Domain Theory in Stochastic Processes (original) (raw)
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A Probabilistic Power Domain Algorithm for Fractal Image Decoding
Stochastics and Dynamics, 2002
A new algorithm, called herein the random power domain algorithm, is discussed; it generates the image corresponding to an iterated function system with probabilities, a technique used in fractal image decoding. A simple complexity analysis for the algorithm is also derived.
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 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 nite 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.
A Fractal Valued Random Iteration Algorithm and Fractal Hierarchy
Fractals, 2005
We describe new families of random fractals, referred to as "Vvariable", which are intermediate between the notions of deterministic and of standard random fractals. The parameter V describes the degree of "variability": at each magnification level any V -variable fractals has at most V key "forms" or "shapes". V -variable random fractals have the surprising property that they can be computed using a forward process. More precisely, a version of the usual Random Iteration Algorithm, operating on sets (or measures) rather than points, can be used to sample each family. To present this theory, we review relevant results on fractals (and fractal measures), both deterministic and random. Then our new results are obtained by constructing an iterated function system (a super IFS) from a collection of standard IFSs together with a corresponding set of probabilities. The attractor of the super IFS is called a superfractal; it is a collection of V -variable random fractals (sets or measures) together with an associated probability distribution on this collection. When the underlying space is for example R 2 , and the transformations are computationally straightforward (such as affine transformations), the superfractal can be sampled by means of the algorithm, which is highly efficient in terms of memory usage. The algorithm is illustrated by some computed examples. Some variants, special cases, generalizations of the framework, and potential applications are mentioned.
A New Probabilistic Approach for Fractal Based Image Compression
Fundamenta …, 2008
Approximation of an image by the attractor evolved through iterations of a set of contractive maps is usually known as fractal image compression. The set of maps is called iterated function system (IFS). Several algorithms, with different motivations, have been suggested towards the solution of this problem. But, so far, the theory of IFS with probabilities, in the context of image compression, has not been explored much. In the present article we have proposed a new technique of fractal image compression using the theory of IFS and probabilities. In our proposed algorithm, we have used a multiscaling division of the given image up to a predetermined level or up to that level at which no further division is required. At each level, the maps and the corresponding probabilities are computed using the gray value information contained in that image level and in the image level higher to that level. A fine tuning of the algorithm is still to be done. But, the most interesting part of the proposed technique is its extreme fastness in image encoding. It can be looked upon as one of the solutions to the problem of huge computational cost for obtaining fractal code of images.
A note on invariance principles for iterated random functions
Journal of Applied Probability, 2003
In this letter, we refer to the papers of Benda (1998) and Wu and Woodroofe (2000). In each paper, a central limit theorem is proved, one for contractive stochastic dynamical systems and the other for iterated random functions, which amount to the same mathematical model. Wu and Woodroofe show that, by slightly strengthening the moment condition on the function g appearing in Benda's central limit theorem, the continuity condition on g can be relaxed essentially. So, for example, the indicator functions of balls are allowed as g. Moreover, using work by Durrett and Resnick (1978), they prove an invariance principle for the central limit theorem. We show that, using work by Heyde and Scott (1973) and Scott (1973), the central results of Benda (1998) and Wu and Woodroofe (2000) can be easily derived. Moreover, we show that, along with an invariance principle for the central limit theorem, such a principle also holds for the law of the iterated logarithm. To illustrate our results, we show that they can be applied to autoregressive processes with an ARCH(1)-noise sequence.
Fractal image compression and the inverse problem of recurrent iterated function systems
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 bi-level images better than current fractal image compression techniques.
Attainable densities for random maps
Journal of Mathematical Analysis and Applications, 2006
We consider a random map T = T (Γ,ω) , where Γ = (τ 1 , τ 2 ,. .. , τ K) is a collection of maps of an interval and ω = (p 1 , p 2 ,. .. , p K) is a collection of the corresponding position dependent probabilities, that is, p k (x) 0 for k = 1, 2,. .. , K and K k=1 p k (x) = 1. At each step, the random map T moves the point x to τ k (x) with probability p k (x). For a fixed collection of maps Γ , T can have many different invariant probability density functions, depending on the choice of the (weighting) probabilities ω. Most of the results in this paper concern random maps where Γ is a family of piecewise linear semi-Markov maps. We investigate properties of the set of invariant probability density functions of T that are attainable by allowing the probabilities in ω to vary in a certain class of functions. We prove that the set of all attainable densities can be determined algorithmically. We also study the duality between random maps generated by transformations and random maps constructed from a collection of their inverse branches. Such representation may be of greater interest in view of new methods of computing entropy [W.
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Stochastic models for fractal processes
Journal of Statistical Planning and Inference, 1999
This paper considers the situation where a stochastic process may display both long-range dependence (LRD) and intermittency. The existence of such a process is established in Anh et al. (1999). Existing works have commonly paid attention either to LRD or intermittency quite separately. This paper o ers a convenient framework to study both e ects simultaneously. A method is given to estimate and separate the two e ects. The wavelet theory plays an essential role in this procedure. Numerical experiments on fractional Brownian motion and multiplicative cascade processes conÿrm the power of the method.