A Unified Approach to Fractal Dimensions

A Unified Approach to Fractal Dimensions

Witold Kinsner (University of Manitoba, Canada)
Copyright: © 2009 |Pages: 22
DOI: 10.4018/978-1-60566-170-4.ch021
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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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1. Introduction

This chapter is concerned with measuring the quality of various multimedia materials used in perception, cognition and evolutionary learning processes. The multimedia materials may include temporal signals such as sound, speech, music, biomedical and telemetry signals, as well as spatial signals such as still images, and spatio-temporal signals such as animation and video. A comprehensive review of the scope of multimedia storage and transmission, as well as quality metrics is presented by Kinsner (2002). Most of such original materials are altered (compressed or enhanced) either to fit the available storage or bandwidth during their transmission, or to enhance perception of the materials. Since the signals may also be contaminated by noise during different stages of their processing and transmission, various denoising techniques must be used to minimize the noise, without affecting the signal itself (Kinsner, 2002). Different classes of colored and fractal noise are described by Kinsner (1994c). A review of approaches to distinguish broadband signals and noise from chaos was provided by Kinsner (2003). The multimedia compression is often lossy in that the signals are altered with respect not only to their redundancy, but also to their perceptual and cognitive relevancy. Since the signals are presented to humans (rather than machines), cognitive processes must be considered in the development of suitable quality metrics. Energy-based metrics are not suitable for such cognitive processes. A very fundamental class of metrics based on entropy was described by Kinsner (2004), with a discussion on its usefulness and limitations in the area of cognitive informatics (CI) as defined in (Wang, 2002; Wang and Kinsner, 2006), and autonomic computing (Kinsner, Potter, & Faghfouri, 2005; Wang, 2007). This chapter is an extension of the single-scale entropy-based metrics to multiscale metrics through fractal dimensions. Many experimental results obtained by the author and his collaborators indicate that quality metrics based on fractal dimensions appear to be most suited for perception. Further research on their suitability for cognition is being conducted.

A topological dimension is by definition a non-negative integer 0, 1, 2,… . The dimension of a general abstract vector space is the number of linearly independent vectors required for a basis, and is n for Rn. An orthonormal basis is by definition a basis which is an orthonormal set. The space-time we live in is often characterized by four Euclidean integer dimensions.

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Table of Contents
Yingxu Wang
Chapter 1
Yingxu Wang
Cognitive Informatics (CI) is a transdisciplinary enquiry of the internal information processing mechanisms and processes of the brain and natural... Sample PDF
The Theoretical Framework of Cognitive Informatics
Chapter 2
Withold Kinsner
This chapter provides a review of Shannon and other entropy measures in evaluating the quality of materials used in perception, cognition, and... Sample PDF
Is Entropy Suitable to Characterize Data and Signals for Cognitive Informatics?
Chapter 3
Ismael Rodríguez, Manuel Núñez, Fernando Rubio
Finite State Machines (FSM) are formalisms that have been used for decades to describe the behavior of systems. They can also provide an intelligent... Sample PDF
Cognitive Processes by using Finite State Machines
Chapter 4
Yingxu Wang
An interactive motivation-attitude theory is developed based on the Layered Reference Model of the Brain (LRMB) and the Object-Attribute-Relation... Sample PDF
On the Cognitive Processes of Human Perception with Emotions, Motivations, and Attitudes
Chapter 5
Qingyong Li, Zhiping Shi, Zhongzhi Shi
Sparse coding theory demonstrates that the neurons in the primary visual cortex form a sparse representation of natural scenes in the viewpoint of... Sample PDF
A Selective Sparse Coding Model with Embedded Attention Mechanism
Chapter 6
Yingxu Wang
Theoretical research is predominately an inductive process, while applied research is mainly a deductive process. Both inference processes are based... Sample PDF
The Cognitive Processes of Formal Inferences
Chapter 7
Douglas Griffith, Frank L. Greitzer
The purpose of this article is to re-address the vision of human-computer symbiosis as originally expressed by J.C.R. Licklider nearly a... Sample PDF
Neo-Symbiosis: The Next Stage in the Evolution of Human Information Interaction
Chapter 8
Ray E. Jennings
Although linguistics may treat languages as a syntactic and/or semantic entity that regulates both language production and comprehension, this... Sample PDF
Language, Logic, and the Brain
Chapter 9
Yingxu Wang, Guenther Ruhe
Decision making is one of the basic cognitive processes of human behaviors by which a preferred option or a course of actions is chosen from among a... Sample PDF
The Cognitive Process of Decision Making
Chapter 10
Tiansi Dong
This chapter proposes a commonsense understanding of distance and orientation knowledge between extended objects, and presents a formal... Sample PDF
A Commonsense Approach to Representing Spatial Knowledge Between Extended Objects
Chapter 11
Natalia López, Manuel Núñez, Fernando L. Pelayo
In this chapter we present the formal language, stochastic process algebra (STOPA), to specify cognitive systems. In addition to the usual... Sample PDF
A Formal Specification of the Memorization Process
Chapter 12
Yingxu Wang
Autonomic computing (AC) is an intelligent computing approach that autonomously carries out robotic and interactive applications based on goal- and... Sample PDF
Theoretical Foundations of Autonomic Computing
Chapter 13
Witold Kinsner
Numerous attempts are being made to develop machines that could act not only autonomously, but also in an increasingly intelligent and cognitive... Sample PDF
Towards Cognitive Machines: Multiscale Measures and Analysis
Chapter 14
Amar Ramdane-Cherif
Cognitive approach through the neural network (NN) paradigm is a critical discipline that will help bring about autonomic computing (AC). NN-related... Sample PDF
Towards Autonomic Computing: Adaptive Neural Network for Trajectory Planning
Chapter 15
Lee Flax
We give an approach to cognitive modelling, which allows for richer expression than the one based simply on the firing of sets of neurons. The... Sample PDF
Cognitive Modelling Applied to Aspects of Schizophrenia and Autonomic Computing
Chapter 16
Yan Zhao, Yiyu Yao
Classification is one of the main tasks in machine learning, data mining, and pattern recognition. Compared with the extensively studied automation... Sample PDF
Interactive Classification Using a Granule Network
Chapter 17
Mehdi Najjar, André Mayers
Encouraging results of last years in the field of knowledge representation within virtual learning environments confirms that artificial... Sample PDF
A Cognitive Computational Knowledge Representation Theory
Chapter 18
Du Zhang
A crucial component of an intelligent system is its knowledge base that contains knowledge about a problem domain. Knowledge base development... Sample PDF
A Fixpoint Semantics for Rule-Base Anomalies
Chapter 19
Christine W. Chan
This chapter presents a method for ontology construction and its application in developing ontology in the domain of natural gas pipeline... Sample PDF
Development of an Ontology for an Industrial Domain
Chapter 20
Václav Rajlich, Shaochun Xu
This article explores the non-monotonic nature of the programmer learning that takes place during incremental program development. It uses a... Sample PDF
Constructivist Learning During Software Development
Chapter 21
Witold Kinsner
Many scientific chapters treat the diversity of fractal dimensions as mere variations on either the same theme or a single definition. There is a... Sample PDF
A Unified Approach to Fractal Dimensions
Chapter 22
Du Zhang, Witold Kinsner, Jeffrey Tsai, Yingxu Wang, Philip Sheu, Taehyung Wang
The 2005 IEEE International Conference on Cognitive Informatics (ICCI’05) was held during August 8th to 10th 2005 on the campus of University of... Sample PDF
Cognitive Informatics: Four Years in Practice
Chapter 23
Yiyu Yao, Zhongzhi Shi, Yingxu Wang, Witold Kinsner, Yixin Zhong, Guoyin Wang
Cognitive informatics (CI) is a cutting-edge and multidisciplinary research area that tackles the fundamental problems shared by modern informatics... Sample PDF
Toward Cognitive Informatics and Cognitive Computers: A Report on IEEE ICCI'06
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