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Topic: Iterated function systems

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	ICTS
		Introduction: Iterated function systems (IFS) are created by the repeated mapping of a set of points in the plane to a new set of locations.
		Iterated coordinate transformation systems (ICTS) are a subset of IFS in which the mappings are limited to elementary coordinate transformations.
		This family of function systems is explored herein and examples of many of the major sub-classes are shown.
	www.geocities.com /leosti/index.html (626 words)

	Fractal Compression
		Alan Sloan is the co- grantee of the patent and co-founder of Iterated Systems.) Barnsley announced his success to the world in the January 1988 issue of BYTE magazine.
		As for the second, it is implicitly understood that the functions of an Iterated Function Systems are contractive.
		It is comprised of portions of the attractor as it emerges during iteration.
	www.mcs.csuhayward.edu /~grewe/CS6825/Mat/Compression/Fract.htm (3068 words)

	Collage Theorem, Iterated Function Systems
		Functions map points to points but also comprise (function) spaces where each is considered an indivisible one.
		Iterated Function Systems (IFSs) is another rewarding topic that capitalizes on the general theory.
		Two rows of edit controls give you a direct access to the values of a,b,e (the first row) and c,d,f (the second row.) Beware: the coordinate system is the usual one for computer graphics.
	www.cut-the-knot.org /ctk/ifs.shtml (1017 words)

	Iterated function - Wikipedia, the free encyclopedia
		In mathematics, iterated functions are the objects of deep study in fractals and dynamical systems.
		The the idea of iteration can be generalized so that the iteration count n becomes a continuous parameter; in this case, such a system is called a flow.
		Eventually, sequential iterations may appear similar to each other to the human eye, but they are not mathematically similar.
	en.wikipedia.org /wiki/Iterated_function (634 words)

	Iterated Function Systems (IFS) (Site not responding. Last check: 2007-10-14)
		An iterated function system is By taking a point and repeatedly applying these mappings you end up with a collection of points on the fractal.
		Iterated function systems can be used to make things such as fractal ferns and trees and are also used in fractal image compression.
		The simplest algorithm to display an IFS is to pick a starting point, randomly select one of the mappings, apply it to generate a new point, plot the new point, and repeat with the new point.
	fractals.iut.u-bordeaux1.fr /sci-faq/IFS.html (247 words)

	Classifier, IFS, L-Systems and Beyond
		A production system is a set of rules that trigger each other to produce action in their environment and as such could be seen as limited models of how cells, organisms or brains work.
		Another approach is the XCS system, where strength is replaced by accuracy, allowing optimized populations of rules to arise in certain circumstances, and by using a 'niched multiobjective genetic algorithm' these are applicable to multiple payoff landscapes also.
		The ability to develop systems that can learn, that can modify themselves, that can act on their environment and also generate novelty opens up a future of intelligent machines with all the philosophical and technical problems that this brings.
	www.calresco.org /lucas/classify.htm (2081 words)

	Fractals and Fractal Architecture - Iterated Function Systems (Site not responding. Last check: 2007-10-14)
		Iterated function systems, the so-called IFS, again belong to the types of linear fractals like the true mathematical fractals.
		Starting the chaos game means to choose any point on the plane and using one of the given functions after the other at random with marking all resulting points.
		This means that if we find the set of functions, the algorithm, of objects we can reduce their quantity of information - this is important in the case of compressing pictures on the computer.
	www.iemar.tuwien.ac.at /modul23/Fractals/subpages/33IFS.html (1187 words)

	Iterated function system - Wikipedia, the free encyclopedia
		The functions are normally contractive which means they bring points closer together and make shapes smaller.
		It consists of picking a random point in the plane, then iteratively applying one of the functions chosen at random from the function system and drawing the point.
		An alternative algorithm is to generate each possible sequence of functions up to a given maximum length, and then to plot the results of applying each of these sequences of functions to an initial point or shape.
	en.wikipedia.org /wiki/Iterated_function_system (653 words)

	The Fractory: Iterated Function Systems (Site not responding. Last check: 2007-10-14)
		Iterate means to repeat an operation, generally using the last result of that operation as the input.
		An iterated funtion system then is a system where a function is repeated over and over again to itself.
		In each iteration the starting shape is replaced by an appropriately scaled, turned, rotated and, if neccessary, flipped over copy of the original figure.
	library.thinkquest.org /3288/ifs.html (332 words)

	Automatic Generation of Iterated Function Systems (Site not responding. Last check: 2007-10-14)
		A set of affine mappings with randomly chosen coefficients is repeatedly iterated numerically using the random iteration algorithm to produce an attractor with fractal characteristics.
		Results of evaluating 7500 iterated function systems, showing that the most visually appealing cases are those with large negative Lyapunov exponents (L) and with correlation dimensions (F) greater than one.
		Examples of iterated function systems produced by pairs of three-dimensional affine maps in which the color is determined by one of the variables.
	sprott.physics.wisc.edu /pubs/paper210.htm (195 words)

	Chaos & Fractals: Iterated Function Systems
		Fractals reproducing realistic shapes, such as mountains, clouds, or plants, can be generated by the iteration of one or more affine transformations.
		An example of an iterated function system is the fl spleenwort fern.
		What is truly amazing is that only 28 numbers are necessary to generate this infinitely complex image: four 2 x 2 transformation matrices, four 2 x 1 translational vectors, and four weighted probabilities for the transformations (each attractor).
	www.pha.jhu.edu /~ldb/seminar/ifs.html (173 words)

	iterated function systems (Site not responding. Last check: 2007-10-14)
		A case with a countable set of functions was considered by Cecilia Åkerlund-Biström.
		The question of uniqueness of the invariant probability measure and ergodicity of the process is truly challenging when some of the functions f are no longer contractions.
		Iterated function systems are widely used for generation of computer images.
	www.abo.fi /~ghognas/ifs.htm (238 words)

	Russell's Research
		The major difference between iteration theory for rational and entire functions is in the proof that, for a function f, the Julia set is the closure of the repelling periodic points.
		The iterated function system machinery is fairly new --- it was primarily developed by Hutchinson [28].
		The general theory of an iterated function system with probabilities is developed in the final section, Section 7.
	www.cfm.brown.edu /people/jackson/research.html (1560 words)

	Adrian's Randim Page (Site not responding. Last check: 2007-10-14)
		Such a function takes points in the plane (or shapes, which are sets of points) and maps them to rotated, skewed, translated versions of themselves.
		(This is the origin of the name iterated function system.) Somehow, from this sequence of random choices, the image-pattern emerges, independent of which point is started from, or what sequence of choices is made.
		Practically, iterated function systems have been applied to image compression and font generation, and theoretically they have relations to dynamical systems and neural networks.
	kamares.ucsd.edu /~arobert/ifs.html (804 words)

	Fractals & Lindenmayer Systems (L-Systems)
		The results demonstrated that this generative system (the combination of L-systems and evolutionary algorithms) produced solutions faster with higher fitness than the non-generative encoding that it was compared against.
		Other systems (such as direct encoding and using graph structure as the generative encoding) were able to produce creatures that had at most 50 components, while this system produced creatures with up to hundreds of parts that were able to behave in an interesting way.
		The fitness of a creature is a function of the distance that the creature’s centre of mass has moved.
	www.itee.uq.edu.au /~pennyd/LandFSummaries.htm?print=1 (3297 words)

	Iterated Function Systems: David Arnold
		The chaos game is actually a particular example of a more general class of processes called iterated functions systems.
		Iterated functions systems follow the same basic rules as the chaos game, but allow you more freedom in selecting and applying your transformations.
		If each transformation is equally likely to be selected at any iteration (each transformation has probability 1/5, or 0.2, of being selected), write a program that produces the image generated by this iterated function system.
	online.redwoods.cc.ca.us /instruct/darnold/ifs (2351 words)

	[No title] (Site not responding. Last check: 2007-10-14)
		The pattern of the brush at a specific moment and where this pattern is displayed will be determined by the output of the difference equations along with options chosen by the user, such as color schemes, sharpness, position of the mouse, and size of the output.
		System Requirements Specification Update Menu Inputs: The inputs to the iterated function systems are updated when the user clicks the Start Painting option in the menu area.
		Action: The function calls the menu option functions and uses the returned values to assign values to the global variables used by the iterated function systems.
	www.cise.ufl.edu /~mrs/cis4930/srs.doc (2973 words)

	Creating Fractals using Iterated Function Systems
		In order to understand what iterated function systems are and why the random iteration algorithm works, we need to be familiar with some mathematical concepts.
		The random iteration algorithm is faster for one iteration, but it needs a lot more iterations for generating the same picture.
		An other property of the random iteration algorithm that might be undesirable in some cases is that every time we run the algorithm, we will never quite get the same picture twice.
	www.cs.mcgill.ca /~ndutil/paper.html (2786 words)

	StdMap by J.D. Meiss
		IFS is a Macintosh application for drawing the fractals created by iterated function systems.
		Iterated function systems are dynamical systems defined by a set of n mappings f
		We use the basic "srand" function built into C for the pseudo-random number generator, so don’t expect sophisticated statistical properties.
	amath.colorado.edu /faculty/jdm/ifs.html (846 words)

	Iterated Function Systems
		One of the more common, and more general, ways to generate fractals is through Iterated Function Systems (IFSs).
		Def: An (hyperbolic) iterated function system is a metric space (X,d) together with a finite set of contraction mappings on that space.
		That is, an iteration consists of applying each transformation to the current set, and then taking the union of the resulting sets.
	www.geom.uiuc.edu /java/IFSoft/IFSs (669 words)

	Nonlinear Iterated Function Systems (Site not responding. Last check: 2007-10-14)
		Nonlinear functions are defined (in analogy to the above approach for affine functions) by grid distortions.
		Above an interactive program for the specification of 2D linear and nonlinear Iterated Function Systems is shown.
		An LR-table controls the application of individual functions during the "stochastic algorithm" that is used for fast visualization of the fractal limit set.
	www.cg.tuwien.ac.at /research/vis-dyn-syst/nifs/nlinifs.html (361 words)

	Iterated Function Systems, Technical Description (Site not responding. Last check: 2007-10-14)
		In fact it is possible to define a METRIC (distance function) on sets and consider a space of sets analogous to the plane as a set of points (a space is a set of elements together with a distance function telling you how far apart any two elements are).
		Just as one calculates the distance between two points in the plane using the formula [(x1-x2)^2 + (y1-y2)^2]^1/2, there is a method for telling how far apart two sets of points are -- which in fact roughly agrees with their similarity in shape.
		This result will in fact simply be another set (consisting of the union of all the possible resulting points), and so the entire system of transformations (henceforth iterated function system, or IFS) may be thought of as a single function mapping points (i.e.
	kamares.ucsd.edu /~arobert/ifstech.html (1042 words)

	Iterated Function System
		The Sierpinski Gasket is a shape which has been generated by an Iterated Function System.
		An Iterated Function System (IFS) is a recursive algorithm for generating a sequence of points, usually in two dimensions.
		We are using Iterated Function Systems so that you will produce some cool graphics.
	www.mtsu.edu /~csjudy/4250/labs/ifs.html (361 words)

	Parabolic Iterated Function Systems - Mauldin, Urba'nski (ResearchIndex) (Site not responding. Last check: 2007-10-14)
		With every parabolic system we associate an infinite hyperbolic conformal iterated function system and we employ it to study geometric and dynamical features (properly defined invariant measures for example) of the limit set.
		10: Conformal iterated function systems with applications to the geometry of continu..
		36 Dimensions and measures in infinite iterated function system..
	citeseer.ist.psu.edu /244819.html (558 words)

	Iterated Function Systems
		For practical image compression, a number of restrictions on the mapping functions are imposed.
		For the next iteration, the new range image then becomes the domain image and the process is repeated.
		The range image converges to what is referred to as the ``attractor image'' as the number of iterations increases.
	www.uoguelph.ca /~rdony/ccece98/paper-node3.html (443 words)

	Iterated Function Systems (Site not responding. Last check: 2007-10-14)
		However, it is not an Iterated Function System yet.
		An initial image is transformed by a set of affine transformations (functions) producing a new image.
		After infinitely many iterations, assuming a contractive transformation, the image will converge to what is called an attractor.
	www.cs.wisc.edu /~ergreen/honors_thesis/IFS.html (541 words)

	Mathematica Notebooks for Iterated Function Systems
		The backward iteration algorithm helps you draw the Julia sets for different kinds of functions.
		I started working with Iterated Function Systems back in 1998 in Argentina under the direction of Professor Horacio A. Caruso.
		He encouraged me to continue studying chaos and fractals and he was an example to me as a person and researcher that I will never forget.
	math.bu.edu /people/smarotta/IFS.htm (515 words)

	IFS Music
		Let me introduce the notion of Iterated Function Systems with a brief mathematical essay. This essay will consider the iterates of sets in R2 for several functions used simultaneously. Such sets are known as iterated function systems.
		First, notice that an expansion function is one to one since it maps each t in the interval [0, end of piece] to a single note.
		iteration of the Koch curve could be a somewhat more complex waveform from which to build a sound than the standard sawtooth or sine waves.
	www.student.math.uwaterloo.ca /~tsherk (775 words)

	Quantification of Determinism in Music using Iterated Function Systems (Site not responding. Last check: 2007-10-14)
		A written score of music is modeled as a dynamical system employing an iterated function system to generate a picture from the music.
		A written score of music cannot be modeled as a dynamical system without certain simplifications.
		Two examples of Iterated Function Systems: (a) with 4 possible input values; (b) using pitch classes as input values.
	sprott.physics.wisc.edu /pubs/paper215.htm (229 words)

	Fractal Questions and Answers - Iterated function systems and compression (Site not responding. Last check: 2007-10-14)
		The basic idea behind fractal image compression is to express the image as an iterated function system (IFS).
		Barnsley's company, Iterated Systems Inc, has a line of products including a Windows viewer, compressor, magnifier program, and hardware assist board.
		Algorithms are illustrated using source code in C. The October 1993 issue of Byte discussed fractal compression.
	www.softlab.ntua.gr /miscellaneous/faq/fractal/faq-doc-11.html (553 words)

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