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A Unified Approach to Fractal Dimensions

A Unified Approach to Fractal Dimensions

Witold Kinsner
Copyright: © 2009 |Pages: 22
ISBN13: 9781605661704|ISBN10: 1605661708|EISBN13: 9781605661711
DOI: 10.4018/978-1-60566-170-4.ch021
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MLA

Kinsner, Witold. "A Unified Approach to Fractal Dimensions." Novel Approaches in Cognitive Informatics and Natural Intelligence, edited by Yingxu Wang, IGI Global, 2009, pp. 304-325. https://doi.org/10.4018/978-1-60566-170-4.ch021

APA

Kinsner, W. (2009). A Unified Approach to Fractal Dimensions. In Y. Wang (Ed.), Novel Approaches in Cognitive Informatics and Natural Intelligence (pp. 304-325). IGI Global. https://doi.org/10.4018/978-1-60566-170-4.ch021

Chicago

Kinsner, Witold. "A Unified Approach to Fractal Dimensions." In Novel Approaches in Cognitive Informatics and Natural Intelligence, edited by Yingxu Wang, 304-325. Hershey, PA: IGI Global, 2009. https://doi.org/10.4018/978-1-60566-170-4.ch021

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Abstract

Many scientific chapters treat the diversity of fractal dimensions as mere variations on either the same theme or a single definition. There is a need for a unified approach to fractal dimensions for there are fundamental differences between their definitions. This chapter presents a new description of three essential classes of fractal dimensions based on: (a) morphology, (b) entropy, and (c) transforms, all unified through the generalized-entropy-based Rényi fractal dimension spectrum. It discusses practical algorithms for computing 15 different fractal dimensions representing the classes. Although the individual dimensions have already been described in the literature, the unified approach presented in this chapter is unique in terms of its progressive development of the fractal dimension concept, similarity in the definitions and expressions, analysis of the relation between the dimensions, and their taxonomy. As a result, a number of new observations have been made, and new applications discovered. Of particular interest are behavioral processes (such as dishabituation), irreversible and birth-death growth phenomena (e.g., diffusion-limited aggregates, DLAs, dielectric discharges, and cellular automata), as well as dynamical nonstationary transient processes (such as speech and transients in radio transmitters), multifractal optimization of image compression using learned vector quantization with Kohonen’s self-organizing feature maps (SOFMs), and multifractal-based signal denoising.

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