Showing papers on "Iterated function system published in 2002"

Quantization dimension for conformal iterated function systems

Abstract: The quantization dimension function is determined for a certain class of probability measures generated by a finite conformal iterated function system satisfying the strong open set condition.

On the Fine Structure of Stationary Measures in Systems which Contract-on-Average

Abstract: Suppose {f1,...,fm} is a set of Lipschitz maps of R d . We form the iterated function system (IFS) by independently choosing the maps so that the map fi is chosen with probability pi ( Pm=1 pi = 1). We assume that the IFS contracts on average. We give an upper bound for the Hausdorff dimension of the invariant measure induced on R d and as a corollary show that the measure will be singular if the modulus of the entropy P i pi logpi is less than d times the modulus of the Lyapunov exponent of the system. Using a version of Shannon's Theorem for random walks on semigroups we improve this estimate and show that it is actually attainable for certain cases of affine mappings ofR.

Fractal vector measures and vector calculus on planar fractal domains

Abstract: We define an abstract framework for self-similar vector-valued Borel measures on a compact space X based upon a formulation of Iterated Function Systems (IFS) on such measures. This IFS method permits the construction of tangent and normal vector measures to planar fractal curves. Line integrals of smooth vector fields over planar fractal curves may then be defined. These line integrals then lead to a formulation of Green's Theorem and the Divergence Theorem for planar regions bounded by fractal curves. The abstract setting also naturally leads to “probability measure”-valued measures. These measures may be used to color the geometric attractor of an IFS in a self-similar way.

Diophantine Analysis of Conformal Iterated Function Systems

Abstract: A formula for the Hausdorff dimension of a subset of the limit set of a conformal infinite iterated function system is derived and expressed as a unique zero of the topological pressure of corresponding strongly Holder family of functions.

A Contractivity Condition for Iterated Function Systems

Abstract: We prove a condition for long-term contractivity and the existence of a unique invariant measure for iterated function systems. We also give an intuitive interpretation of the condition in terms of weighted derivatives and weighted Wasserstein metrics. We use our condition in order to show some results for stochastic population models based on the logistic and Ricker maps.

Approximating Distribution Functions by iterated function systems

Abstract: In this small note an iterated function system on the space of distribution functions is built. The inverse problem is introduced and studied by convex optimization problems. Applications of this method to approximation of distribution functions and estimations are presented.

Iterated Function Systems for Lossless Data Compression

Abstract: Iterated Function Systems with place-dependent probabilities are considered. Fascinating geometrical invariants, that apply even when there is no unique invariant measure, are presented. Furthermore, it is shown that the invariant measure of a stationary stochastic process, when it contains no atoms and is fully supported, can sometimes be associated with an IFS with probabilities, and with an associated dynamical system. This leads to the idea of an ergodic transform of a string of symbols, with respect to a given string; this is introduced and shown to be useful; it provides a unifying geometrical approach to the description of data compression algorithms such as Huffman, arithmetic, and the Burrows-Wheeler transform.

Hausdorff dimension estimates for infinite conformal IFSs

Abstract: We show that in order to estimate the Hausdorff dimension of conformal infinite iterated function systems (IFSs), it is completely sufficient to consider finite subsystems with N

The boundary of the attractor of a recurrent iterated function system

Abstract: We prove for a subclass of recurrent iterated function systems (also called graph-directed iterated function systems) that the boundary of their attractor is again the attractor of a recurrent IFS. Our method is constructive and permits computation of the Hausdorff dimension of the attractor and its boundary.

Level sets of harmonic functions on the Sierpiński gasket

Abstract: We give a detailed description of nonconstant harmonic functions and their level sets on the Sierpinski gasket. We introduce a parameter, calledeccentricity, which classifies these functions up to affine transformationsh→ah+b. We describe three (presumably) distinct measures that describe how the eccentricities are distributed in the limit as we subdivide the gasket into smaller copies (cells) and restrict the harmonic function to the small cells. One measure simply counts the number of small cells with eccentricity in a specified range. One counts the contribution to the total energy coming from those cells. And one counts just those cells that intersect a fixed generic level set. The last measure yields a formula for the box dimension of a generic level set. All three measures are defined by invariance equations with respect to the same iterated function system, but with different weights. We also give a construction for a rectifiable curve containing a given level set. We exhibit examples where the curve has infinite winding number with respect to some points.

Convolutions of equicontractive self-similar measures on the line

Abstract: Let μ be a self-similar measure on R generated by an equicontractive iterated function system. We prove that the Hausdorff dimension of μ∗n tends to 1 as n tends to infinity, where μ∗n denotes the n-fold convolution of μ. Similar results hold for the Lq dimension and the entropy dimension of μ∗n.

Exploring the Effect of Direction on Vector-Based Fractals

Abstract: This paper investigates an approach to begin the study of fractals in architectural design. Vector-based fractals are studied to determine if the modification of vector direction in either the generator or the initiator will develop alternate fractal forms. The Koch Snowflake is used as the demonstrating fractal. Introduction A fractal is an object or quantity that displays self-similarity on all scales. The object need not exhibit exactly the same structure at all scales, but the same “type” of structures must appear on all scales [7]. Fractals were first discussed by Mandelbrot in the 1970s [4], but the idea was identified as early as 1925. Fractals have been investigated for their visual qualities as art, their relationship to explain natural processes, music, medicine, and in mathematics [5]. Javier Barrallo classified fractals [2] into six main groups depending on their type: 1. Fractals derived from standard geometry by using iterative transformations on an initial common figure. 2. IFS (Iterated Function Systems), this is a type of fractal introduced by Michael Barnsley. 3. Strange attractors. 4. Plasma fractals. Created with techniques like fractional Brownian motion. 5. L-Systems, also called Lindenmayer systems, were not invented to create fractals but to model cellular growth and interactions. 6. Fractals created by the iteration of complex polynomials. From the mathematical point of view, we can classify fractals into three major categories. The first, IFS, iterated function system, like Koch Snowflake, Cantor set, Barnsley's Fern and the Dragon Curve shown in Figure 4. This method can generate a fractal from any set of vectors or any defined curve. The second is the complex number fractals. They can be two-dimensional, three-dimensional, or multipledimensional. They represent a single case of the IFS that is using the complex numbers or the hyper complex numbers in a Cartesian plane to plot the fractals. The Mandelbrot set and Julia set are examples of these. The third is orbit fractals. They are generated by plotting an orbit in two or three-dimensional space. Examples include the Bifurcation orbit, Lorenz Attractors, Rossler Attractors, Henon Attractors, Pickover Attractors, Gingerbreadman, and Martin Attractors. These are associated with chaos theory. By examining these fractals, we were also able to classify them into two major categories depending on the method they could be created, or the mathematical method used to calculate them. From the drawing method point of view, the first method is a line or vector fractal. These are generated from the recursive replacement of vectors, like the Dragon Curve in Figure 1a. The second are fractals that are generated as a group of points in the complex plane, such as the Mandelbrot set and the Julia set, as in Figure 1b. Some fractals that exhibit a vector quality can also be generated by point plotting methods. Figure 1. Fractal types In this research, we are interested in vector-based fractals and the use of the replacement concept and the iterated function system as a way to generate them. These fractals have directional and geometrical properties that make them possibly suitable for applications in architecture. Vector-based fractals can be described in terms of vertices and the lines connecting them. This has the potential to be used directly as architectural elements or to simply use the vertices to define the locations of such elements. Chris Yessios with Peter Eisenman [8] were among the first to write about utilizing fractals and fractal geometry in architecture. Yessios described a way computers can be introduced to architectural design as an explorer and generator of architectural forms. He used the fractal geometry, arabesque ornamentation and DNA/RNA biological processes as fractal generators. A fractal program was developed that enabled him to use several generators on the same base and to go many steps forward in the iteration process, as well as, backward. The project that was used in this investigation was a studio project to design a building for a competition given a specific architectural program. More recently, S. Durmisevic and O. Ciftcioglu [3] discussed another approach on how fractals can be used in architectural design. They proposed to use a fractal tree form as an indicator of a road infrastructure and another fractal to determine the type of architectural forms to place along this transportation spine. Fractal geometry was used for architectural forms and urban planning. No other similar studies could be found where fractals were used as architectural form generators to suggest three-dimensional forms. The results from these two suggest that the explicit self-similarity and the repetition of the same shape may distract from developing interesting architectural forms, so both developed means to modify the replication. For this particular study we intend to investigate only the directional property found in generating vector-based fractals. b. Mandelbrot set a. Dragon fractal a. Dragon curve b. Mandelbrot set Creating a fractal A vector-base fractal, Figure 2, is composed of two parts: the initiator and the generator. For example, the Koch Snowflake starts with an equilateral triangle as the initiator. The generator is a line that is divided into three equal segments. The middle segment forms an equilateral triangle. By replacing every line of the initiator with the full generator, we get the first iteration of the snowflake. By iterating this operation again and again, replacing every line of the new initiator with the full generator, we end with a figure that approximates a snowflake. The iteration process can continue to infinity to generate a real Koch Snowflake fractal, but as we are interested in the evolving form, we only iterate the function for some finite number of times. Figure 3a displays the Koch Snowflake for each of three iterations. Normally, an alternative version can be created if the generator is changed, inverted. It then becomes the Koch Antisnowflake, Figure 3b. Figure 2. Generator and initiator Figure 3. The Koch Snowflake and the Antisnowflake Shape of generators and initiators There is a group of fractals, including Mandelbrot set variations that have been formally identified in his book “The Fractal Geometry of Nature” [4] as depending on the concept of replacement. Some of the IFS fractals are: Cantor Set, Barnsley's Fern, Koch Antisnowflake, Koch Snowflake, Box Fractal, Cantor Square Fractal, Cesaro Fractal, Dragon Curve, Gosper Island Fractal, H-Fractal, Sierpinski Curve, Minkowski Sausage. Some of these are displayed in Figure 4. Initiator Generator

Supports of Locally Linearly Independent M -Refinable Functions, Attractors of Iterated Function Systems and Tilings

Abstract: This paper is devoted to a study of supports of locally linearly independent M-refinable functions by means of attractors of iterated function systems, where M is an integer greater than (or equal to) 2. For this purpose, the local linear independence of shifts of M-refinable functions is required. So we give a complete characterization for this local linear independence property by finite matrix products, strictly in terms of the mask. We do this in a more general setting, the vector refinement equations. A connection between self-affine tilings and L 2 solutions of refinement equations without satisfying the basic sum rule is pointed out, which leads to many further problems. Several examples are provided to illustrate the general theory.

On some structure–morphology problems by the extended partitioned iterated function system (PIFS-SF)

Abstract: Using a newly developed technique of coding 2D sets (PIFS-SF) we have shown how to analyse the family of sets (images) given with different resolutions. Based on the number of codes (PIFS-SF) we have been able to describe quantitatively the high-resolution structure (HRS) and low-resolution structure (LRS), as well as supramolecular structure (SMS). Following the essence of aggregation processes (crystallization, gelation, clustering) we showed how different SMS, LRS and morphologies are generated. In a sense “by occasion” we also approached the problems of uniqueness of coding and data compression. To produce required images with different resolution we have employed a process of deterministic aggregation of cells (first generation aggregates – FGA) made of different generators (molecules) with the Sierpinski gasket as an attractor. An example of some real sets analysis has also been provided.

Approximating Continuous Functions by Iterated Function Systems and Optimization

Abstract: In this paper some new contractive operators on C([a,b]) of IFS type are built. Inverse problems are introduced and studied by convex optimization problems. A stability result and some optimality conditions are given.

Introduction to IMA fractal proceedings

Abstract: This volume describes the status of fractal imaging research and looks to future directions. It is to be useful to researchers in the areas of fractal image compression, analysis, and synthesis, iterated function systems, and fractals in education. In particular it includes a vision for the future of these areas.

Methods of Fractal Image for Simulating the Natural Sight

Abstract: Fractal L system simulates botanic structure. The pictures of fractal about simulating plant are made by 3-DL system. The ideas and arithmetic of fractal iterated function system to render virtual scenes are described. The image of art is formed by Julia set. An approach to the form of fractal mountain terrain models is introduced. Render natural scenes are achieved and simulating the natural sight comparison amony three methods is made.

Rendering of virtual scenes based on IFS

Abstract: The fractal geometry can be used to display the natural scene. The ideas and arithmetic of fractal iterated function system to render virtual scenes are described and the multi-IFS and nested-IFS to render natural scenes are discussed and researched. The random iterated arithmetic based on probability is used to render scenes true to nature.

Nonlinear thoughts about linear signal processing

Abstract: Recent work on modelling digital channels using iterated function systems suggests a general approach to the theory of signal processing in digital communications which uses so-called delay methods developed for deterministic nonlinear timeseries analysis. Here we make the connection between this work and the more conventional approach to digital communications by casting linear channel models as iterated function systems and showing how the use of delay methods gives a nice connection with the theory of observability in the control of linear systems.

On the continuity and differentiability of a kind of fractal interpolation function

Abstract: The sufficient conditions of Holder continuity of two kinds of fractal interpolation functions defined by IFS (Iterated Function System) were obtained. The sufficient and necessary condition for its differentiability was proved. Its derivative was a fractal interpolation function generated by the associated IFS, if it is differentiable.

Fractal image compression using MNLPC, MIC and H-MPC network library

Abstract: The partitioned iterated function systems (PIFS) fractal image compression technique provides very competitive rate-distortion curves and fast decoding. However, it suffers from complicated encoding computation. Three novel neural network techniques, mixture of nonlinear principal components (MNLPC), mixture of independent components (MIC) and high-dimensional mixture of principal components (H-MPC) are developed to reduce the encoding complexity of the PIFS fractal coding. Applying these new techniques, the potential best range-domain matching search is confined to a relatively small size domain block pool. Using the new techniques, the encoding time is shortened dramatically, and the compression performance is improved as well.

Approximating continuous functions by iterated function system and optimization

Abstract: In this paper some new contractive operators on C([a, b]) of IFS type arebuilt. Inverse problems are introduced and studied by convex optimizationproblems. A stability result and some optimality conditions are given.

Fractal Image Coding as Cross-Scale Wavelet Coefficient Prediction

Abstract: Fractal image compression techniques, introduced by Barnsley and Jacquin [3], are the product of the study of iterated function systems (IFS)[2]. These techniques involve an approach to compression quite different from standard transform coder-based methods. Transform coders model images in a very simple fashion, namely, as vectors drawn from a wide-sense stationary random process. They store images as quantized transform coefficients. Fractal block coders, as described by Jacquin, assume that “image redundancy can be efficiently exploited through selftransformability on a blockwise basis” [16]. They store images as contraction maps of which the images are approximate fixed points. Images are decoded via iterative application of these maps. In this chapter we examine the connection between the fractal block coders as introduced by Jacquin [16] and transform coders. We show that fractal coding is a form of wavelet subtree quantization. The basis used by the Jacquin-style block coders is the Haar basis. Our wavelet based analysis framework leads to improved understanding of the behavior of fractal schemes. We describe a simple generalization of fractal block coders that yields a substantial improvement in performance, giving results comparable to the best reported for fractal schemes. Finally, our wavelet framework reveals some of the limitations of current fractal coders. The term “fractal image coding” is defined rather loosely and has been used to describe a wide variety of algorithms. Throughout this chapter, when we discuss fractal image coding, we will be referring to the block-based coders of the form described in [16] and [11].

Identification and modeling of fractal signals with iterated function systems

Abstract: We analyze a minimal model of iterated function systems (IFS) and show possibilities for controlling properties of the fractal signals they generate. Applications for signal modeling and spread-spectrum communications illustrate the potentialities of IFS for signal processing.

A probabilistic power domain algorithm for fractal image decoding

Abstract: A new algorithm, called herein the random power domain algorithm, is discussed; it generates the image corresponding to an iterated function system with probabilities, a technique used in fractal image decoding. A simple complexity analysis for the algorithm is also derived.

Euroattractor: a brief introduction to Iterated Function Systems

Abstract: In this work we propose a definition of an Euroattractor: an attracting invariant measure of a certain iterated functions system (IFS). An IFS is defined by specifying a set of functions, defined in subsets of R^N or in a classical phase space, which act randomly on the initial point, so it may be considered as a generalization of the notion of classical dynamical system. If the functions are sufficiently contracting, there exists an invariant measure of the system, often concentrated on a fractal set. We investigate invariant measures of a certain class of generalized or weakly contracting IFS's.

Magnifying partial image using fractal method

Abstract: A new method is proposed, in this paper, to magnify a portion of an image based on the fact that fractal attractor has details at every scale The formulae for magnifying image portions with fractal method are deduced The fractal codes, the coefficients of a set of contraction mappings, are determined by encoding the portion to be magnified The fractal codes are then modified according to the formulae deduced in this paper, and the magnified image is obtained by decoding the modified fractal codes A portion of Lenna image is enlarged by a factor of 8 at both horizontal and vertical directions with 2 methods respectively, one is the new method described and the other is pixel duplication Experimental results show that the new method is good for partial image magnification with no block effect