a respect able predictor. Six variables from previous psychophysical literature known to effect complexity judgments were compared with these fractal variables: symmetry, moments of spatial distribu tion, angular variance, number of sides, P2/A, and Leeuwenberg codes. The latter three provided reliable predictive value and were highly correlated with recursive depth, fractal dimension, and number of segments in the generator, respectively. Thus, the measures from the previous litera ture and those of fractal parameters provide equal predictive value in judgments of these stimuli. Fractals are mathematicalobjectsthat have recently cap determine the fractional dimension by dividing the loga tured the imaginations of artists, computer graphics en rithm of the number of unit lengths in the generator by gineers, and psychologists. Synthesized and popularized the logarithm of the number of unit lengths across the ini by Mandelbrot (1977, 1983), with ever-widening appeal tiator. Since there are five segments in this generator and (e.g., Peitgen & Richter, 1986), fractals have many curi three unit lengths across the initiator, the fractionaldimen ous and fascinating properties. Consider four. sion is log(5)/log(3), or about 1.47. -
Role of Nonlinear Dynamics and Chaos in Applied Sciences
v.;.;.:.:.:.;.;.^ ROLE OF NONLINEAR DYNAMICS AND CHAOS IN APPLIED SCIENCES by Quissan V. Lawande and Nirupam Maiti Theoretical Physics Oivisipn 2000 Please be aware that all of the Missing Pages in this document were originally blank pages BARC/2OOO/E/OO3 GOVERNMENT OF INDIA ATOMIC ENERGY COMMISSION ROLE OF NONLINEAR DYNAMICS AND CHAOS IN APPLIED SCIENCES by Quissan V. Lawande and Nirupam Maiti Theoretical Physics Division BHABHA ATOMIC RESEARCH CENTRE MUMBAI, INDIA 2000 BARC/2000/E/003 BIBLIOGRAPHIC DESCRIPTION SHEET FOR TECHNICAL REPORT (as per IS : 9400 - 1980) 01 Security classification: Unclassified • 02 Distribution: External 03 Report status: New 04 Series: BARC External • 05 Report type: Technical Report 06 Report No. : BARC/2000/E/003 07 Part No. or Volume No. : 08 Contract No.: 10 Title and subtitle: Role of nonlinear dynamics and chaos in applied sciences 11 Collation: 111 p., figs., ills. 13 Project No. : 20 Personal authors): Quissan V. Lawande; Nirupam Maiti 21 Affiliation ofauthor(s): Theoretical Physics Division, Bhabha Atomic Research Centre, Mumbai 22 Corporate authoifs): Bhabha Atomic Research Centre, Mumbai - 400 085 23 Originating unit : Theoretical Physics Division, BARC, Mumbai 24 Sponsors) Name: Department of Atomic Energy Type: Government Contd...(ii) -l- 30 Date of submission: January 2000 31 Publication/Issue date: February 2000 40 Publisher/Distributor: Head, Library and Information Services Division, Bhabha Atomic Research Centre, Mumbai 42 Form of distribution: Hard copy 50 Language of text: English 51 Language of summary: English 52 No. of references: 40 refs. 53 Gives data on: Abstract: Nonlinear dynamics manifests itself in a number of phenomena in both laboratory and day to day dealings. -
An Introduction Into Ecological Modelling for Students, Teachers & Scientists
Modelling Complex Ecological Dynamics . Fred Jopp l Hauke Reuter l Broder Breckling Editors Modelling Complex Ecological Dynamics An Introduction into Ecological Modelling for Students, Teachers & Scientists Title Drawings by Melanie Trexler Foreword by Sven Erik Jørgensen & Donald L. DeAngelis Editors Dr. Fred Jopp Dr. Hauke Reuter Department of Biology Department of Ecological Modelling University of Miami Leibniz Center for Tropical Marine Ecology P.O. Box 249118 GmbH (ZMT) Coral Gables, FL 33124 Fahrenheitstraße 6 USA 28359 Bremen [email protected] Germany [email protected] Dr. Broder Breckling General and Theoretical Ecology Center for Environmental Research and Sustainable Technology (UFT) University of Bremen Leobener St. 28359 Bremen Germany [email protected] ISBN 978-3-642-05028-2 e-ISBN 978-3-642-05029-9 DOI 10.1007/978-3-642-05029-9 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011921703 # Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protec- tive laws and regulations and therefore free for general use. -
Fractal Dimension of Self-Affine Sets: Some Examples
FRACTAL DIMENSION OF SELF-AFFINE SETS: SOME EXAMPLES G. A. EDGAR One of the most common mathematical ways to construct a fractal is as a \self-similar" set. A similarity in Rd is a function f : Rd ! Rd satisfying kf(x) − f(y)k = r kx − yk for some constant r. We call r the ratio of the map f. If f1; f2; ··· ; fn is a finite list of similarities, then the invariant set or attractor of the iterated function system is the compact nonempty set K satisfying K = f1[K] [ f2[K] [···[ fn[K]: The set K obtained in this way is said to be self-similar. If fi has ratio ri < 1, then there is a unique attractor K. The similarity dimension of the attractor K is the solution s of the equation n X s (1) ri = 1: i=1 This theory is due to Hausdorff [13], Moran [16], and Hutchinson [14]. The similarity dimension defined by (1) is the Hausdorff dimension of K, provided there is not \too much" overlap, as specified by Moran's open set condition. See [14], [6], [10]. REPRINT From: Measure Theory, Oberwolfach 1990, in Supplemento ai Rendiconti del Circolo Matematico di Palermo, Serie II, numero 28, anno 1992, pp. 341{358. This research was supported in part by National Science Foundation grant DMS 87-01120. Typeset by AMS-TEX. G. A. EDGAR I will be interested here in a generalization of self-similar sets, called self-affine sets. In particular, I will be interested in the computation of the Hausdorff dimension of such sets. -
Fractal Geometry and Applications in Forest Science
ACKNOWLEDGMENTS Egolfs V. Bakuzis, Professor Emeritus at the University of Minnesota, College of Natural Resources, collected most of the information upon which this review is based. We express our sincere appreciation for his investment of time and energy in collecting these articles and books, in organizing the diverse material collected, and in sacrificing his personal research time to have weekly meetings with one of us (N.L.) to discuss the relevance and importance of each refer- enced paper and many not included here. Besides his interdisciplinary ap- proach to the scientific literature, his extensive knowledge of forest ecosystems and his early interest in nonlinear dynamics have helped us greatly. We express appreciation to Kevin Nimerfro for generating Diagrams 1, 3, 4, 5, and the cover using the programming package Mathematica. Craig Loehle and Boris Zeide provided review comments that significantly improved the paper. Funded by cooperative agreement #23-91-21, USDA Forest Service, North Central Forest Experiment Station, St. Paul, Minnesota. Yg._. t NAVE A THREE--PART QUE_.gTION,, F_-ACHPARToF:WHICH HA# "THREEPAP,T_.<.,EACFi PART" Of:: F_.AC.HPART oF wHIct4 HA.5 __ "1t4REE MORE PARTS... t_! c_4a EL o. EP-.ACTAL G EOPAgTI_YCoh_FERENCE I G;:_.4-A.-Ti_E AT THB Reprinted courtesy of Omni magazine, June 1994. VoL 16, No. 9. CONTENTS i_ Introduction ....................................................................................................... I 2° Description of Fractals .................................................................................... -
Fractal-Based Methods As a Technique for Estimating the Intrinsic Dimensionality of High-Dimensional Data: a Survey
INFORMATICA, 2016, Vol. 27, No. 2, 257–281 257 2016 Vilnius University DOI: http://dx.doi.org/10.15388/Informatica.2016.84 Fractal-Based Methods as a Technique for Estimating the Intrinsic Dimensionality of High-Dimensional Data: A Survey Rasa KARBAUSKAITE˙ ∗, Gintautas DZEMYDA Institute of Mathematics and Informatics, Vilnius University Akademijos 4, LT-08663, Vilnius, Lithuania e-mail: [email protected], [email protected] Received: December 2015; accepted: April 2016 Abstract. The estimation of intrinsic dimensionality of high-dimensional data still remains a chal- lenging issue. Various approaches to interpret and estimate the intrinsic dimensionality are deve- loped. Referring to the following two classifications of estimators of the intrinsic dimensionality – local/global estimators and projection techniques/geometric approaches – we focus on the fractal- based methods that are assigned to the global estimators and geometric approaches. The compu- tational aspects of estimating the intrinsic dimensionality of high-dimensional data are the core issue in this paper. The advantages and disadvantages of the fractal-based methods are disclosed and applications of these methods are presented briefly. Key words: high-dimensional data, intrinsic dimensionality, topological dimension, fractal dimension, fractal-based methods, box-counting dimension, information dimension, correlation dimension, packing dimension. 1. Introduction In real applications, we confront with data that are of a very high dimensionality. For ex- ample, in image analysis, each image is described by a large number of pixels of different colour. The analysis of DNA microarray data (Kriukien˙e et al., 2013) deals with a high dimensionality, too. The analysis of high-dimensional data is usually challenging. -
An Investigation of Fractals and Fractal Dimension
An Investigation of Fractals and Fractal Dimension Student: Ian Friesen Advisor: Dr. Andrew J. Dean April 10, 2018 Contents 1 Introduction 2 1.1 Fractals in Nature . 2 1.2 Mathematically Constructed Fractals . 2 1.3 Motivation and Method . 3 2 Basic Topology 5 2.1 Metric Spaces . 5 2.2 String Spaces . 7 2.3 Hausdorff Metric . 8 3 Topological Dimension 9 3.1 Refinement . 9 3.2 Zero-dimensional Spaces . 9 3.3 Covering Dimension . 10 4 Measure Theory 12 4.1 Outer/Inner Lebesgue Measure . 12 4.2 Carath´eodory Measurability . 13 4.3 Two dimensional Lebesgue Measure . 14 5 Self-similarity 14 5.1 Ratios . 14 5.2 Examples and Calculations . 15 5.3 Chaos Game . 17 6 Fractal Dimension 18 6.1 Hausdorff Measure . 18 6.2 Packing Measure . 21 6.3 Fractal Definition . 24 7 Fractals and the Complex Plane 25 7.1 Julia Sets . 25 7.2 Mandelbrot Set . 26 8 Conclusion 27 1 1 Introduction 1.1 Fractals in Nature There are many fractals in nature. Most of these have some level of self-similarity, and are called self-similar objects. Basic examples of these include the surface of cauliflower, the pat- tern of a fern, or the edge of a snowflake. Benoit Mandelbrot (considered to be the father of frac- tals) thought to measure the coastline of Britain and concluded that using smaller units of mea- surement led to larger perimeter measurements. Mathematically, as the unit of measurement de- creases in length and becomes more precise, the perimeter increases in length. -
Irena Ivancic Fraktalna Geometrija I Teorija Dimenzije
SveuˇcilišteJ.J. Strossmayera u Osijeku Odjel za matematiku Sveuˇcilišninastavniˇckistudij matematike i informatike Irena Ivanˇci´c Fraktalna geometrija i teorija dimenzije Diplomski rad Osijek, 2019. SveuˇcilišteJ.J. Strossmayera u Osijeku Odjel za matematiku Sveuˇcilišninastavniˇckistudij matematike i informatike Irena Ivanˇci´c Fraktalna geometrija i teorija dimenzije Diplomski rad Mentor: izv. prof. dr. sc. Tomislav Maroševi´c Osijek, 2019. Sadržaj 1 Uvod 3 2 Povijest fraktala- pojava ˇcudovišnih funkcija 4 2.1 Krivulje koje popunjavaju ravninu . 5 2.1.1 Peanova krivulja . 5 2.1.2 Hilbertova krivulja . 5 2.1.3 Drugi primjeri krivulja koje popunjavaju prostor . 6 3 Karakterizacija 7 3.1 Fraktali su hrapavi . 7 3.2 Fraktali su sami sebi sliˇcnii beskonaˇcnokompleksni . 7 4 Podjela fraktala prema stupnju samosliˇcnosti 8 5 Podjela fraktala prema naˇcinukonstrukcije 9 5.1 Fraktali nastali iteracijom generatora . 9 5.1.1 Cantorov skup . 9 5.1.2 Trokut i tepih Sierpi´nskog. 10 5.1.3 Mengerova spužva . 11 5.1.4 Kochova pahulja . 11 5.2 Kochove plohe . 12 5.2.1 Pitagorino stablo . 12 5.3 Sustav iteriranih funkcija (IFS) . 13 5.3.1 Lindenmayer sustavi . 16 5.4 Fraktali nastali iteriranjem funkcije . 17 5.4.1 Mandelbrotov skup . 17 5.4.2 Julijevi skupovi . 20 5.4.3 Cudniˇ atraktori . 22 6 Fraktalna dimenzija 23 6.1 Zahtjevi za dobru definiciju dimenzije . 23 6.2 Dimenzija šestara . 24 6.3 Dimenzija samosliˇcnosti . 27 6.4 Box-counting dimenzija . 28 6.5 Hausdorffova dimenzija . 28 6.6 Veza izmedu¯ dimenzija . 30 7 Primjena fraktala i teorije dimenzije 31 7.1 Fraktalna priroda geografskih fenomena . -
Chaos Theory and Its Application in the Atmosphere
Chaos Theory and its Application in the Atmosphere by XubinZeng Department of Atmospheric Science Colorado State University Fort Collins, Colorado Roger A. Pielke, P.I. NSF Grant #ATM-8915265 CHAOS THEORY AND ITS APPLICATION IN THE ATMOSPHERE Xubin Zeng Department of Atmospheric Science CoJorado State University Fort Collins, Colorado Summer, 1992 Atmospheric Science Paper No. 504 \llIlll~lIl1ll~I""I1~II~'I\1 U16400 7029194 ABSTRACT CHAOS THEORY AND ITS APPLICATION IN THE ATMOSPHERE Chaos theory is thoroughly reviewed, which includes the bifurcation and routes to tur bulence, and the characterization of chaos such as dimension, Lyapunov exponent, and Kolmogorov-Sinai entropy. A new method is developed to compute Lyapunov exponents from limited experimental data. Our method is tested on a variety of known model systems, and it is found that our algorithm can be used to obtain a reasonable Lyapunov exponent spectrum from only 5000 data points with a precision of 10-1 or 10-2 in 3- or 4-dimensional phase space, or 10,000 data points in 5-dimensional phase space. On 1:he basis of both the objective analyses of different methods for computing the Lyapunov exponents and our own experience, which is subjective, this is recommended as a good practical method for estiIpating the Lyapunov-exponent spectrum from short time series of low precision. The application of chaos is divided into three categories: observational data analysis, llew ideas or physical insights inspired by chaos, and numerical model output analysis. Corresponding with these categories, three subjects are studied. First, the fractal dimen sion, Lyapunov-exponent spectrum, Kolmogorov entropy, and predictability are evaluated from the observed time series of daily surface temperature and pressure over several regions of the United States and the North Atlantic Ocean with different climatic signal-to-noise ratios.