The Fractal Construction by Iterated Function Systems
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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. -
Harmonious Hilbert Curves and Other Extradimensional Space-Filling Curves
Harmonious Hilbert curves and other extradimensional space-filling curves∗ Herman Haverkorty November 2, 2012 Abstract This paper introduces a new way of generalizing Hilbert's two-dimensional space-filling curve to arbitrary dimensions. The new curves, called harmonious Hilbert curves, have the unique property that for any d0 < d, the d-dimensional curve is compatible with the d0-dimensional curve with respect to the order in which the curves visit the points of any d0-dimensional axis-parallel space that contains the origin. Similar generalizations to arbitrary dimensions are described for several variants of Peano's curve (the original Peano curve, the coil curve, the half-coil curve, and the Meurthe curve). The d-dimensional harmonious Hilbert curves and the Meurthe curves have neutral orientation: as compared to the curve as a whole, arbitrary pieces of the curve have each of d! possible rotations with equal probability. Thus one could say these curves are `statistically invariant' under rotation|unlike the Peano curves, the coil curves, the half-coil curves, and the familiar generalization of Hilbert curves by Butz and Moore. In addition, prompted by an application in the construction of R-trees, this paper shows how to construct a 2d-dimensional generalized Hilbert or Peano curve that traverses the points of a certain d-dimensional diagonally placed subspace in the order of a given d-dimensional generalized Hilbert or Peano curve. Pseudocode is provided for comparison operators based on the curves presented in this paper. 1 Introduction Space-filling curves A space-filling curve in d dimensions is a continuous, surjective mapping from R to Rd. -
Homeomorphisms Group of Normed Vector Space: Conjugacy Problems
HOMEOMORPHISMS GROUP OF NORMED VECTOR SPACE: CONJUGACY PROBLEMS AND THE KOOPMAN OPERATOR MICKAEL¨ D. CHEKROUN AND JEAN ROUX Abstract. This article is concerned with conjugacy problems arising in the homeomorphisms group, Hom(F ), of unbounded subsets F of normed vector spaces E. Given two homeomor- phisms f and g in Hom(F ), it is shown how the existence of a conjugacy may be related to the existence of a common generalized eigenfunction of the associated Koopman operators. This common eigenfunction serves to build a topology on Hom(F ), where the conjugacy is obtained as limit of a sequence generated by the conjugacy operator, when this limit exists. The main conjugacy theorem is presented in a class of generalized Lipeomorphisms. 1. Introduction In this article we consider the conjugacy problem in the homeomorphisms group of a finite dimensional normed vector space E. It is out of the scope of the present work to review the problem of conjugacy in general, and the reader may consult for instance [13, 16, 29, 33, 26, 42, 45, 51, 52] and references therein, to get a partial survey of the question from a dynamical point of view. The present work raises the problem of conjugacy in the group Hom(F ) consisting of homeomorphisms of an unbounded subset F of E and is intended to demonstrate how the conjugacy problem, in such a case, may be related to spectral properties of the associated Koopman operators. In this sense, this paper provides new insights on the relations between the spectral theory of dynamical systems [5, 17, 23, 36] and the topological conjugacy problem [51, 52]1. -
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. -
Iterated Function Systems, Ruelle Operators, and Invariant Projective Measures
MATHEMATICS OF COMPUTATION Volume 75, Number 256, October 2006, Pages 1931–1970 S 0025-5718(06)01861-8 Article electronically published on May 31, 2006 ITERATED FUNCTION SYSTEMS, RUELLE OPERATORS, AND INVARIANT PROJECTIVE MEASURES DORIN ERVIN DUTKAY AND PALLE E. T. JORGENSEN Abstract. We introduce a Fourier-based harmonic analysis for a class of dis- crete dynamical systems which arise from Iterated Function Systems. Our starting point is the following pair of special features of these systems. (1) We assume that a measurable space X comes with a finite-to-one endomorphism r : X → X which is onto but not one-to-one. (2) In the case of affine Iterated Function Systems (IFSs) in Rd, this harmonic analysis arises naturally as a spectral duality defined from a given pair of finite subsets B,L in Rd of the same cardinality which generate complex Hadamard matrices. Our harmonic analysis for these iterated function systems (IFS) (X, µ)is based on a Markov process on certain paths. The probabilities are determined by a weight function W on X. From W we define a transition operator RW acting on functions on X, and a corresponding class H of continuous RW - harmonic functions. The properties of the functions in H are analyzed, and they determine the spectral theory of L2(µ).ForaffineIFSsweestablish orthogonal bases in L2(µ). These bases are generated by paths with infinite repetition of finite words. We use this in the last section to analyze tiles in Rd. 1. Introduction One of the reasons wavelets have found so many uses and applications is that they are especially attractive from the computational point of view. -
Redalyc.Self-Similarity of Space Filling Curves
Ingeniería y Competitividad ISSN: 0123-3033 [email protected] Universidad del Valle Colombia Cardona, Luis F.; Múnera, Luis E. Self-Similarity of Space Filling Curves Ingeniería y Competitividad, vol. 18, núm. 2, 2016, pp. 113-124 Universidad del Valle Cali, Colombia Available in: http://www.redalyc.org/articulo.oa?id=291346311010 How to cite Complete issue Scientific Information System More information about this article Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Journal's homepage in redalyc.org Non-profit academic project, developed under the open access initiative Ingeniería y Competitividad, Volumen 18, No. 2, p. 113 - 124 (2016) COMPUTATIONAL SCIENCE AND ENGINEERING Self-Similarity of Space Filling Curves INGENIERÍA DE SISTEMAS Y COMPUTACIÓN Auto-similaridad de las Space Filling Curves Luis F. Cardona*, Luis E. Múnera** *Industrial Engineering, University of Louisville. KY, USA. ** ICT Department, School of Engineering, Department of Information and Telecommunication Technologies, Faculty of Engineering, Universidad Icesi. Cali, Colombia. [email protected]*, [email protected]** (Recibido: Noviembre 04 de 2015 – Aceptado: Abril 05 de 2016) Abstract We define exact self-similarity of Space Filling Curves on the plane. For that purpose, we adapt the general definition of exact self-similarity on sets, a typical property of fractals, to the specific characteristics of discrete approximations of Space Filling Curves. We also develop an algorithm to test exact self- similarity of discrete approximations of Space Filling Curves on the plane. In addition, we use our algorithm to determine exact self-similarity of discrete approximations of four of the most representative Space Filling Curves. -
Recursion, Writing, Iteration. a Proposal for a Graphics Foundation of Computational Reason Luca M
Recursion, Writing, Iteration. A Proposal for a Graphics Foundation of Computational Reason Luca M. Possati To cite this version: Luca M. Possati. Recursion, Writing, Iteration. A Proposal for a Graphics Foundation of Computa- tional Reason. 2015. hal-01321076 HAL Id: hal-01321076 https://hal.archives-ouvertes.fr/hal-01321076 Preprint submitted on 24 May 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. 1/16 Recursion, Writing, Iteration A Proposal for a Graphics Foundation of Computational Reason Luca M. Possati In this paper we present a set of philosophical analyses to defend the thesis that computational reason is founded in writing; only what can be written is computable. We will focus on the relations among three main concepts: recursion, writing and iteration. The most important questions we will address are: • What does it mean to compute something? • What is a recursive structure? • Can we clarify the nature of recursion by investigating writing? • What kind of identity is presupposed by a recursive structure and by computation? Our theoretical path will lead us to a radical revision of the philosophical notion of identity. The act of iterating is rooted in an abstract space – we will try to outline a topological description of iteration. -
Fractal Initialization for High-Quality Mapping with Self-Organizing Maps
Neural Comput & Applic DOI 10.1007/s00521-010-0413-5 ORIGINAL ARTICLE Fractal initialization for high-quality mapping with self-organizing maps Iren Valova • Derek Beaton • Alexandre Buer • Daniel MacLean Received: 15 July 2008 / Accepted: 4 June 2010 Ó Springer-Verlag London Limited 2010 Abstract Initialization of self-organizing maps is typi- 1.1 Biological foundations cally based on random vectors within the given input space. The implicit problem with random initialization is Progress in neurophysiology and the understanding of brain the overlap (entanglement) of connections between neu- mechanisms prompted an argument by Changeux [5], that rons. In this paper, we present a new method of initiali- man and his thought process can be reduced to the physics zation based on a set of self-similar curves known as and chemistry of the brain. One logical consequence is that Hilbert curves. Hilbert curves can be scaled in network size a replication of the functions of neurons in silicon would for the number of neurons based on a simple recursive allow for a replication of man’s intelligence. Artificial (fractal) technique, implicit in the properties of Hilbert neural networks (ANN) form a class of computation sys- curves. We have shown that when using Hilbert curve tems that were inspired by early simplified model of vector (HCV) initialization in both classical SOM algo- neurons. rithm and in a parallel-growing algorithm (ParaSOM), Neurons are the basic biological cells that make up the the neural network reaches better coverage and faster brain. They form highly interconnected communication organization. networks that are the seat of thought, memory, con- sciousness, and learning [4, 6, 15]. -
Bézier Curves Are Attractors of Iterated Function Systems
New York Journal of Mathematics New York J. Math. 13 (2007) 107–115. All B´ezier curves are attractors of iterated function systems Chand T. John Abstract. The fields of computer aided geometric design and fractal geom- etry have evolved independently of each other over the past several decades. However, the existence of so-called smooth fractals, i.e., smooth curves or sur- faces that have a self-similar nature, is now well-known. Here we describe the self-affine nature of quadratic B´ezier curves in detail and discuss how these self-affine properties can be extended to other types of polynomial and ra- tional curves. We also show how these properties can be used to control shape changes in complex fractal shapes by performing simple perturbations to smooth curves. Contents 1. Introduction 107 2. Quadratic B´ezier curves 108 3. Iterated function systems 109 4. An IFS with a QBC attractor 110 5. All QBCs are attractors of IFSs 111 6. Controlling fractals with B´ezier curves 112 7. Conclusion and future work 114 References 114 1. Introduction In the late 1950s, advancements in hardware technology made it possible to effi- ciently manufacture curved 3D shapes out of blocks of wood or steel. It soon became apparent that the bottleneck in mass production of curved 3D shapes was the lack of adequate software for designing these shapes. B´ezier curves were first introduced in the 1960s independently by two engineers in separate French automotive compa- nies: first by Paul de Casteljau at Citro¨en, and then by Pierre B´ezier at R´enault. -
Summary of Unit 1: Iterated Functions 1
Summary of Unit 1: Iterated Functions 1 Summary of Unit 1: Iterated Functions David P. Feldman http://www.complexityexplorer.org/ Summary of Unit 1: Iterated Functions 2 Functions • A function is a rule that takes a number as input and outputs another number. • A function is an action. • Functions are deterministic. The output is determined only by the input. x f(x) f David P. Feldman http://www.complexityexplorer.org/ Summary of Unit 1: Iterated Functions 3 Iteration and Dynamical Systems • We iterate a function by turning it into a feedback loop. • The output of one step is used as the input for the next. • An iterated function is a dynamical system, a system that evolves in time according to a well-defined, unchanging rule. x f(x) f David P. Feldman http://www.complexityexplorer.org/ Summary of Unit 1: Iterated Functions 4 Itineraries and Seeds • We iterate a function by applying it again and again to a number. • The number we start with is called the seed or initial condition and is usually denoted x0. • The resulting sequence of numbers is called the itinerary or orbit. • It is also sometimes called a time series or a trajectory. • The iterates are denoted xt. Ex: x5 is the fifth iterate. David P. Feldman http://www.complexityexplorer.org/ Summary of Unit 1: Iterated Functions 5 Time Series Plots • A useful way to visualize an itinerary is with a time series plot. 0.8 0.7 0.6 0.5 t 0.4 x 0.3 0.2 0.1 0.0 0 1 2 3 4 5 6 7 8 9 10 time t • The time series plotted above is: 0.123, 0.189, 0.268, 0.343, 0.395, 0.418, 0.428, 0.426, 0.428, 0.429, 0.429. -
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 ....................................................................................