Fractals

Applications in Biological Signalling and Image Processing

Dinesh Kumar, Sridhar P. Arjunan, Behzad Aliahmad

• Publisher: CRC Press
• Year: 2017
• eBook ISBN: 9781351678384
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

The book provides an insight into the advantages and limitations of the use of fractals in biomedical data. It begins with a brief introduction to the concept of fractals and other associated measures and describes applications for biomedical signals and images. Properties of biological data in relations to fractals and entropy, and the association with health and ageing are also covered. The book provides a detailed description of new techniques on physiological signals and images based on the fractal and chaos theory.

The aim of this book is to serve as a comprehensive guide for researchers and readers interested in biomedical signal and image processing and feature extraction for disease risk analyses and rehabilitation applications. While it provides the mathematical rigor for those readers interested in such details, it also describes the topic intuitively such that it is suitable for audience who are interested in applying the methods to healthcare and clinical applications. The book is the outcome of years of research by the authors and is comprehensive and includes other reported outcomes.

CHAPTER 1

Introduction

ABSTRACT

Biomedical signals and images have been found to be non-deterministic but describable as chaotic and having fractal properties. These are measured by obtaining the fractal dimension (FD) of the signal or image. The FD of biological signals has been associated with various health and age related factors. This chapter introduces the reader to the brief history of fractal analysis and examines this in terms of biological signals, images and data. The chapter sets the stage for detailed analysis in the subsequent chapters.

1.1 Introduction

Science attempts to model observations in terms of definitive laws and rules. It deals with supposedly predictable phenomena such as gravity, electricity, and biological processes. When these studies are undertaken, the system is simplified into a number of independent components, each described in deterministic terms. Such models are generally suitable for describing a large number of observations and most of our technology has evolved from such exercises. For example, the earth’s surface was first thought to be flat; however, detailed analysis shows that the earth is round. Further analysis now demonstrates the relationship between the surface of earth and galaxies far away. While we all now know that earth is not flat, for many day to day applications, it is sufficient to model and explain most observations made by the naked eyes and for us human to perform many of our daily activities such as walking or driving. It also allows us to build our buildings and perform our other activities. However, it does not allow us to explore the Universe.

The three important laws of nature were discovered by Newton, though later were found to be inaccurate. Though these laws have been found to be inaccurate, they can still be used to explain most of the phenomena that are observed by us during our daily life and thus these laws cannot be considered

to be incorrect. However, these laws are unable to provide the precision and clarity that may be necessary for certain purposes. One such example is the understanding of weather patterns. Seemingly similar conditions can lead to very different outcomes and deterministic computational models predictions appear to be very far from real observations.

Small causes can sometimes have abnormally large effects. This has been observed by philosophers, historians and scientists since time immemorial. Evolution in science is a result of continuous improvement in the experimental methodology and the ability to perform more exact measurements. Very often, the laws that describe the observations create methods and instruments, which on being used enable measurements to be performed more accurately, thereby negating the laws themselves. With the evolution of science, we know that the earth is not flat, and that there is an uncertainty in all measurements. However, our traditional mathematics that models the environment is designed to provide deterministic models, while many phenomena such as the weather and biological systems are not deterministic.

The term “chaos” had been used since antiquity to describe various forms of randomness, but in the late 1970s it became specifically tied to the phenomenon of sensitive dependence on initial conditions. Chaos is the science which explains when the outcomes do not appear to follow the natural laws leading to the unpredictable. It teaches us to expect the unexpected. The underpinning mathematics of Chaos theory allows the description of observations that appear to be unexplainable, even though the system seems to be well understood.

There are a number of reasons why many times the predicted outcomes appear to be very different from the observations. One reason is because the assumption of independence between different parts of the system is not accurate and there is a complex relationship between many seemingly independent elements. If we consider the body which is made up of a number of organs and each organ is made from individual cells. Each of these are independent, however they are also dependent on the rest of the body. For example, each cell requires the flow of blood, which requires a number of different organs. But these cells are also independent and many of these will continue to live after the body dies.

The second cause of large differences between the predictions and the observations are due to the variability in the initial conditions of the system. In most biological and natural systems, it is difficult to accurately identify the point in time that can be considered to be the starting point, thus determining the initial conditions accurately is impossible. While the definition of the start of life is given for the sake of legal or cultural reasons, it is near impossible to determine this from a scientific view point.

Chaos Theory describes nonlinearity and complexity of events and phenomena that are effectively impossible to predict or control, such as weather, or turbulence of a jet engine, or the states of the body. These phenomena are often described by fractal mathematics, which captures the infinite complexity of nature. Most natural systems and events exhibit fractal properties, including landscapes, clouds, trees, organs, rivers. Also many of the systems in which we live exhibit complex, chaotic behavior. Recognizing the chaotic, fractal nature of our world can give us new insight, power, and wisdom. For example, by understanding the complex, chaotic dynamics of the atmosphere, a balloon pilot can “steer” a balloon to a desired location. By understanding that our ecosystems, our social systems, and our economic systems are interconnected, we can hope to avoid actions which may end up being detrimental to our long-term well-being.

Biological systems and most natural systems can often be treated as systems within systems. Similar to the concept of seeing the Universe as a giant atom, biological systems can, to an extent, be treated as having scale dependence on the observer. These properties may be in terms of spatial, temporal or other dimensions and can be referred to by their self-affinity and are described by fractal geometry.

According to Edward Norton Lorenz [1], the entire universe is connected, and the movement of air due to the fluttering of the wings of a butterfly in one location can be the cause of a storm in a place that is very remote. Similarly, biological systems may appear to have separate organs, but the entire body is a single entity and a small change in one part of the body could lead to major changes in a different part of the body. Often, this cause and effect may not be evident when we look at an individual organ or part of the body. It is important that the entire system should be considered as a whole along with considering the individual parts for accurate diagnosis and predictions.

Biological systems are known to be unpredictable and it is often difficult to predict the outcome or response of the body to treatment or to a change in circumstances. Chaos theory has demonstrated that small difference in the initial condition can change the outcome of an experiment very significantly. As the initial conditions are difficult to know precisely, and it is often difficult to identify what should be considered as the starting point, the final outcome is very unpredictable in biological systems. Often, what appears to be disorder and random behavior is not because of lack of order but due to this unpredictability and interconnectivity.

While outside the scope of this book, there is the concept of the difference in the psychological response between different people. The fundamental laws that govern all people are the same, but the behavior of different people to the same situation can be very different. It has been found that even identical twins brought up under identical conditions can behave very differently. This is now understood in terms of chaos theory, which explains that seemingly similar initial conditions would have sufficient differences that leads to large differences, and makes the two people behave extremely differently.

Fractals are an ever continuing pattern, a pattern that is infinitely complex and is based on self-similarity, with the underlying process being simple. Fractals show a system that is seemingly based on very simple principles but leads to very complex structures. Fractals are suited to describe the chaotic behavior and are effective in describing systems such as biological systems. They have also been adapted to describe and develop music and art, to study natural objects, and have also been used in attempts at giving rigor to concepts such as beauty.

1.2 History of Fractal Analysis

The concepts of fractals and Chaos have been discussed by scientists and philosophers for a long period of time. However, it was the availability of computers, especially with graphical displays that allowed the formulation of the fractal concepts. Such displays have also provided the strength of imagery and based on these displays, the abstract concepts have become easier to understand for even lay people, and are now commonly accepted.

Fractal geometry, associated math and analysis techniques are largely attributed to Benoit Mandelbrot, a Polish born French mathematician (20 November 1924–14 October 2010). While the fundamental concepts had been discussed far earlier in physics, and perhaps these concepts have been discussed in Chinese and Indian philosophies a few thousand years ago, he was responsible for formulating the concept and providing the rigor, though in an unconventional style. He studied and demonstrated the scale invariant properties in nature. He created the associated concepts of self-similarity and later in 1975 coined the term, Fractals, and also associated this with the concept of roughness. The word, fractals, is from Latin and means, fractured, and implies the inherent complexity at changed scales. He demonstrated this concept using graphical displays of fractals which showed how visual complexity could be created from underlying simple rules.

1.3 Fundamentals of Fractals

A pattern, with the repetition embedded in it is called Fractal, and may be from a natural phenomenon or a numerical set. It is also known as expanding symmetry or evolving symmetry and has been shown to describe the power law. If the replication is exactly the same at every scale, it is called a self-similar pattern. There are number of computational examples that provide the visualisation of this phenomenon such as Menger Sponge and Koch Curve. However, in real life, the phenomena are not exactly self-similar, but nearly identical at different scales and are called Fractals.

Geometric figures such as a rectangle or any other polygon has the area change by the square of the factor by which the lengths of the sides were changed. Thus, if the length of the side of a square is doubled, its area increases by 4, or 2ⁿ, where n = 2, the dimension of the polygon. However, when the length of the fractal is doubled, the area increases by a factor which is not an integer, or n is not an integer but a fraction and is the fractal dimension of the objects exhibiting fractal geometry.

Shapes or functions that describe fractal geometry (or data) are generally not differentiable. A line is generally considered to be 1 dimensional. However, a line corresponding to fractal dimension would have resemblance to a surface and would have a dimension greater than 1.

1.4 Definition of Fractal

There is no precise definition for fractals. Perhaps the best description for fractals is that these have (1) self-similarity, (2) iterated, and (3) fractal dimensions. These are not limited to geometric patterns or mathematical expressions, with many examples in nature that may describe functions of time.

Conceptually, fractals can be considered to be associated with repetitive procedures, and while the fundamental procedure is simple, the final output is very complex. A simple example of a fractal is based on an equilateral triangle which has each sides of 1 unit length. Further, on each edge of the triangle, if a new (smaller) equilateral triangle is included, sides of these triangles will be 1/3 units. Repeat this process to each of the new triangle sides, and this time the length of the sides of the new triangles will be 1/9 units. If this process is continued till infinitum, we now have the fractal. It can be observed from the Fig. 1.1 that magnification (or reduction) along any edge results in the same shape.

Figure 1.1. Fractal object with an equilateral triangle.

The area of a fractal shaped object or figure cannot be computed using concepts of polygon geometry, but needs to be obtained by summing the area of the individual areas, in this case, the individual triangles. Based on the visualisation of this figure and geometric summation, the area of the figure will converge to a finite number. However, the length of the line, or perimeter, could continue to increase as the number of iterations increases, and theoretically, this could increase to infinity.

Using geometric series, the length of each side would increase by a factor of 1/3 for each iteration, or the perimeter = 3*(4/3)^m, where m = number of iterations. It has been empirically found that the distributions of a wide variety of natural, biological, and many man-made phenomena approximately follow a three power law. This explains number of observations as when there is relatively large reduction in the size of some object such as the cross-section of a pipe, the reduction in throughput is relatively small. This is described by the relationship A^3/4 law and has been found to be associated with the self-similarity within itself.

1.5 Complexity of Biological Systems

No two bodies are identical and within the body, there is never exact symmetry. The body is nearly laterally symmetrical and the heart nearly beats regularly. However, the important word is; nearly. But within the short observations, and in controlled conditions, our physiology may be described as if deterministic and for most measurements the body does appear to be symmetrical. It is often the small differences that exist in different dimensions of the body or in different selections within the body that makes it robust, resilient, and perhaps beautiful.

Bodies are complex structures which can be described as fractal. Whether we see our skin surface, the neurons, the circulatory system, or our retinal vasculature, all of these follow a tree structure. These follow an approximate self-similarity and can be defined in terms of their fractal properties. One common outcome of such a structure is that it allows for packing very long or wide structures in a small volume. Thus, based on such branching tree structure, the lungs maximize the surface area, thereby enhancing our capabilities for exchange of gases, while enclosed in a small volume. The lungs have an area of a full tennis court while in a few cubic centimeters of space. Such enhanced area allows for greater exchange of gases, making breathing more efficient.

Another section of our body that packs a large number of connectivity in a small volume is the brain. One aspect of evolution is the high degree ...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Preface
• List of Figures
• 1. Introduction
• 2. Physiology, Anatomy and Fractal Properties
• 3. Fractal Dimension of Biosignals
• 4. Fractals Analysis of Electrocardiogram
• 5. Fractals Analysis of Surface Electromyogram
• 6. Fractals Analysis of Electroencephalogram
• 7. Fractal Analysis of Biomedical Images
• 8. Fractal Dimension of Retinal Vasculature
• 9. Fractal Dimension of Mammograms
• 10. Fractal Dimension of Skin Lesions
• 11. Case study I: Age Associated Change of Complexity
• 12. Case Study 2: Health, Well-being and Fractal Properties
• Index