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90
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...
A WaveletBased Analysis of Fractal Image Compression
 IEEE Trans. Image Processing
, 1997
"... Why does fractal image compression work? What is the implicit image model underlying fractal block coding? How can we characterize the types of images for which fractal block coders will work well? These are the central issues we address. We introduce a new waveletbased framework for analyzing block ..."
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Cited by 42 (2 self)
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Why does fractal image compression work? What is the implicit image model underlying fractal block coding? How can we characterize the types of images for which fractal block coders will work well? These are the central issues we address. We introduce a new waveletbased framework for analyzing blockbased fractal compression schemes. Within this framework we are able to draw upon insights from the wellestablished transform coder paradigm in order to address the issue of why fractal block coders work. We show that fractal block coders of the form introduced by Jacquin[1] are a Haar wavelet subtree quantization scheme. We examine a generalization of this scheme to smooth wavelets with additional vanishing moments. The performance of our generalized coder is comparable to the best results in the literature for a Jacquinstyle coding scheme. Our wavelet framework gives new insight into the convergence properties of fractal block coders, and leads us to develop an unconditionally convergen...
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.
Solving the Inverse Problem for Function and Image Approximation Using Iterated Function Systems
, 1994
"... This paper is concerned with function approximation and image representation using a new formulation of Iterated Function Systems (IFS) over the general function spaces L p (X; ¯): An Nmap IFS with grey level maps (IFSM), to be denoted as (w; \Phi), is a set w of N contraction maps w i : X ! X o ..."
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Cited by 29 (10 self)
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This paper is concerned with function approximation and image representation using a new formulation of Iterated Function Systems (IFS) over the general function spaces L p (X; ¯): An Nmap IFS with grey level maps (IFSM), to be denoted as (w; \Phi), is a set w of N contraction maps w i : X ! X over a compact metric space (X; d) (the "base space") with an associated set \Phi of maps OE i : R ! R. Associated with each IFSM is an operator T which, under certain conditions, may be contractive with unique fixed point u 2 L p (X; ¯). A rigorous solution to the following inverse problem is provided: Given a target v 2 L p (X; ¯) and an ffl ? 0, find an IFSM whose attractor satisfies k u \Gamma v k p ! ffl. An algorithm for the construction of IFSM approximations of arbitary accuracy to a target set in L 2 (X; ¯), where X ae R D and ¯ = m (D) (Lebesgue measure), is also given. The IFSM formulation can easily be generalized to include the "local IFSM" (LIFSM) which considers the...
Using Fractal Compression Scheme to Embed a Digital Signature Into an Image
, 1996
"... With the increase in the number of digital networks and recording devices, digital images appear to be a material, especially still images, whose ownership is widely threatened due to the availability of simple, rapid and perfect duplication and distribution means. It is in this context that several ..."
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Cited by 27 (0 self)
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With the increase in the number of digital networks and recording devices, digital images appear to be a material, especially still images, whose ownership is widely threatened due to the availability of simple, rapid and perfect duplication and distribution means. It is in this context that several European projects are devoted to finding a technical solution which, as it applies to still images, introduces a code or Watermark into the image data itself. This Watermark should not only allow one to determine the owner of the image, but also respect its quality and be difficult to remove. An additional requirement is that the code should be retrievable by the only mean of the protected information. In this paper, we propose a new scheme based on fractal coding and decoding. In general terms, a fractal coder exploits the spatial redundancy within the image by establishing a relationship between its different parts. We describe a way to use this relationship as a means of embedding a Watermark. Tests have been performed in order to measure the robustness of the technique against JPEG conversion and low pass filtering. In both cases, very promising results have been obtained.
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 ...
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
Uniqueness Of Invariant Measures For PlaceDependent Random Iterations
 IMA Vol. Math. Appl
, 2002
"... . We give a survey of some results within the convergence theory for iterated random functions with an emphasis on the question of uniqueness of invariant probability measures for placedependent random iterations with finitely many maps. Some problems for future research are pointed out. 1. ..."
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Cited by 8 (1 self)
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. We give a survey of some results within the convergence theory for iterated random functions with an emphasis on the question of uniqueness of invariant probability measures for placedependent random iterations with finitely many maps. Some problems for future research are pointed out. 1.
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 7 (1 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