Topic
Fractal analysis
About: Fractal analysis is a research topic. Over the lifetime, 6829 publications have been published within this topic receiving 141205 citations.
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Book•
01 Jan 1989TL;DR: A Starting Point for the Randomwalk What Use are Fractals? Delinquent Coins and Staggering Drunks Fractal Systems Generated by Randomwalks in Two-dimensional Space Vanishing Carpets, Fractal Felts and Dendritic Capture Trees.
Abstract: A Starting Point for the Randomwalk What Use are Fractals? Delinquent Coins and Staggering Drunks Fractal Systems Generated by Randomwalks in Two-Dimensional Space Vanishing Carpets, Fractal Felts and Dendritic Capture Trees An Exploration of the Physical Significance of Fractal Structures in Three-Dimensional Space Fractal Fingers and Floods Fracture, Fragments and Fractals Signposts to More Rambling Explorations of Fractal Space.
534 citations
Book•
11 May 1995
TL;DR: Introduction to wavelet analysis over IR Discretizing and periodizing the half-plane Multi-resolution analysis Fractal analysis and wavelet transforms Group theory as unifying language.
Abstract: Introduction to wavelet analysis over IR Discretizing and periodizing the half-plane Multi-resolution analysis Fractal analysis and wavelet transforms Group theory as unifying language.
525 citations
TL;DR: In this paper, the impact of fractal dimension on adsorption capacity has been discussed based on the physical description of the fractal surfaces, and the authors showed that fractal geometries with fractal dimensions ranging from 2.68 to 2.83 were obtained from the nitrogen adsorptions data using the Frenkel-Halsey-Hill method.
493 citations
TL;DR: In this paper, the wavelet transform modulus maxima is used to describe the scaling properties of singular measures of fractal objects, and it is shown that the generalized fractal dimensions D q and the f (α) singularity spectrum can be determined from the scaling behavior of these partition functions.
Abstract: The multifractal formalism originally introduced to describe statistically the scaling properties of singular measures is revisited using the wavelet transform. This new approach is based on the definition of partition functions from the wavelet transform modulus maxima. We demonstrate that very much like thermodynamic functions, the generalized fractal dimensions D q and the f ( α ) singularity spectrum can be readily determined from the scaling behavior of these partition functions. We show that this method provides a natural generalization of the classical box-counting techniques to fractal signals, the wavelets playing the role of “generalized boxes”. We illustrate our theoretical considerations on pedagogical examples, e.g., devil's staircases and fractional Brownian motions. We also report the results of some recent application of the wavelet transform modulus maxima method to fully developed turbulence data. That we emphasize the wavelet transform as a mathematical microscope that can be further used to extract microscopic informations about the scaling properties of fractal objects. In particular, we show that a dynamical system which leaves invariant such an object can be uncovered form the space-scale arrangement of its wavelet transform modulus maxima. We elaborate on a wavelet based tree matching algorithm that provides a very promising tool for solving the inverse fractal problem. This step towards a statistical mechanics of fractals is illustrated on discrete period-doubling dynamical systems where the wavelet transform is shown to reveal the renormalization operation which is essential to the understanding of the universal properties of this transition to chaos. Finally, we apply our technique to analyze the fractal hierarchy of DLA azimuthal Cantor sets defined by intersecting the inner frozen region of large mass off-lattice diffusion-limited aggregates (DLA) wit a circle. This study clearly lets out the existence of an underlying multiplicative process that is likely to account for the Fibonacci structural ordering recently discovered in the apparently disordered arborescent DLA morphology.
468 citations
TL;DR: It is shown that fuzzy set approach produces more consistent models (in terms of their performance), and how the power law of granularity helps construct mappings between system's variables in rule-based models.
465 citations