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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 50 (11 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...
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 31 (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.
Fast Hierarchical Codebook Search for Fractal Coding of Still Images
 EOS/SPIE Visual Communication and PACS for Medical Applications 93
, 1993
"... This paper presents a method for fast encoding of still images based on iterated function systems (IFSs). The major disadvantage of this coding approach, usually referred to as fractal coding, is the high computational effort of the encoding process compared to e.g. the JPEG algorithm [1]. This is m ..."
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Cited by 15 (2 self)
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This paper presents a method for fast encoding of still images based on iterated function systems (IFSs). The major disadvantage of this coding approach, usually referred to as fractal coding, is the high computational effort of the encoding process compared to e.g. the JPEG algorithm [1]. This is mainly due to the costly "full search" of the transform parameters within a fractal codebook. We therefore propose an hierarchical encoding scheme which is based upon a two level codebook search and a structural classification of its entries. By this way only a small subset of the codebook has to be considered, which increases encoding speed significantly. Refining the initial codebook and applying a second search even increases the reconstruction quality compared to the full search but with a fraction of its computational effort. 1.
On the convergence of fractal transforms
 in Proc. IEEE Int. Conf. Acoustics, Speech and Signal Processing
, 1994
"... ..."
Solving the Inverse Problem for Measures Using Iterated Function Systems: A New Approach
 Adv. Appl. Prob
, 1995
"... We present a systematic method of approximating, to an arbitrary accuracy, a probability measure ¯ on [0; 1] q ; q 1, with invariant measures for Iterated Function Systems by matching its moments. There are two novel features in our treatment: (1) An infinite number of fixed affine contraction ma ..."
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Cited by 12 (6 self)
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We present a systematic method of approximating, to an arbitrary accuracy, a probability measure ¯ on [0; 1] q ; q 1, with invariant measures for Iterated Function Systems by matching its moments. There are two novel features in our treatment: (1) An infinite number of fixed affine contraction maps on X; W = fw 1 ; w 2 ; : : :g, subject to an "fflcontractivity" condition, is employed. Thus, only an optimization over the associated probabilities p i is required. (2) We prove a Collage Theorem for Moments which reduces the moment matching problem to that of minimizing the "collage distance" between moment vectors. The minimization procedure is a standard quadratic programming problem in the p i which can be solved in a finite number of steps. Some numerical calculations for the approximation of measures on [0,1] are presented. AMS Subject Classifications: 28A, 41A, 58F 1. Introduction This paper is concerned with the approximation of probability measures on a compact metric space X ...
Fractalwavelet image denoising
 Proceedings of IEEE International Conference on Image Processing, (ICIP 2002), I836 – I839
, 2002
"... I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be electronically available to the public. ii The need for image enhancement and restoration is encounter ..."
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Cited by 10 (2 self)
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I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be electronically available to the public. ii The need for image enhancement and restoration is encountered in many practical applications. For instance, distortion due to additive white Gaussian noise (AWGN) can be caused by poor quality image acquisition, images observed in a noisy environment or noise inherent in communication channels. In this thesis, image denoising is investigated. After reviewing standard image denoising methods as applied in the spatial, frequency and wavelet domains of the noisy image, the thesis embarks on the endeavor of developing and experimenting with new image denoising methods based on fractal and wavelet transforms. In particular, three new image denoising methods are proposed: contextbased wavelet thresholding, predictive fractal image denoising and fractalwavelet image denoising. The proposed contextbased thresholding strategy adopts localized hard and soft thresholding operators which take in consideration the content of an immediate neighborhood of a wavelet coefficient before thresholding it. The two fractalbased predictive schemes are based on a simple yet effective algorithm for estimating the fractal code of the original noisefree image from the noisy one. From this predicted code, one can then reconstruct a fractally denoised estimate of the original image. This fractalbased denoising algorithm can be applied in the pixel and the wavelet domains of the noisy image using standard fractal and fractalwavelet schemes, respectively. Furthermore, the cycle spinning idea was implemented in order to enhance the quality of the fractally denoised estimates. Experimental results show that the proposed image denoising methods are competitive, or sometimes even compare favorably with the existing image denoising techniques reviewed in the thesis. This work broadens the application scope of fractal transforms, which have been used mainly for image coding and compression purposes. iii
Signal Modeling With Iterated Function Systems
, 1993
"... this memory requirement issue may become a factor, in which case the Random Iteration Algorithm could be adapted to overcome the shortcomings mentioned here with some simple checks on the path of the calculations. 2.3 Conclusion ..."
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Cited by 9 (0 self)
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this memory requirement issue may become a factor, in which case the Random Iteration Algorithm could be adapted to overcome the shortcomings mentioned here with some simple checks on the path of the calculations. 2.3 Conclusion
Individual GP: an Alternative Viewpoint for the Resolution of Complex Problems.
"... An unususal GP implementation is proposed, based on a more "economic" exploitation of the GP algorithm: the "individual" approach, where each individual of the population embodies a single function rather than a set of functions. The nal solution is then a set of individuals. Exa ..."
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Cited by 8 (3 self)
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An unususal GP implementation is proposed, based on a more "economic" exploitation of the GP algorithm: the "individual" approach, where each individual of the population embodies a single function rather than a set of functions. The nal solution is then a set of individuals. Examples are presented where results are obtained more rapidly than with the conventional approach, where all individuals of the nal generation but one are discarded.
FOURIER FREQUENCIES IN AFFINE ITERATED FUNCTION SYSTEMS
, 2006
"... We examine two questions regarding Fourier frequencies for a class of iterated function systems (IFS). These are iteration limits arising from a fixed finite families of affine and contractive mappings in R d, and the “IFS” refers to such a finite system of transformations, or functions. The itera ..."
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Cited by 8 (8 self)
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We examine two questions regarding Fourier frequencies for a class of iterated function systems (IFS). These are iteration limits arising from a fixed finite families of affine and contractive mappings in R d, and the “IFS” refers to such a finite system of transformations, or functions. The iteration limits are pairs (X, µ) where X is a compact subset of R d, (the support of µ) and the measure µ is a probability measure determined uniquely by the initial IFS mappings, and a certain strong invariance axiom. The two questions we study are: (1) existence of an orthogonal Fourier basis in the Hilbert space L²(X, µ); and (2) the interplay between the geometry of (X, µ) on the one