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...

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-based fractal compression schemes. Within this framework we are able to draw upon insights from the well-established 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 Jacquin-style coding scheme. Our wavelet framework gives new insight into the convergence properties of fractal block coders, and leads us to develop an unconditionally convergen...

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 N-map 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...

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 domain-theoretic 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 everywh...

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.

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 "ffl-contractivity" 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 ...

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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In this paper we review some recent results concerning the approximations of distribution functions and measures on [0, 1] based on iterated function systems. The two di#erent approaches available in the literature are considered and their relation are investigated in the statistical perspective. In the second part of the paper we propose a new class of estimators for the distribution function and the related characteristic and density functions. Glivenko-Cantelli, LIL properties and local asymptotic minimax e#ciency are established for some of the proposed estimators. Via Monte Carlo analysis we show that, for small sample sizes, the proposed estimator can be as e#cient or even better than the empirical distribution function and the kernel density estimator respectively. This paper is to be considered as a first attempt in the construction of new class of estimators based on fractal objects. Pontential applications to survival analysis with random censoring are proposed at the end of the paper. 1

. 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 place-dependent random iterations with finitely many maps. Some problems for future research are pointed out. 1.