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A Genetic Algorithm Applied to the Fractal Inverse Problem

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A Genetic Algorithm Applied to the Fractal Inverse Problem UNLV Retrospective Theses & Dissertations 1-1-1993 A genetic algorithm applied to the fractal inverse problem Hiroo Miyamoto University of Nevada, Las Vegas Follow this and additional works at: https://digitalscholarship.unlv.edu/rtds Repository Citation Miyamoto, Hiroo, "A genetic algorithm applied to the fractal inverse problem" (1993). UNLV Retrospective Theses & Dissertations. 329. http://dx.doi.org/10.25669/6d8g-4bse This Thesis is protected by copyright and/or related rights. It has been brought to you by Digital Scholarship@UNLV with permission from the rights-holder(s). You are free to use this Thesis in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s) directly, unless additional rights are indicated by a Creative Commons license in the record and/ or on the work itself. This Thesis has been accepted for inclusion in UNLV Retrospective Theses & Dissertations by an authorized administrator of Digital Scholarship@UNLV. For more information, please contact [email protected]. INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back of the book. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6" x 9" black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. University Microfilms International A Bell & Howell Information Company 300 North Zeeb Road. Ann Arbor. Ml 48106-1346 USA 313/761-4700 800/521-0600 Order Number 1356712 A genetic algorithm applied to the fractal inverse problem Miyamoto, Hiroo, M.S. University of Nevada, Las Vegas, 1993 UMI 300 N. ZeebRd. Ann Arbor, MI 48106 A GENETIC ALGORITHM APPLIED TO THE FRACTAL INVERSE PROBLEM by Hiroo Miyamoto A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mathematical Sciences Department of Mathematical Sciences University of Nevada, Las Vegas August 1993 The Thesis of Hiroo Miyamoto for the degree of Master of Science in Mathematical Sciences is approved. .r.ovc y Chairperson, Benjamin Goertzel, Ph. D. Examining Committee Member, Rohan Dalpatadu, Ph. D. 2. * Examining Committee Member, George J. Miel, Ph. D. \ y Q __________ _________ Graduate Faculty Representative, William R. Wells, Ph. D. u . Dean of the Graduate College, Ronald W. Smith, Ph. D. University of Nevada, Las Vegas August 1993 ABSTRACT The theory of iterated function systems (IFS) allows one to construct a fractal which depends on a finite number of parameters. The associated inverse problem, finding a set of parameters which reconstructs a given fractal, is much more difficult. This paper explores the possibility of solving the inverse program with the genetic algorithm. TABLE OF CONTENTS ABSTRACT ..................................................................................................................... iii LIST OF FIG U R ES........................................................................................................... vii LIST OF TABLES......................................................................................................... viii ACKNOWLEDGEMENT............................................................................................... ix CHAPTER 1 INTRODUCTION ................................................................................... 1 CHAPTER 2 FRACTALS............................................................................................... 4 Definition 1 ......................................................................................................... 4 Definition 2 ......................................................................................................... 4 Definition 3 ......................................................................................................... 5 Example 1 ........................................................................................................... 5 Example 2 ........................................................................................................... 5 Deterministic Algorithm ...................................................................................... 6 Random Iteration Algorithm .............................................................................. 6 Contraction Mapping Theorem ......................................................................... 6 Hausdorff M etric .................................................................................................. 7 Exam ple.................................................................................................... 8 Theorem 1 ........................................................................................................... 8 Hutchinson M etric ............................................................................................... 9 Measures .............................................................................................................. 9 Theorem 2 ......................................................................................................... 10 Markov Operator ............................................................................................... 10 Exam ple.................................................................................................. 10 L em m a................................................................................................................ 13 Measure Theorem ............................................................................................. 13 Invariant Measure ............................................................................................. 14 CHAPTER 3 GENETIC ALGORITHM .................................................................... 15 Procedure Genetic Algorithm ......................................................................... 16 Reproduction ....................................................................................................... 17 C rossover ........................................................................................................... 17 M utation .............................................................................................................. 18 iv Population S iz e .................................................................................................. 19 A Simple Example............................................................................................. 19 CHAPTER 4 THE FRACTAL INVERSE PROBLEM ............................................ 23 Initialization ...................................................................................................... 23 Population ......................................................................................................... 24 F itn ess ................................................................................................................ 24 Reproduction ...................................................................................................... 26 C rossover ........................................................................................................... 27 M utation .............................................................................................................. 27 One-Dimensional Experiments ......................................................................... 27 Attractor 1 (The Original Cantor S e t) ............................................................. 30 Attractor 2 (The Best of the First Experiment) ............................................ 31 Attractor 3 (From the Second Experiment) ................................................... 32 Attractor 4 (The Original for the Third Experiment) ..................................... 33 Attractor 5 (The Best of the Third Experiment)............................................ 34 Attractor 6 (The Best of the Fourth Experiment) .......................................... 35 Second and Third trials of the First Experiment .............................................. 36 Attractor 7 (Generation 1, Fitness = 0 .279) ................................................... 36 Attractor 8 (Generation 21, Fitness = 0 .4 3 0 ) ................................................. 38 Attractor 9 (Generation 53, fitness = 0.765) ................................................. 39 Attractor 10 (Generation 202, Fitness = 0.951)
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