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19
Dynamical systems, Measures and Fractals via Domain Theory
 Information and Computation
, 1995
"... We introduce domain theory in dynamical systems, iterated function systems (fractals) and measure theory. For a discrete dynamical system given by the action of a continuous map f:X X on a metric space X, we study the extended dynamical systems (l/X,l/f), (UX, U f) and (LX, Lf) where 1/, U and L ar ..."
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Cited by 69 (19 self)
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We introduce domain theory in dynamical systems, iterated function systems (fractals) and measure theory. For a discrete dynamical system given by the action of a continuous map f:X X on a metric space X, we study the extended dynamical systems (l/X,l/f), (UX, U f) and (LX, Lf) where 1/, U and L are respectively the Vietoris hyperspace, the upper hyperspace and the lower hyperspace functors. We show that if (X, f) is chaotic, then so is (UX, U f). When X is locally compact UX, is a continuous bounded complete dcpo. If X is second countable as well, then UX will be omegacontinuous and can be given an effective structure. We show how strange attractors, attractors of iterated function systems (fractals) and Julia sets are obtained effectively as fixed points of deterministic functions on UX or fixed points of nondeterministic functions on CUX where C is the convex (Plotkin) power domain. We also show that the set, M(X), of finite Borel measures on X can be embedded in PUX, where P is the probabilistic power domain. This provides an effective framework for measure theory. We then prove that the invariant measure of an hyperbolic iterated function system with probabilities can be obtained as the unique fixed point of an associated continuous function on PUX.
Domains for Computation in Mathematics, Physics and Exact Real Arithmetic
 Bulletin of Symbolic Logic
, 1997
"... We present a survey of the recent applications of continuous domains for providing simple computational models for classical spaces in mathematics including the real line, countably based locally compact spaces, complete separable metric spaces, separable Banach spaces and spaces of probability dist ..."
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Cited by 48 (10 self)
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We present a survey of the recent applications of continuous domains for providing simple computational models for classical spaces in mathematics including the real line, countably based locally compact spaces, complete separable metric spaces, separable Banach spaces and spaces of probability distributions. It is shown how these models have a logical and effective presentation and how they are used to give a computational framework in several areas in mathematics and physics. These include fractal geometry, where new results on existence and uniqueness of attractors and invariant distributions have been obtained, measure and integration theory, where a generalization of the Riemann theory of integration has been developed, and real arithmetic, where a feasible setting for exact computer arithmetic has been formulated. We give a number of algorithms for computation in the theory of iterated function systems with applications in statistical physics and in period doubling route to chao...
Image Coding By Block Prediction Of Multiresolution Subimages
 IEEE Transactions on Image Processing
"... The redundancy of the multiresolution representation has been clearly demonstrated in the case of fractal images, but it has not been fully recognized and exploited for general images. Recently, fractal block coders have exploited the selfsimilarity among blocks in images. In this work we devise ..."
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Cited by 33 (2 self)
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The redundancy of the multiresolution representation has been clearly demonstrated in the case of fractal images, but it has not been fully recognized and exploited for general images. Recently, fractal block coders have exploited the selfsimilarity among blocks in images. In this work we devise an image coder in which the causal similarity among blocks of different subbands in a multiresolution decomposition of the image is exploited. In a pyramid subband decomposition, the image is decomposed into a set of subbands which are localized in scale, orientation and space. The proposed coding scheme consists of predicting blocks in one subimage from blocks in lower resolution subbands with the same orientation. Although our prediction maps are of the same kind of those used in fractal block coders, which are based on an iterative mapping scheme, our coding technique does not impose any contractivity constraint on the block maps. This makes the decoding procedure very simple and...
The Object Instancing Paradigm for Linear Fractal Modeling
 IN PROC. OF GRAPHICS INTERFACE
, 1992
"... The recurrent iterated function system and the Lsystem are two powerful linear fractal models. The main drawback of recurrent iterated function systems is a difficulty in modeling whereas the main drawback of Lsystems is inefficient geometry specification. Iterative and recursive structures ext ..."
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Cited by 32 (4 self)
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The recurrent iterated function system and the Lsystem are two powerful linear fractal models. The main drawback of recurrent iterated function systems is a difficulty in modeling whereas the main drawback of Lsystems is inefficient geometry specification. Iterative and recursive structures extend the object instancing paradigm, allowing it to model linear fractals. Instancing models render faster and are more intuitive to the computer graphics community. A preliminary section briefly introduces the object instancing paradigm and illustrates its ability to model linear fractals. Two main sections summarize recurrent iterated function systems and Lsystems, and provide methods with examples for converting such models to the object instancing paradigm. Finally, a short epilogue describes a particular use of color in the instancing paradigm and the conclusion outlines directions for further research.
Power domains and iterated function systems
 Information and Computation
, 1996
"... We introduce the notion of weakly hyperbolic iterated function system (IFS) on a compact metric space, which generalises that of hyperbolic IFS. Based on a domaintheoretic model, which uses the Plotkin power domain and the probabilistic power domain respectively, we prove the existence and uniquene ..."
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Cited by 30 (10 self)
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We introduce the notion of weakly hyperbolic iterated function system (IFS) on a compact metric space, which generalises that of hyperbolic IFS. Based on a domaintheoretic model, which uses the Plotkin power domain and the probabilistic power domain respectively, we prove the existence and uniqueness of the attractor of a weakly hyperbolic IFS and the invariant measure of a weakly hyperbolic IFS with probabilities, extending the classic results of Hutchinson for hyperbolic IFSs in this more general setting. We also present finite algorithms to obtain discrete and digitised approximations to the attractor and the invariant measure, extending the corresponding algorithms for hyperbolic IFSs. We then prove the existence and uniqueness of the invariant distribution of a weakly hyperbolic recurrent IFS and obtain an algorithm to generate the invariant distribution on the digitised screen. The generalised Riemann integral is used to provide a formula for the expected value of almost everywhere continuous functions with respect to this distribution. For hyperbolic recurrent IFSs and Lipschitz maps, one can estimate the integral up to any threshold of accuracy.] 1996 Academic Press, Inc. 1.
Fractal Views: A FractalBased Method for Controlling Information Display
 ACM Transactions on Information Systems
"... ing methods; fractals; information visualization; program display; UI theory 1. INTRODUCTION As computer systems evolve, the capability of restoring and managing information increases more and more. At the same time, computer users must view increasing amounts of information through video displays ..."
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Cited by 21 (1 self)
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ing methods; fractals; information visualization; program display; UI theory 1. INTRODUCTION As computer systems evolve, the capability of restoring and managing information increases more and more. At the same time, computer users must view increasing amounts of information through video displays which are physically limited in size. Displaying information 1 effectively is a main concern in many software applications. For example, in visual programming systems[Shu 1988], graphic representations become very complex if the number of visual elements increases. In hypertext 1 The word "information" is used as a structured set of primitive elements which is specific to each application. Author's address: 481 Minor Hall, School of Optometry, University of California, Berkeley, CA 947202020. email: koike@milo.berkeley.edu; (permanent address: Graduate School of Information Systems, University of ElectroCommunications, 151, Chofugaoka, Chofu, Tokyo 182, Japan. email: koike@cas.uec.a...
Fractal Image Compression and the Inverse Problem of Recurrent Iterated Function Systems
 Directions for Fractal Modeling in Computer Graphics. SIGGRAPH '94 Course Notes
, 1996
"... Fractal image compression currently relies on the partitioning of an image into both coarse #domain" segments and #ne #range" segments, and for each range element, determines the domain element that best transforms into the range element. Under normal circumstances, this algorithm produces a stru ..."
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Cited by 7 (3 self)
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Fractal image compression currently relies on the partitioning of an image into both coarse #domain" segments and #ne #range" segments, and for each range element, determines the domain element that best transforms into the range element. Under normal circumstances, this algorithm produces a structure equivalent to a recurrent iterated function system. This equivalence allows recent innovations to fractal image compression to be applied to the general inverse problem of recurrent iterated function systems. Additionally, the RIFS representation encodes bitmaps #bilevel images# better than current fractal image compression techniques. Keywords: bitmap, block coding, compression, fractals, imaging, recurrent iterated function system. 1 1 Introduction Fractal geometry provides a basis for modeling the in#nite detail found in nature #Mandelbrot, 1982#. Fractal methods are quite popular in the modeling of natural phenomena in computer graphics, ranging from random fractal models ...
Escapetime visualization method for languagerestricted iterated function systems
 IN PROC. OF GRAPHICS INTERFACE
, 1992
"... The escapetime method was introduced to generate images of Julia and Mandelbrot sets, then applied to visualize attractors of iterated function systems. This paper extends it further to languagerestricted iterated function systems (LRIFS's). They generalize the original definition of IFS's by prov ..."
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Cited by 6 (0 self)
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The escapetime method was introduced to generate images of Julia and Mandelbrot sets, then applied to visualize attractors of iterated function systems. This paper extends it further to languagerestricted iterated function systems (LRIFS's). They generalize the original definition of IFS's by providing means for restricting the sequences of applicable transformations. The resulting attractors include sets that cannot be generated using ordinary IFS's. The concepts of this paper are expressed using the terminology of formal languages and finite automata.
Stylized multiresolution image representation
 JOURNAL OF ELECTRONIC IMAGING 17(1), 013009 (JANâ€“MAR 2008)
, 2008
"... We integrate stylized rendering with an efficient multiresolution image representation, enabling a user to control how compression affects the aesthetic appearance of an image. We adopt a pointbased rendering approach to progressive image transmission and compression. We use a novel, adaptive far ..."
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Cited by 6 (2 self)
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We integrate stylized rendering with an efficient multiresolution image representation, enabling a user to control how compression affects the aesthetic appearance of an image. We adopt a pointbased rendering approach to progressive image transmission and compression. We use a novel, adaptive farthest point sampling algorithm to represent the image at progressive levels of detail, balancing global coverage with local precision. A progressively generated discrete Voronoi diagram forms the common foundation for our sampling and rendering framework. This framework allows us to extend traditional photorealistic methods of image reconstruction by scattered data interpolation to encompass nonphotorealistic rendering. It supports a wide variety of artistic rendering styles based on geometric subdivision or parametric procedural textures. Genetic programming enables the user to create original rendering styles through interactive evolution by aesthetic selection. We compare our results with conventional compression, and we discuss the implications of using nonphotorealistic representations for highly compressed imagery.
Advances in fractal compression in multimedia applications
, 1995
"... Fractal image compression is a promising new technology but is not without problems. Most critically, fast encoding is required for it to find wide use in multimedia applications. This is now within reach: recent methods are five orders of magnitude faster than early attempts. Beginning with the bas ..."
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Cited by 5 (0 self)
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Fractal image compression is a promising new technology but is not without problems. Most critically, fast encoding is required for it to find wide use in multimedia applications. This is now within reach: recent methods are five orders of magnitude faster than early attempts. Beginning with the basic ideas and problems, this paper explains how to accelerate fractal image compression.