Table Of ContentFRACTALS
Applications in Biological Signalling and
Image Processing
Design, FFaRbAricCaTtAioLnS, Properties
and Applications of Smart and
ApplicatiAondsv ian nBcioelodg iMcaal tSeigrniaalllsing and
Image Processing
DINESH K. KUMAR
RMIT University, Melbourne
Editor
VIC, Australia
Xu Hou
SRIDHAR P. ARJUNAN
Harvard University
RMIT University, Melbourne
School of Engineering and Applied Sciences
VIC, Australia
Cambridge, MA, USA
and
BEHZAD ALIAHMAD
RMIT University, Melbourne
VIC, Australia
p,
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Preface
It has been well established that healthy and stable natural systems are
chaotic in nature. For example heart-rate variability and not heart-rate, is
an important indicator of the healthy heart of the person. While there may
be large differences in the resting heart-rate of two healthy individuals,
it is important that this is not remaining monotonous but has significant
variability. Over the past four decades, numerous formulas have been
developed to measure and quantify such variability. This variability is
often referred to as the complexity of the parameters and explained using
Chaos Theory.
There are thousands of scientific publications on the application of
Chaos Theory for the analysis of biomedical signals and images. We have
attended many conferences and meetings where the relationship between
the fractal dimension (FD) of biomedical signals and images with disease
conditions, have been discussed. Many authors have demonstrated that
there is change in the values of FD with factors such as age and health.
The aim of this book is not to capture the details of these publications;
because we are certain that the readers can access those papers directly
and without our help. In our current world of information overload, we
do not see the purpose for writing any book to be repeating publications
that are already available.
When reading the numerous publications on the topic, one common
shortcoming was observed; the authors gave numbers, formulas and in
some cases, statistics. What they have missed out is the explanation to the
concepts. The aim of this book is to provide the conceptual framework for
fractal dimension of biomedical signals and images. We have begun by
explaining the concepts of chaos, complexity and fractal properties of the
signal in plain language and then discussed some examples to explain the
concepts. We are aware that there are many more examples and research
outcomes than are covered in this book. While we have attempted to
discuss current research and examples, this book is not a replacement of
your literature review on the topic.
We are hopeful that this book will help the reader understand the
concepts and develop new applications. Once the fundamentals are
vi FRACTALS: Applications in Biological Signalling and Image Processing
understood, the human body could be recognised in terms of its chaotic
properties. In such a situation, the measurements are not just numbers but
quantification of the physical phenomena. We hope that this would be useful
for engineers, physiologists, clinicians and lay persons.
Content
Preface v
List of Figures xiii
1. Introduction 1
Abstract 1
1.1 Introduction 1
1.2 History of Fractal Analysis 4
1.3 Fundamentals of Fractals 4
1.4 Definition of Fractal 5
1.5 Complexity of Biological Systems 6
1.6 Fractal Dimension 7
1.7 Summary of this Book 7
References 7
2. Physiology, Anatomy and Fractal Properties 8
Abstract 8
2.1 Introduction 8
2.2 Conceptual Understanding 10
2.3 Chaos, Complexity, Fractals and Entropy 10
2.4 Chaos Theory 11
2.5 Complex Systems 13
2.6 Entropy 14
2.7 Fractal and Fractal Dimension 16
2.8 Computing Fractal Dimension 16
2.8.1 Box-counting 17
2.8.2 Power spectrum fractal dimension 18
2.9 Relationship of Fractals and Self-similarity 18
2.9.1 Sierpinski triangle 18
viii FRACTALS: Applications in Biological Signalling and Image Processing
2.9.2 Fractal dimension of the Menger Sponge 19
2.10 Fractals in Biology 19
2.11 Properties of Natural and Synthetic Objects 20
2.12 Human Physiology 21
2.12.1 Fractals and Electrocardiogram (ECG), 21
Electromyogram (EMG) and Electroencephalogram (EEG)
2.12.2 Fractal dimension for human movement and gait analysis 22
2.13 Summary 22
References 22
3. Fractal Dimension of Biosignals 24
Abstract 24
3.1 Introduction 24
3.2 Fractal Dimension and Self-similarity 25
3.2.1 Self-similarity 26
Exact self-similarity 26
Approximate self-similarity 26
Statistical self-similarity 27
3.2.2 Fractal dimension 27
3.3 Different Methods to Estimate Fractal Dimension of a Waveform 29
3.3.1 Box-counting method 29
3.3.2 Katz’s algorithm 30
3.3.3 Higuchi’s algorithm 31
3.3.4 Petrosian’s algorithm 31
3.3.5 Sevcik’s algorithm 32
3.3.6 Correlation dimension 33
3.3.7 Adapted box fractal dimension 33
3.3.8 Fractal dimension estimate based on power law function 33
3.4 Fractals and Electrocardiogram (ECG), 34
Electromyogram (EMG) and Electroencephalogram (EEG)
3.5 Fractal Dimension for Gait Analysis 36
3.5.1 Example 37
3.6 Summary 38
References 39
4. Fractals Analysis of Electrocardiogram 42
Abstract 42
4.1 Introduction 42
4.1.1 Recording cardiac activity 44
Content ix
4.2 Heart Rate Variability 46
4.2.1 Computing heart rate variability 47
4.3 Fractal Properties of ECG 48
4.4 An Example 49
4.5 Poincaré Plot of Heart-rate Variability 49
4.6 Application—ECG and Heart Rate Variability 51
Time domain analysis 53
Frequency domain analysis 54
Poincaré analysis 55
Fractal dimension 57
4.7 Summary 57
References 58
5. Fractals Analysis of Surface Electromyogram 60
Abstract 60
5.1 Introduction 60
5.2 Surface Electromyogram (sEMG) 62
5.2.1 Principles of sEMG 63
5.2.2 Factors that influence sEMG 63
5.2.3 Signal features of sEMG 64
Amplitude analysis 64
Spectral analysis 64
Statistical and chaos based features 65
5.3 Fractal Analysis of sEMG 65
5.3.1 Self-similarity of sEMG 65
5.3.2 Algorithms to compute fractal dimension of sEMG 67
Signals in the time domain 67
Signals in the phase space domain 67
5.3.3 Fractal features of sEMG 67
5.4 Summary 71
References 72
6. Fractals Analysis of Electroencephalogram 74
Abstract 74
6.1 Introduction 74
6.1.1 History of EEG 75
6.1.2 Fundamentals of EEG 75
6.2 Techniques for EEG Analysis 76
6.3 Fractal Properties of EEG 78
