Fractal Initialization for High-Quality Mapping with Self-Organizing Maps
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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. -
Measuring the Fractal Dimensions of Empirical Cartographic Curves
MEASURING THE FRACTAL DIMENSIONS OF EMPIRICAL CARTOGRAPHIC CURVES Mark C. Shelberg Cartographer, Techniques Office Aerospace Cartography Department Defense Mapping Agency Aerospace Center St. Louis, AFS, Missouri 63118 Harold Moellering Associate Professor Department of Geography Ohio State University Columbus, Ohio 43210 Nina Lam Assistant Professor Department of Geography Ohio State University Columbus, Ohio 43210 Abstract The fractal dimension of a curve is a measure of its geometric complexity and can be any non-integer value between 1 and 2 depending upon the curve's level of complexity. This paper discusses an algorithm, which simulates walking a pair of dividers along a curve, used to calculate the fractal dimensions of curves. It also discusses the choice of chord length and the number of solution steps used in computing fracticality. Results demonstrate the algorithm to be stable and that a curve's fractal dimension can be closely approximated. Potential applications for this technique include a new means for curvilinear data compression, description of planimetric feature boundary texture for improved realism in scene generation and possible two-dimensional extension for description of surface feature textures. INTRODUCTION The problem of describing the forms of curves has vexed researchers over the years. For example, a coastline is neither straight, nor circular, nor elliptic and therefore Euclidean lines cannot adquately describe most real world linear features. Imagine attempting to describe the boundaries of clouds or outlines of complicated coastlines in terms of classical geometry. An intriguing concept proposed by Mandelbrot (1967, 1977) is to use fractals to fill the void caused by the absence of suitable geometric representations. -
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. -
Fractals, Self-Similarity & Structures
© Landesmuseum für Kärnten; download www.landesmuseum.ktn.gv.at/wulfenia; www.biologiezentrum.at Wulfenia 9 (2002): 1–7 Mitteilungen des Kärntner Botanikzentrums Klagenfurt Fractals, self-similarity & structures Dmitry D. Sokoloff Summary: We present a critical discussion of a quite new mathematical theory, namely fractal geometry, to isolate its possible applications to plant morphology and plant systematics. In particular, fractal geometry deals with sets with ill-defined numbers of elements. We believe that this concept could be useful to describe biodiversity in some groups that have a complicated taxonomical structure. Zusammenfassung: In dieser Arbeit präsentieren wir eine kritische Diskussion einer völlig neuen mathematischen Theorie, der fraktalen Geometrie, um mögliche Anwendungen in der Pflanzen- morphologie und Planzensystematik aufzuzeigen. Fraktale Geometrie behandelt insbesondere Reihen mit ungenügend definierten Anzahlen von Elementen. Wir meinen, dass dieses Konzept in einigen Gruppen mit komplizierter taxonomischer Struktur zur Beschreibung der Biodiversität verwendbar ist. Keywords: mathematical theory, fractal geometry, self-similarity, plant morphology, plant systematics Critical editions of Dean Swift’s Gulliver’s Travels (see e.g. SWIFT 1926) recognize a precise scale invariance with a factor 12 between the world of Lilliputians, our world and that one of Brobdingnag’s giants. Swift sarcastically followed the development of contemporary science and possibly knew that even in the previous century GALILEO (1953) noted that the physical laws are not scale invariant. In fact, the mass of a body is proportional to L3, where L is the size of the body, whilst its skeletal rigidity is proportional to L2. Correspondingly, giant’s skeleton would be 122=144 times less rigid than that of a Lilliputian and would be destroyed by its own weight if L were large enough (cf. -
FRACTAL CURVES 1. Introduction “Hike Into a Forest and You Are Surrounded by Fractals. the In- Exhaustible Detail of the Livin
FRACTAL CURVES CHELLE RITZENTHALER Abstract. Fractal curves are employed in many different disci- plines to describe anything from the growth of a tree to measuring the length of a coastline. We define a fractal curve, and as a con- sequence a rectifiable curve. We explore two well known fractals: the Koch Snowflake and the space-filling Peano Curve. Addition- ally we describe a modified version of the Snowflake that is not a fractal itself. 1. Introduction \Hike into a forest and you are surrounded by fractals. The in- exhaustible detail of the living world (with its worlds within worlds) provides inspiration for photographers, painters, and seekers of spiri- tual solace; the rugged whorls of bark, the recurring branching of trees, the erratic path of a rabbit bursting from the underfoot into the brush, and the fractal pattern in the cacophonous call of peepers on a spring night." Figure 1. The Koch Snowflake, a fractal curve, taken to the 3rd iteration. 1 2 CHELLE RITZENTHALER In his book \Fractals," John Briggs gives a wonderful introduction to fractals as they are found in nature. Figure 1 shows the first three iterations of the Koch Snowflake. When the number of iterations ap- proaches infinity this figure becomes a fractal curve. It is named for its creator Helge von Koch (1904) and the interior is also known as the Koch Island. This is just one of thousands of fractal curves studied by mathematicians today. This project explores curves in the context of the definition of a fractal. In Section 3 we define what is meant when a curve is fractal. -
Turtlefractalsl2 Old
Lecture 2: Fractals from Recursive Turtle Programs use not vain repetitions... Matthew 6: 7 1. Fractals Pictured in Figure 1 are four fractal curves. What makes these shapes different from most of the curves we encountered in the previous lecture is their amazing amount of fine detail. In fact, if we were to magnify a small region of a fractal curve, what we would typically see is the entire fractal in the large. In Lecture 1, we showed how to generate complex shapes like the rosette by applying iteration to repeat over and over again a simple sequence of turtle commands. Fractals, however, by their very nature cannot be generated simply by repeating even an arbitrarily complicated sequence of turtle commands. This observation is a consequence of the Looping Lemmas for Turtle Graphics. Sierpinski Triangle Fractal Swiss Flag Koch Snowflake C-Curve Figure 1: Four fractal curves. 2. The Looping Lemmas Two of the simplest, most basic turtle programs are the iterative procedures in Table 1 for generating polygons and spirals. The looping lemmas assert that all iterative turtle programs, no matter how many turtle commands appear inside the loop, generate shapes with the same general symmetries as these basic programs. (There is one caveat here, that the iterating index is not used inside the loop; otherwise any turtle program can be simulated by iteration.) POLY (Length, Angle) SPIRAL (Length, Angle, Scalefactor) Repeat Forever Repeat Forever FORWARD Length FORWARD Length TURN Angle TURN Angle RESIZE Scalefactor Table 1: Basic procedures for generating polygons and spirals via iteration. Circle Looping Lemma Any procedure that is a repetition of the same collection of FORWARD and TURN commands has the structure of POLY(Length, Angle), where Angle = Total Turtle Turning within the Loop Length = Distance from Turtle’s Initial Position to Turtle’s Final Position within the Loop That is, the two programs have the same boundedness, closing, and symmetry. -
Introduction to Fractal Geometry: Definition, Concept, and Applications
University of Northern Iowa UNI ScholarWorks Presidential Scholars Theses (1990 – 2006) Honors Program 1992 Introduction to fractal geometry: Definition, concept, and applications Mary Bond University of Northern Iowa Let us know how access to this document benefits ouy Copyright ©1992 - Mary Bond Follow this and additional works at: https://scholarworks.uni.edu/pst Part of the Geometry and Topology Commons Recommended Citation Bond, Mary, "Introduction to fractal geometry: Definition, concept, and applications" (1992). Presidential Scholars Theses (1990 – 2006). 42. https://scholarworks.uni.edu/pst/42 This Open Access Presidential Scholars Thesis is brought to you for free and open access by the Honors Program at UNI ScholarWorks. It has been accepted for inclusion in Presidential Scholars Theses (1990 – 2006) by an authorized administrator of UNI ScholarWorks. For more information, please contact [email protected]. INTRODUCTION TO FRACTAL GEOMETRY: DEFINITI0N7 CONCEPT 7 AND APP LI CA TIO NS MAY 1992 BY MARY BOND UNIVERSITY OF NORTHERN IOWA CEDAR F ALLS7 IOWA • . clouds are not spheres, mountains are not cones~ coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line, . • Benoit B. Mandelbrot For centuries, geometers have utilized Euclidean geometry to describe, measure, and study the world around them. The quote from Mandelbrot poses an interesting problem when one tries to apply Euclidean geometry to the natural world. In 1975, Mandelbrot coined the term ·tractal. • He had been studying several individual ·mathematical monsters.· At a young age, Mandelbrot had read famous papers by G. Julia and P . Fatou concerning some examples of these monsters. When Mandelbrot was a student at Polytechnique, Julia happened to be one of his teachers. -
Efficient Neighbor-Finding on Space-Filling Curves
Universitat¨ Stuttgart Efficient Neighbor-Finding on Space-Filling Curves Bachelor Thesis Author: David Holzm¨uller* Degree: B. Sc. Mathematik Examiner: Prof. Dr. Dominik G¨oddeke, IANS Supervisor: Prof. Dr. Miriam Mehl, IPVS October 18, 2017 arXiv:1710.06384v3 [cs.CG] 2 Nov 2019 *E-Mail: [email protected], where the ¨uin the last name has to be replaced by ue. Abstract Space-filling curves (SFC, also known as FASS-curves) are a useful tool in scientific computing and other areas of computer science to sequentialize multidimensional grids in a cache-efficient and parallelization-friendly way for storage in an array. Many algorithms, for example grid-based numerical PDE solvers, have to access all neighbor cells of each grid cell during a grid traversal. While the array indices of neighbors can be stored in a cell, they still have to be computed for initialization or when the grid is adaptively refined. A fast neighbor- finding algorithm can thus significantly improve the runtime of computations on multidimensional grids. In this thesis, we show how neighbors on many regular grids ordered by space-filling curves can be found in an average-case time complexity of (1). In 풪 general, this assumes that the local orientation (i.e. a variable of a describing grammar) of the SFC inside the grid cell is known in advance, which can be efficiently realized during traversals. Supported SFCs include Hilbert, Peano and Sierpinski curves in arbitrary dimensions. We assume that integer arithmetic operations can be performed in (1), i.e. independent of the size of the integer. -
Math Morphing Proximate and Evolutionary Mechanisms
Curriculum Units by Fellows of the Yale-New Haven Teachers Institute 2009 Volume V: Evolutionary Medicine Math Morphing Proximate and Evolutionary Mechanisms Curriculum Unit 09.05.09 by Kenneth William Spinka Introduction Background Essential Questions Lesson Plans Website Student Resources Glossary Of Terms Bibliography Appendix Introduction An important theoretical development was Nikolaas Tinbergen's distinction made originally in ethology between evolutionary and proximate mechanisms; Randolph M. Nesse and George C. Williams summarize its relevance to medicine: All biological traits need two kinds of explanation: proximate and evolutionary. The proximate explanation for a disease describes what is wrong in the bodily mechanism of individuals affected Curriculum Unit 09.05.09 1 of 27 by it. An evolutionary explanation is completely different. Instead of explaining why people are different, it explains why we are all the same in ways that leave us vulnerable to disease. Why do we all have wisdom teeth, an appendix, and cells that if triggered can rampantly multiply out of control? [1] A fractal is generally "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole," a property called self-similarity. The term was coined by Beno?t Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured." A mathematical fractal is based on an equation that undergoes iteration, a form of feedback based on recursion. http://www.kwsi.com/ynhti2009/image01.html A fractal often has the following features: 1. It has a fine structure at arbitrarily small scales. -
Fractal Modeling and Fractal Dimension Description of Urban Morphology
Fractal Modeling and Fractal Dimension Description of Urban Morphology Yanguang Chen (Department of Geography, College of Urban and Environmental Sciences, Peking University, Beijing 100871, P. R. China. E-mail: [email protected]) Abstract: The conventional mathematical methods are based on characteristic length, while urban form has no characteristic length in many aspects. Urban area is a measure of scale dependence, which indicates the scale-free distribution of urban patterns. Thus, the urban description based on characteristic lengths should be replaced by urban characterization based on scaling. Fractal geometry is one powerful tool for scaling analysis of cities. Fractal parameters can be defined by entropy and correlation functions. However, how to understand city fractals is still a pending question. By means of logic deduction and ideas from fractal theory, this paper is devoted to discussing fractals and fractal dimensions of urban landscape. The main points of this work are as follows. First, urban form can be treated as pre-fractals rather than real fractals, and fractal properties of cities are only valid within certain scaling ranges. Second, the topological dimension of city fractals based on urban area is 0, thus the minimum fractal dimension value of fractal cities is equal to or greater than 0. Third, fractal dimension of urban form is used to substitute urban area, and it is better to define city fractals in a 2-dimensional embedding space, thus the maximum fractal dimension value of urban form is 2. A conclusion can be reached that urban form can be explored as fractals within certain ranges of scales and fractal geometry can be applied to the spatial analysis of the scale-free aspects of urban morphology. -
Algorithms for Scientific Computing
Algorithms for Scientific Computing Space-Filling Curves Michael Bader Technical University of Munich Summer 2017 Start: Morton Order / Cantor’s Mapping 00 01 0000 0001 0100 0101 0010 0011 0110 0111 1000 1001 1100 1101 10 11 1010 1011 1110 1111 Questions: • Can this mapping lead to a contiguous “curve”? • i.e.: Can we find a continuous mapping? • and: Can this continuous mapping fill the entire square? Michael Bader j Algorithms for Scientific Computing j Space-Filling Curves j Summer 2017 2 Morton Order and Cantor’s Mapping Georg Cantor (1877): 0:0110 ::: 0:01111001 ::: ! 0:1101 ::: • bijective mapping [0; 1] ! [0; 1]2 • proved identical cardinality of [0; 1] and [0; 1]2 • provoked the question: is there a continuous mapping? (i.e. a curve) Michael Bader j Algorithms for Scientific Computing j Space-Filling Curves j Summer 2017 3 History of Space-Filling Curves 1877: Georg Cantor finds a bijective mapping from the unit interval [0; 1] into the unit square [0; 1]2. 1879: Eugen Netto proves that a bijective mapping f : I ! Q ⊂ Rn can not be continuous (i.e., a curve) at the same time (as long as Q has a smooth boundary). 1886: rigorous definition of curves introduced by Camille Jordan 1890: Giuseppe Peano constructs the first space-filling curves. 1890: Hilbert gives a geometric construction of Peano’s curve; and introduces a new example – the Hilbert curve 1904: Lebesgue curve 1912: Sierpinski curve Michael Bader j Algorithms for Scientific Computing j Space-Filling Curves j Summer 2017 4 Part I Space-Filling Curves Michael Bader j Algorithms for Scientific Computing j Space-Filling Curves j Summer 2017 5 What is a Curve? Definition (Curve) n As a curve, we define the image f∗(I) of a continuous mapping f : I! R .