Krzysztof Leśniak, Nina Snigireva, Filip Strobin et al. · arXiv (Cornell University) · 2022
Iterated function systems (IFSs) and their attractors have been central to the theory of fractal geometry almost from its inception. And contractivity of the functions in the IFS has been central to the theory of iterated functions systems. If the functions in the IFS are contractions, then the IFS is guaranteed to have a unique attractor. Recently, however, there has been an interest in what occurs to the attractor at the boundary between contractvity and expansion of the IFS. That is the subject of this paper. For a family $F_t$ of IFSs depending on a real parameter $t>0$, the existence and properties of two types of transition attractors, called the lower transition attractor $A_{\bullet}$ and the upper transition attractor $A^{\bullet}$, are investigated. A main theorem states that, for a wide class of IFS families, there is a threshold $t_0$ such that the IFS $F_t$ has a unique attractor $A_t$ for $t<t_0$ and no attractor for $t>t_0$. At the threshold $t_0$, there is an $F_{t_0}$-invariant set $A^{\bullet}$ such that $A^{\bullet} = \lim_{t\rightarrow t_0} A_t$.
Cited by 5Open access
Mamoon Ahmed, D. Easwaramoorthy · International Journal of Analysis and Applications · 2025
Nowadays, most of the real time problems have been attempted by using well-known fixed point theorems. Especially the Banach contraction theorem is a well-posted tool to solve many dynamical problems of applied mathematics. This paper explores an idea in generalizing fixed point theorem to generate a proposed fractal type set called Controlled Strong b−Kannan Fractal (CSbK-Fractal) through the dynamical system of Kannan contractivity function in the Controlled Strong b−Metric Space (CSbMS). Furthermore, the collage type theorem is proved on CSbK-Fractal. In this context, the interesting results and consequences of newly developing iterated function system and its fractal attractor in the controlled strong b−metric space are discussed with examples. This theory can provide a novel direction to construct a new kind of fractal set in generalized spaces.
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Erik L.J. Bohez, Tissa Senevirathne, J.A. van Winden · Electronics Letters · 1992
A new phoneme recognition method is presented which is based on fractal theory. The fractal dimension of the associated waveform and the IFS parameters that represent the waveform are used as the parameters for recognition.
Cited by 9
Amol Potgantwar, Dr.S.G. Bhirud · International Journal of Computer Applications · 2010
Many conventional methods of face recognition depend solely on appearance and model based. However there is inbuilt degree of self-similarity in the image of faces, which can be efficiently utilized through representation exploiting self-transformations, known as Iterated Function System (IFS).
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Mamta Rani, Renu Chugh · 2014
The classical Cantor set discovered and introduced by the famous mathematicians Henery Smith and George Cantor has m any interesting properties in the field of set theory, Topology and fractal geometry. One of the way in which the behavior of Cantor sets can be spe cified is through the use of Iterated Function System. In (2008), Gerald Edgar i n his book gave a systematic study of classical Cantor ternary set in iterated f unction system and introduced some beautiful properties. Our goal in this paper ipresent two new examples of Cantor variants using iterated function system.
Cited by 1
Jonathan Rice · Canada Human-Computer Communications Society · 1996
An algorithm is presented which, given the parameters of an Iterated Function System (IFS) which uses affine maps, constructs a closed ball which completely contains the attractor set of the IFS. These bounding balls are almost always smaller than those computed by existing methods, and are sometimes much smaller. The algorithm is numerical in form, involving the optimisation of centre-point and radius relationships between the overall bounding ball and a set of smaller, contained balls which are derived by analysis of the contractive maps of the IFS. The algorithm is well-behaved, in that although it converges toward an optimal ball which it only achieves in the limit, the process may still be stopped after any finite number of steps, with a guarantee that the sub-optimal ball which is returned will still bound the attractor.
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Krzysztof Leśniak, Nina Snigireva, Filip Strobin et al. · Nonlinearity · 2022
Abstract Iterated function systems (IFSs) and their attractors have been central in fractal geometry. If the functions in the IFS are contractions, then the IFS is guaranteed to have a unique attractor. Two natural questions concerning contractivity arise. First, whether an IFS needs to be contractive to admit an attractor? Second, what occurs to the attractor at the boundary between contractivity and expansion of an IFS? The first question is addressed in the paper by providing examples of highly noncontractive IFSs with attractors. The second question leads to the study of two types of transition phenomena associated with an IFS family that depend on a real parameter. These are called lower and upper transition attractors. Their existence and properties are the main topic of this paper. Lower transition attractors are related to the semiattractors, introduced by Lasota and Myjak in 1990s. Upper transition attractors are related to the problem of continuous dependence of an attractor upon the IFS. A main result states that, for a wide class of IFS families, there is a threshold such that the IFSs in the one-parameter family have an attractor for parameters below the threshold and they have no attractor for parameters above the threshold. At the threshold there exists a unique upper transition attractor.
Cited by 1Open access
Vian Hafid Suny, Kosala Dwidja Purnomo, Firdaus Ubaidillah · Majalah Ilmiah Matematika dan Statistika · 2020
Fractals have two types, namely fractals sets (artificial fractals) and natural fractals. Each type of fractal has a variety of fractal objects. One of the fractal objects is the Dragon Curve. Fractal objects can be generated through two methods, namely the Lindenmayer System (L-System) and the Iterated Function System (IFS). In previous studies, the Dragon curve can be generated through the L-System approach. The method is to start from determining the rotation angle, then determining the initial string, and the last one, which is determining the production rules. In this study, the Dragon curve is generated using IFS with Affine Transformation. The Affine transformation used in this study is dilation and rotation. Some variation is given on the scale of dilation and rotation angle. The variation is using a fixed angle with a variety of scale and using a fixed scale with a variation of angle. Each variation gives a different effect. This influence results in a varied visualization of the Naga curve. If the scale and angle that is varied approach a scale of one and an angle of 90° then the fractal formed approaches the Dragon curve of a scale of one with an angle of 90°. Conversely, if the scale and angle are varied away from one scale and angle of 90°, the fractal formed away from the Dragon curve of scale one with an angle of 90°. Keywords: Affine transformation, dragon curve, IFS method.
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Jie Yang · Mathematical Problems in Engineering · 2013
Recent literature highlights the multiple description coding (MDC) as a promising method to solve the problem of resilient image coding over error-prone networks, where packet losses occur. In this paper, we introduce a novel multiple description wavelet-based image coding scheme using fractal. This scheme exploits the fractal’s ability, which is to describe the different resolution scale similarity (redundancy) among wavelet coefficient blocks. When one description is lost, the lost information can be reconstructed by the proposed iterated function system (IFS) recovering scheme with the similarity and some introduced information. Compared with the referenced methods, the experimental results suggest that the proposed scheme can achieve better performance. Furthermore, it is substantiated to be more robust for images transmission and better subjective quality in reconstructed images even with high packet loss ratios.
Cited by 6Open access
Zachary Alex, A Elizabeth Bradley, B Joshua Garl et al. · 2016
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. a)Electronic Address:
Cited by 0
J.B. Waite · 1990
Presents a tutorial introduction to the theory of iterated function systems (IFSs) as applied to digital images and digital image compression, first developed by Jacquin (see ICASSP '90). There have been a number of excellent mathematical accounts of IFS theory applied to images but the aim is to show, using only elementary methods, why the IFS approach to digital image compression works. The author describes how an arbitrary image can be encoded using an IFS starting with a very brief overview of IFS based coding. >
Cited by 6
A.W.H. Lee, L.M. Cheng · 2002
Artificial neural networks (ANNs) are widely used on many research fields because of their high speed for execution and low cost for hardware implementation. In the paper, ANNs are used in fractal image compression because most of the well-known methodologies in fractal transformation are rather computational intensive. One of them is the iterated function system (IFS). This algorithm consists of few parameters, the IFS codes, which can be used to generate the IFS attractors but also consume most of the time extracting suitable attractors to represent the target image. In order to solve this problem, ANNs are investigated as a practical alternative approach.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">></ETX>
Cited by 1
Izabella Abraham · Mediterranean Journal of Mathematics · 2024
Abstract The aim of this paper is to answer one of the open questions raised in Strobin [Qual. Theory Dyn. Syst. 19 , 85 (2020)] of whether there exists an invariant (Hutchinson) measure for generalized iterated function systems of any order, consisting of a countably infinite number of maps. Our results likewise strengthen those obtained in Secelean [Mediterr. J. Math. 11 , 361–372 (2014)], where the existence of the invariant measure is ascertained only for the case of generalized iterated function systems of order 2, consisting of functions which satisfy a particular contractive condition.
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David Canright · 1993
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Nevart A. Minas, Faten Hassan Al-Qadhee · Tikrit Journal of Pure Science · 2023
Large video files processing involves a huge volume of data. The codec, storage systems and network needs resource utilization, so it becomes important to minimize the used memory space and time to distribute these videos over the Internet using compression techniques. Fractal image and video compression falls under the category of lossy compression. It gives best results when used for natural images. This paper presents an efficient method to compress an AVI (Audio Video Interleaved) file with fractal video compression(FVC). The video first is separated into a sequence of frames that are a color bitmap images, then images are transformed from RGB color space to Luminance/Chrominance components (YIQ) color space; each of these components is compressed alone with Enhanced Partition Iterated Function System (EPIFS), then fractal codes are saved. The classical IFS suffers from a very long encoding time that needed to find the best matching for each range block when compared with the domain image blocks. In this work, the (FVC) is enhanced by enhancing the IFS of the fractal image compression using a classification scheme based on the Discrete Cosine Transform (DCT). Experimentation is performed by considering different block sizes and jump steps to reduce number of the tested domain blocks. Results shows a significant reduction in the encoding time with good quality and high compression ratio for different video files.
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Dao-Xin Ding · Journal of Geometry and Physics · 2011
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Chol-Hui Yun, Hee-Hwan Choe, Mi-Gyong Ri · ArXiv.org · 2025
In nature, there are many phenomena with both irregularity and uncertainty. Therefore, a fuzzy-valued fractal interpolation is more useful for modeling them than fuzzy interpolation or fractal interpolation. We construct fractal interpolation functions that interpolate a given data set of fuzzy numbers using an iterated function system and study the Holder continuity of the constructed function. Firstly, we construct an iterated function system (IFS) using the given data set and construct a fuzzy-valued fractal interpolation function whose graph is the attractor of the constructed IFS. Next, we find the relationship between the fuzzy-valued fractal interpolation function and the fractal interpolation function of the level set of the data set. Finally, we prove the Holder continuity of the fuzzy-valued fractal interpolation function.
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Orion Sky Lawlor · ISRN Computer Graphics · 2012
Nonlinear functions, including nonlinear iterated function systems, have interesting fixed points. We present a non-Lipschitz theoretical approach to nonlinear function system fixed points which generalizes to noncontractive functions, compare several methods for evaluating such fixed points on modern graphics hardware, and present a nonlinear generalization of Barnsley’s Deterministic Iteration Algorithm. Unlike the many existing randomized rendering algorithms, this deterministic method avoids noncoherent branching and memory access and takes advantage of programmable texture mapping hardware. Together with the performance potential of modern graphics hardware, this allows us to animate high-quality and high-definition fixed points in real time.
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Shivam Dubey, Mrinal Kanti Roychowdhury, Saurabh Verma · ArXiv.org · 2025
For a given $r\in (0, +\infty)$, the quantization dimension of order $r$, if it exists, denoted by $D_r(μ)$, of a Borel probability measure $μ$ on ${\mathbb R}^d$ represents the speed how fast the $n$th quantization error of order $r$ approaches to zero as the number of elements $n$ in an optimal set of $n$-means for $μ$ tends to infinity. If $D_r(μ)$ does not exists, we call $\underline D_r(μ)$ and $\overline D_r(μ)$, the lower and upper quantization dimensions of $μ$ of order $r$. In this paper, we estimate the quantization dimension of condensation measures associated with condensation systems $(\{f_i\}_{i=1}^N, (p_i)_{i=0}^N, ν)$, where the mappings $f_i$ are bi-Lipschitz and the measure $ν$ is an image measure of an ergodic measure with bounded distortion supported on a conformal set. In addition, we determine the optimal quantization for an infinite discrete distribution, and give an example which shows that the quantization dimension of a Borel probability measure can be positive with zero quantization coefficient.
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Shivam Dubey, Mrinal Roychowdhury, SAURABH VERMA · SSRN Electronic Journal · 2026
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