Showing papers on "Iterated function system published in 2011"
TL;DR: The main theorem of as discussed by the authors states that the chaos game algorithm almost surely yields the attractor of an iterated function system, and the theorem holds in a very general setting, even for non-contractive iterated functions.
Abstract: The main theorem of this paper establishes conditions under which the ‘chaos game’ algorithm almost surely yields the attractor of an iterated function system. The theorem holds in a very general setting, even for non-contractive iterated function systems, and under weaker conditions on the random orbit of the chaos game than obtained previously.
75 citations
TL;DR: In this article, the authors consider a class of planar self-affine sets which they call "box-like" and compute the packing and box-counting dimensions by means of a pressure type formula based on the singular values of the maps.
Abstract: We consider a class of planar self-affine sets which we call "box-like". A box-like self-affine set is the attractor of an iterated function system (IFS) of affine maps where the image of the unit square, [0,1]^2, under arbitrary compositions of the maps is a rectangle with sides parallel to the axes. This class contains the Bedford-McMullen carpets and the generalisations thereof considered by Lalley-Gatzouras, Bara\'nski and Feng-Wang as well as many other sets. In particular, we allow the mappings in the IFS to have non-trivial rotational and reflectional components. Assuming a rectangular open set condition, we compute the packing and box-counting dimensions by means of a pressure type formula based on the singular values of the maps.
43 citations
TL;DR: The notion of uniform scaling scenery as discussed by the authors is related to existing notions of self-similarity: it is satisfied by the occupation measure of a typical Brownian motion, as well as by the measures associated to attractors of affine iterated function systems.
Abstract: We introduce a property of measures on Euclidean space, termed ‘uniform scaling scenery’. For these measures, the empirical distribution of the measure-valued time series, obtained by rescaling around a point, is (almost everywhere) independent of the point. This property is related to existing notions of self-similarity: it is satisfied by the occupation measure of a typical Brownian motion (which is ‘statistically’ self-similar), as well as by the measures associated to attractors of affine iterated function systems (which are ‘exactly’ self-similar). It is possible that different notions of self-similarity are unified under this property. The proofs trace a connection between uniform scaling scenery and Furstenberg’s ‘CP processes’, a class of natural, discrete-time, measure-valued Markov processes, useful in fractal geometry.
30 citations
TL;DR: It is shown that fractal features obtained from Iterated Function System allow a successful face recognition and outperform the classical approaches and a new fractal feature extraction algorithm based on genetic algorithms to speed up the feature extraction step is proposed.
Abstract: In this article, we present an automatic face recognition system. We show that fractal features obtained from Iterated Function System allow a successful face recognition and outperform the classical approaches. We propose a new fractal feature extraction algorithm based on genetic algorithms to speed up the feature extraction step. In order to capture the more important information that is contained in a face with a few fractal features, we use a bi-dimensional principal component analysis. We have shown with experimental results using two databases as to how the optimal recognition ratio and the recognition time make our system an effective tool for automatic face recognition.
29 citations
25 Feb 2011
TL;DR: In this article, the pattern of attractors of the iterated function systems (IFS) through Ishikawa iterative scheme is studied and the results obtained are illustrated through figures generated by Matlab programs.
Abstract: Fractal geometry is an exciting area of interest with diverse applications in various disciplines of engineering and applied sciences. There is a plethora of papers on its versatility in the literature. The basic aim of this paper is to study the pattern of attractors of the iterated function systems (IFS) through Ishikawa iterative scheme. Several recent results are also obtained as special cases. The results obtained are illustrated through figures generated by Matlab programs.
27 citations
TL;DR: In this paper, the spectral dimension of a class of one-dimensional self-similar measures defined by iterated function systems with overlaps and satisfying a family of second-order selfsimilar identities is computed.
Abstract: We set up a framework for computing the spectral dimension of a class of one-dimensional self-similar measures that are defined by iterated function systems with overlaps and satisfy a family of second-order self-similar identities. As applications of our result we obtain the spectral dimension of important measures such as the infinite Bernoulli convolution associated with the golden ratio and convolutions of Cantor-type measures. The main novelty of our result is that the iterated function systems we consider are not post-critically finite and do not satisfy the well-known open set condition.
25 citations
TL;DR: In this paper, a necessary and sufficient condition for a fractal transformation from the attractor of one overlapping IFS to the anchor of another IFS is found. And the topological entropy of the dynamical system associated with an overlapping function is derived.
Abstract: The term "overlapping" refers to a certain fairly simple type of piecewise continuous function from the unit interval to itself and also to a fairly simple type of iterated function system (IFS) on the unit interval. A correspondence between these two classes of objects is used (1) to find a necessary and sufficient condition for a fractal transformation from the attractor of one overlapping IFS to the attractor of another overlapping IFS to be a homeomorphism and (2) to find a formula for the topological entropy of the dynamical system associated with an overlapping function.
24 citations
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TL;DR: The mathematics of fractal transformations is generalized and it is illustrated how it leads to a new approach to the representation and processing of digital images, and consequent novel methods for filtering, watermarking, and encryption.
Abstract: We present a general theory of fractal transformations and show how it leads to a new type of method for filtering and transforming digital images. This work substantially generalizes earlier work on fractal tops. The approach involves fractal geometry, chaotic dynamics, and an interplay between discrete and continuous representations. The underlying mathematics is established and applications to digital imaging are described and exemplified.
23 citations
TL;DR: In this paper, the authors generalize the mathematics of fractal transformations and illustrate how it leads to a new approach to the representation and processing of digital images, and consequent novel methods for filtering, watermarking, and encryption.
Abstract: We generalize the mathematics of fractal transformations and illustrate how it leads to a new approach to the representation and processing of digital images, and consequent novel methods for filtering, watermarking, and encryption. This work substantially generalizes earlier work on fractal tops. The approach involves fractal geometry, chaotic dynamics, and an interplay between discrete and continuous representations. The underlying mathematics is established and some applications to digital imaging are described and exemplified.
23 citations
TL;DR: In this paper, the authors investigate multifractal regularity for infinite conformal iterated function systems (cIFS) and obtain an Exhausting Principle for infinite cIFS allowing them to carry over results for finite to infinite systems.
Abstract: We investigate multifractal regularity for infinite conformal iterated function systems (cIFS). That is we determine to what extent the multifractal spectrum depends continuously on the cIFS and its thermodynamic potential. For this we introduce the no- tion of regular convergence for families of cIFS not necessarily sharing the same index set, which guarantees the convergence of the multifractal spectra on the interior of their domain. In particular, we obtain an Exhausting Principle for infinite cIFS allowing us to carry over results for finite to infinite systems, and in this way to establish a multifractal analysis without the usual regularity conditions. Finally, we discuss the connections to the λ-topology introduced by Roy and Urba´ nski.
19 citations
TL;DR: In this paper, the authors employed the self-similarity properties of the power-line communications (PLC) orthogonal frequency-division multiplexing (OFDM) signals, and proposed a new signal reconstruction solution based on the fractals.
Abstract: By employing the self-similarity properties of the power-line communications (PLC) orthogonal frequency-division multiplexing (OFDM) signals, this paper presents a novel PLC OFDM signal reconstruction solution based on the fractals. This new signal reconstruction solution consists of two methods that try to solve two main difficulties faced by the fractals interpolation. The first method is based on the affine transform, trying to determine the vertical scaling factors by a set of the middle points in each affine mapping, and solving the difficulty of determining the vertical scaling factors, which is one of the most difficult challenges faced by the fractal interpolation. The second method tries to solve the problem where the ordinary fractals interpolation cannot obtain the value of any arbitrary point directly, a very novel algorithm called the pointed point algorithm, based on iterated function systems, is proposed to reconstruct the PLC OFDM signals with the expected interpolation error. Numerical experiments have shown that great accuracy in the reconstruction of the OFDM signals has been found, leading to a significant improvement over other signal reconstruction methods.
TL;DR: In this paper, a generalized countable iterated function system (GCIFS) is proposed to generate fractals by considering contractions from X £ X into X instead of contractions on the metric space X to itself, where (X;d) is a compact metric space.
Abstract: One of the most common and most general way to generate fractals is by using iterated function systems which consists of a flnite or inflnitely many maps. Generalized countable iterated function systems (GCIFS) are a generalization of countable iterated function systems by considering contractions from X £ X into X instead of contractions on the metric space X to itself, where (X;d) is a compact metric space. If all contractions of a GCIFS are Lipschitz with respect to a parameter and the supremum of the Lipschitz constants is flnite, then the associated attractor depends continuously on the respective parameter.
TL;DR: In this article, it was shown that any fractal measure μ obtained from an affine iterated function system possesses a sequence of complex exponentials which forms a Riesz basic sequence, or more generally a Bessel sequence, in L 2(μ) such that the frequencies have positive Beurling dimension.
TL;DR: For a homogeneous symmetric Cantor set C, this article considered all real numbers t such that the intersection C ∩(C + t) is a self-similar set and investigated the form of the corresponding iterated function systems.
Abstract: For a homogeneous symmetric Cantor set C, we consider all real numbers tsuch that the intersection C∩(C + t)is a self-similar set and investigate the form of the corresponding iterated function systems. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
TL;DR: In this article, the definition of a generalized continuous frame for a separable Hilbert space was given, and a class of generalized continuous frames was constructed by using an iterated function system.
06 Jan 2011
TL;DR: In this article, the authors give a new approach to the study of conformal iterated function systems with arbitrary overlaps and provide lower and upper estimates for the Hausdorff dimension of the limit sets of such systems, expressed in terms of the topological pressure and the function d, counting overlaps.
Abstract: We give a new approach to the study of conformal iterated function systems with arbitrary overlaps. We provide lower and upper estimates for the Hausdorff dimension of the limit sets of such systems; these are expressed in terms of the topological pressure and the function d, counting overlaps. In the case when the function d is constant, we get an exact formula for the Hausdorff dimension. We also prove that in certain cases this formula holds if and only if the function d is constant.
TL;DR: In this paper, the authors consider limit sets of conformal iterated function systems, and introduce classes of subsets of these limit sets, with the property that the classes are closed under countable intersections and that all sets in the classes have a large Hausdorff dimension.
Abstract: We consider limit sets of conformal iterated function systems, and introduce classes of subsets of these limit sets, with the property that the classes are closed under countable intersections and that all sets in the classes have a large Hausdorff dimension. Using these classes we determine the Hausdorff dimension and large intersection properties of some sets occurring in ergodic theory, Diophantine approximation and complex dynamics.
TL;DR: In this paper, the connectivity of the Sierpinski family of fractals is studied and conditions for the construction of closed paths in the case of multiply-connected relatives are presented.
Abstract: This paper presents a study of the connectivity of the class of fractals known as the Sierpinski relatives. These fractals all have the same fractal dimension, but different topologies. Some are totally disconnected, some are disconnected but with paths, some are simply-connected, and some are multiply-connected. Conditions for these four cases are presented. Constructions of paths, including non-contractible closed paths in the case of multiply-connected relatives, are presented. Examples of specific relatives are provided to illustrate the four cases.
TL;DR: The role of superior iterative procedure to find the orbital picture under an IFS consisting of non-contractive or non-expansive transformations is introduced.
Abstract: An orbital picture is a mathematical structure depicting the path of an object under Iterated Function System. Orbital and V-variable orbital pictures initially developed by Barnsley (2006) have utmost importance in computer graphics, image compression, biological modeling and other areas of fractal geometry. These pictures have been generated for linear and contractive transformations using function and superior iterative procedures. In this paper, the authors introduce the role of superior iterative procedure to find the orbital picture under an IFS consisting of non-contractive or non-expansive transformations. A mild comparison of the computed figures indicates the usefulness of study in computational mathematics and fractal image processing. A modified algorithm along with program code is given to compute a 2-variable superior orbital picture.
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TL;DR: Fractal homeomorphisms between the attractors of two bi-affine iterated function systems have been constructed in this paper, where conditions are provided under which a Bi-Affine Iterated Function System is contractive, thus guaranteeing an attractor.
Abstract: The paper concerns fractal homeomorphism between the attractors of two bi-affine iterated function systems. After a general discussion of bi-affine functions, conditions are provided under which a bi-affine iterated function system is contractive, thus guaranteeing an attractor. After a general discussion of fractal homeomorphism, fractal homeomorphisms are constructed for a specific type of bi-affine iterated function system.
TL;DR: In this article, the authors describe some basic results for Quantum Stochastic Processes and present some new results about a certain class of processes which are associated to Quantum Iterated Function Systems (QIFS).
Abstract: We describe some basic results for Quantum Stochastic Processes and present some new results about a certain class of processes which are associated to Quantum Iterated Function Systems (QIFS). We discuss questions related to the Markov property and we present a de nition of entropy which is induced by a QIFS. This definition is a natural generalization of the Shannon-Kolmogorov entropy from Ergodic Theory.
TL;DR: In this paper, a new methodology allowing to generate a vegetal set by a fractal approach is proposed, which is applied and validated on fifteen deciduous tree and two of them are fully developed: Quercus prinus and Populous tremuloides.
TL;DR: This paper introduces a new fractal recognition method based on a dependence graph obtained from the partitioned iterated function system with division that improves the recognition rate of fractal shapes.
Abstract: One of the approaches in pattern recognition is the use of fractal geometry. The property of self-similarity of fractals has been used as a feature in several pattern recognition methods. All fractal recognition methods use global analysis of the shape. In this paper we present some drawbacks of these methods and propose fractal local analysis using partitioned iterated function systems with division. Moreover, we introduce a new fractal recognition method based on a dependence graph obtained from the partitioned iterated function system. The proposed method uses local analysis of the shape, which improves the recognition rate. The effectiveness of our method is shown on two test databases. The first one was created by the authors and the second one is the MPEG7 CE-Shape-1 PartB database. The obtained results show that the proposed methodology has led to a significant improvement in the recognition rate.
26 Aug 2011
TL;DR: The fuzzy contraction properties of the Hutchinson-Barnsley operator on the fuzzy hyperspace with respect to the Hausdorff fuzzy metrics are presented and theorems generalize and extend some recent results related with Hutchinson- Barnesley operator in the metric spaces.
Abstract: The purpose of this paper is to present the fuzzy contraction properties of the Hutchinson-Barnsley operator on the fuzzy hyperspace with respect to the Hausdorff fuzzy metrics. Also we discuss about the relationships between the Hausdorff fuzzy metrics on the fuzzy hyperspaces. Our theorems generalize and extend some recent results related with Hutchinson-Barnsley operator in the metric spaces. Keywords—Fractals, Iterated Function System, HutchinsonBarnsley Operator, Fuzzy Metric Space, Hausdorff Fuzzy Metric.
01 Jan 2011
29 Sep 2011
TL;DR: By calculating local attractor's coarse convex-hull and selecting rotative matching between IFSs, a new IFS correspondence method is designed and a group of matching is found, and maximum similar matching is selected as correspondence of I FSs.
Abstract: Morphing IFS fractal is a significant way to create new fractal mode ls. Many works have exploited this issue for various purposes, but without considering the appearance of fractal feature degeneration during the morphing procedure between two IFSs. In this paper, by calculating local attractor's coarse convex-hull and selecting rotative matching between IFSs, we design a new IFS correspondence method. Based on this method, a group of matching is found, and maximum similar matching is selected as correspondence of IFSs. Finally we linearly interpolate the parameters of the iterated function to finish the morphing procedure of two IFS's fractal attractors and perform the fractal morphing with fractal feature preserved.
TL;DR: In this paper, the eigenvalue problem of a linear function L is generalized to an iterated function system F consisting of possibly an infinite number of linear or affine functions, and the main result is that an irreducible, linear iterated F has a unique eigen value λ equal to the joint spectral radius of the functions in F and a corresponding eigenset S that is centrally symmetric, star-shaped, and full dimensional.
TL;DR: This paper investigates the use of iterated function system (IFS) models for data analysis and detects the sequence of regime switches under the assumption that each map is continuous.
Abstract: We investigate the use of iterated function system (IFS) models for data analysis. An IFS is a discrete dynamical system in which each time step corresponds to the application of one of a finite collection of maps. The maps, which represent distinct dynamical regimes, may act in some pre-determined sequence or may be applied in random order. An algorithm is developed to detect the sequence of regime switches under the assumption of continuity. This method is tested on a simple IFS and applied to an experimental computer performance data set. This methodology has a wide range of potential uses: from change-point detection in time-series data to the field of digital communications.