CHAPTER 1 CONCEPT OF FRACTAL

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1 CHAPTER 1 CONCEPT OF FRACTAL

2 1.1 Introduction Fractal geometry is a new language used to describe, model and analyze complex forms found in nature. The termji-actal is commonly used to describe the family of nondifferentiable functions that are infinite in length. Over the past few years fractal geometry has been used as a language in theoretical, numerical and experimental investigations. It provides a set of abstract forms that can be used to represent a wide range of irregular objects. Fractal objects contain structures that are nested within one another. Each smaller structure is a miniature form of the entire structure. The use of fractals as a descriptive tool is diffusing into various scientific fields, from astronomy to biology. Fractal concepts can be used not only to describe the iltegular structures but also to study the dynamic properties of these structures. The applications of fractals can be divided into two groups. One is to analyze data sets with completely iltegular structures and the other is to generate data by recursively duplicating certain patterns. In ~ image processing and analysis, fdctal techniques are applied to image compression, Image coding, modeling of objects, representation and classification of images ( Falconer, 1992 ; Sato et ai., 1996 ; Li et ai., 1997; Cochran et ai., 1996; Lee and Lee, 1998 ;Luo, 1998). 1.2 Mathematical fractals- self-similarity Objects considered in Euclidean geometry are sets embedded in Euclidean space and object's dimension is the dimension of the embedding space. A point has a dimension of 0, a line has dimension of 1, a square is 2-dimensional, and a cube is 3 dimensional. The topological dimension is preserved when the objects are transformed by homomorphism. In the case of fi"actals, Hausdorff-Besikovitch dimension is used. topological dimension cannot be used, instead The fornlal definition of a fractal is stated as an object for which the fractal dimension is greater than the topological dimension. This definition is too restrictive to model natural phenomena. The alternative definition uses the concept of self-similarity. A fractal is an object made oillarts which are similar to the whole. The notion of self similarity is the basic property of 1''1ctal objects. Self-similarity

3 means that the pattern of the whole shape is similar to the pattern of an arbitrary small part of the shape. In principle, a theoretical or mathematically generated fractal is self similar over an infinite range of scales, while natural fractal images have a limited range of self similarity. Example of a fractal c. The Koch Curve The Koch curve was introduced by the Swedish mathematician, Helge Von Koch, in early 1900's. The von Koch's curve can be generated by a step-by-step procedure that takes a simple initial figure and turns it into an increasingly crinkly form. The first six stages in constructing a Koch snowflake is given in Figure 1.1 (Stevens, 1995). (a) _/\'----- (b) (c) (d) (e) 2

4 (t) Figure 1.1 : Representation of Koch curve in varying scales For constructing Koch's curve, an initiator and generator are needed. For the Koch curve, initiator is a line segment. The generator is a set of four segments, each segment, one-third the length of the initiator, arranged as shown in Figure 1.1. After the first replacement, the new figure is 4/3 as long as the original line and has an undifferentiable point at the peak of the equilateral triangle. After the second replacement, the figure is 4/3 the length of the figure created by the first replacement and has four undifferentiable points. Eventually, after infinite replacements, the figure has so many triangle peaks that every point is undifferentiable. 1.3 Natural fractals - statistical self-similarity The similarity method for calculating fractal dimension works for a mathematical fractal, like Koch curve which is composed of a certain number identical versions of itself. This is not the case with natural objects. Such objects show only statistical selfsimilarity. A mathematical fractal has an infinite amount of details. This means that magnifying it adds additional details, thereby increasing overall size. In non fractals, the size remains the same in spite of applied magnification. The graph of log (fractal's size) against log (magnification factor) gives a straight line. If the object is nonfractal, then this line is horizontal since the size does not change. If the object is fractal, the line is no longer horizontal since the size increases with magnification. The geometric method of calculating fractal dimension is by computing the slope of the above plotted line. 3

5 1.4 Concept offractal dimension Fractal geometry characterizes the way in which a quantitative dataset grows in mass, with linear size. The fractal dimension (D) is a measure of non-linear growth, which reflects the degree of irregularity over multiple scales. It is very often a non integer and is the basic measure of fractals. For a D-dimensional object, the number of identical parts, N divided by a scale ratio, r can be calculated from N...(1.1) Conversely, the self-similarity dimension 0 can be obtained as o 10g(N) I log( lir)...(1.2) Here 0 is also known as fractal dimension (Luo, 1998) 1.5 Applications of fractals Fractals can be used to model the underlying process in a variety of applications. A range of fractal analytical methods are used to characterise the fractal behaviour of the World Wide Web traffic. A realistic queuing model of Web traffic is developed based on fractal theory that provides analytical indications of network bandwidth dimensioning for Internet service providers ( Another application is to characterize the fractal nature of the entire system working in a LAN environment. Research is currently going on in the design of an airborne conformal antenna usmg It'actal structure that offers multiband operation ( Fractal geometry is used for understanding and planning the physical form of cities. It helps to simulate cities through computer graphics. The structural properties of fractals can be used in the architectural designs and also to model the morphology of surface growth. Fractal theory can be applied to a wide range of issues in chemical sciences like aggregatit1l1 phenomena,deposition and diffusion processes, chemical reactivity etc. The geological!catures like rock breakage, 4

6 ore and petroleum concentrations, seismic activity and tectonics, and volcanic eruptions can be studied using their fractal characteristics. In biology, it has been found that the DNA of plants and animal cells does not contain a complete description of all growth patterns, but cop.tains a set of instructions for cell development that follows a fractal pattern. Fractal geometry can be used in an analytical way to predict outcomes, to generate hypotheses, and to design experiments in biological systems in which fractal properties are most seen. Man-made objects are well defined using Euclidean geometry whereas natural objects are better modelled by fractal geometry. After the introduction of fractal geometry, its effect on various natural phenomena were studied. Goodchild(1980) studied the relation between fractal and geographical measure and pointed out that the fractal dimension can be used to predict the effect of cartographic generalization and spatial sampling. In cartography, Dutton (1981) used the properties of irregularity and self similarity of fractal to develop an algorithm to enhance the detail of digitized curves by altering their dimensionality in a parametrically controlled self similar fashion. Batty (1985) showed a number of examples of simulated landscape, mountainscape and other graphics generated by using the property of 'self-similarity' of fractals. Fractals are widely used in biosignal analysis and pattern recognition. They are used to study the structure, complexity and chaos in tumors. Aging, immunological response, autoimmune and chronic diseases can be better analysed through fractal geometry (Klonowski, 2000). The study of turbulence in flows is adapted to fractals. Turbulent flows are chaotic and are very difficult to model correctly. A fractal representation ofthem helps engineers and physicists to better understand complex flows. Fractal techniques are used in histopathology to interpret histological images to make a diagnosis and selection oftreatment. (Gabriel,1996). Fractal models have been found to be useful in describing and predicting the location and timing of earthquakes (Hastings and Sugihara, 1993). Astronomy, in particular cosmology, was one of the fields where fractals were found and applied to study various phenomena. The largest subsystem studied by means of fractals is the distribution ofgalaxies. The map of the distribution of 5

7 the galaxies obeys the postulation ofa power-law decreasing behavior for the density in concentric spheres as a function of radius. This is the characteristic behavior expected in heirarchical fractals (Perdang, 1990; Heck and Perdang, 1991; Elmegreen and Elmegreen, 2001). 1.6 Applications of fractals in image processing tasks Fractals are widely used in different image processing tasks. They have proved to be successful in the detection of edge points. The boundaries between homogeneous regions do not fit well into a fractal model. The boundaries give rise to nonfractal intensity surface and this provides an efficient method of detecting the edge points from an image. It can be identified when the measured fractal dimension becomes less than the topological dimension. The detection of tumor region from medical images IS an example. Another use of fractal is in the application of I-D fractal analysis to 2-D patterns. It is possible to transform a 2-D pattern in such a way to obtain a I-D pattern, which is then analysed using methods applied for signal analysis. For example, grey level images are segmented to produce the corresponding binary image. Then strips can be taken of the binary image of total length, N pixels and height, M pixels, with N several times greater than M. At each point ofthe long axis, the fraction of 'white' pixels in the column orthogonal to the long axis, denoted as t [1,N] is calculated by [0,1]...(1.3) where M)(t) denotes the number of white pixels in the t-th column. The resulting series of N rational numbers x,(t) serves as input for the subsequent 'signal analysis' (Klonowski, 2000). Such an approach was adapted by Mattfeld (1997 ) for analysis of histological texture of tumors. He analysed microscopic images of fibrous mastopathy and of invasive 6

8 ductal mammary cancer, with epithelial component. Produced signals were analysed using linear methods which are based on autocorrelation and power spectra. The methods based on chaos theory makes it possible to differentiate between two kinds of tumors and these differences have biological justifications. Another application of fractals is segmentation where the image is divided into a number of blocks and the fractal dimension of each block is computed. A histogram is plotted for the fractal dimension values. The valley at which the histogram is broken is chosen as the threshold and the image is segmented. Besides these image processing applications, fractals are used for classification of textures, determination of shape from texture, estimation of 3D roughness from image data, image compression etc In this thesis, discussion is focused on fractal techniques in compression, analysis and classification methods Fractal compression Fractal techniques can be used for data compression. Fractal image compression algorithms find self similarity at different scales and eliminate repeated description. Though, this compression technique is time consuming, the compression ratio can be as, high to 90 and the image may be decompressed quickly using iterative methods. Decompression speed, resolution independence and the compression ratios distinguish fractal image compression from other compression methods. In medical application, the electroencephalography (EEG) data points are condensed by computing the fractal dimension of the data points (Klonowski, 2000). 7

9 1.6.2 Fractal techniques in ID and 2D analysis a. 1-D fractals and signal analysis Signal processing methods have been greatly improved with the introduction of new tools stemming from fractal geometry. The main advantage of these approaches is that they make it possible to take into account fine local smoothness properties of signals. The Fourier analysis which is a popular tool does not provide an easy means of relating the local smoothness ofa function to the behaviour of its coefficients. The fractal analysis concems itself with the measurement of local smoothness of the signals. The significant infonnation in a signal does not result in its amplitude, but in the local variations of its iltegularities. An important application of 10 signal analysis is in biomedical signals. Biomedical signals are generated by complex self-regulating systems. The physiological time series may have fractal or multifractal temporal structure, though it is extremely inhomogenous and non-stationary. A characteristic feature of nonlinear (as opposed to linear) process is the interaction (coupling) of different modes, which may lead to nonrandom signal phase structure. Such collective phase properties of the signal cannot be detected by linear spectral methods. Fractal dimension can be an useful measure for the characterization ofelectrophysiological time series (Klonowski, 2000). b. 2-D fractals and image analysis A digitized image is a pattem stored as a rectangular data matrix. It is distinguished between binary images, grayscale images and color images. The ultimate goal of image analysis is the identification of a scene and all objects in the image. Image analysis can be describcd as a set of techniques required to cxtract symbolic infollllation from the image data. Different techniqucs arc availablc for performing image analysis, of which fractal methods are promising in certain applications. The il11ages can be studied by comparing the fractal dimcnsions of the original and transforjiwd images. Another 8

10 approach which has been put forth is the local fractal operator method. In this method, the fractal dimension corresponding to each pixel of the image is computed. This is followed by segmentation technique to extract the region of interest from the image. Marchette et ai., (1997 ) have employed fractal based techniques in digital mammography to detect tumorous tissues. The tumor region was better identified when segmentation boundaries were incorporated into the calculation. Similar techniques can be employed for the detection of an object from remotely sensed images or in the recognition of biological structures from plant images Fractal techniques in classification Fractal geometry is widely used in the study of image characteristics. For recognition ofregions and objects in natural scenes, there is always need for features that are invariant and provide a good set of descriptive values for the region. There are many fractal features that can be generated from an image (Chaudhuri and Sarkar, 1995). The most commonly used fractal feature is the fractal dimension. In some applications, fractal dimension alone is capable of discriminating one object from one another. But in certain applications, fractal dimension alone may not be sufficient in identifying the desired object. Here, a set of fractal features obtained on simple image transformations can be used as the feature set. Another technique is based on fractal signature. Fractal signature is designed using epsilon blanket method. The fractal signature assigns a unique signature to each sample. Samples belonging to each class have similar signature which makes them distinguish from one another easily. Fractal features are commonly used III textures. Images that possess homogeneous regions can be easily classified using these features. Fractal features are used in the recognition of skin samples to determine the age of a person. They are also used in fingerprint recognition and fabrics processing. 9

11 1.7 Conclusion The fractal theory is still descriptive rather than inferential. There are a lot of applications for fractals in different"fields. The basic concepts and applications of fractals are discussed in this chapter. In this thesis, the fractal techniques in image compression,analysis and classification are studied in detail. 10

Fractal Geometry. LIACS Natural Computing Group Leiden University

Fractal Geometry. LIACS Natural Computing Group Leiden University Fractal Geometry Contents Introduction The Fractal Geometry of Nature Self-Similarity Some Pioneering Fractals Dimension and Fractal Dimension Cellular Automata Particle Systems Scope of Fractal Geometry