Fractal Geometry in Image Processing

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IRACST- International Journal of Research in Management & Technology (IJRMT), ISSN: 2249-9563 Vol. 2, No. 1, 2012 Conversely, the self-similarity dimension D can be obtained as D = log(N) / log(l/r) ……………… (1.2) Here D is also known as fractal dimension (Luo, 1998) 1.5 Applications of fractals 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 self-similarity. 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 /actor) 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. 1.4 Concept of fractal 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 = l/r D ……………… (1.1) 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 (http://vvww.cs.bu.edu/facultv/crovella/paper-archive/selfsim/paper.html). 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 using fractal structure that offers multiband operation (http://www.fractenna.com/nca_faq.html). 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 aggregation phenomena Reposition and diffusion processes, chemical reactivity etc. The geological features like rock breakage, 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 contains 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 111
IRACST- International Journal of Research in Management & Technology (IJRMT), ISSN: 2249-9563 Vol. 2, No. 1, 2012 property of 'self-similarity' of fractals. (Klonowski, 2000). Such an approach was adapted by Mattfeld (1997) Fractals are widely used in biosignal analysis and for analysis of histological texture of tumors. He analysed pattern recognition. They are used to study the structure, microscopic images of fibrous mastopathy and of invasive complexity and chaos in tumors. Aging, immunological ductal mammary cancer, with epithelial component. response, autoimmune and chronic diseases can be better Produced signals were analysed using linear methods which analysed through fractal geometry (Klonowski, 2000). The are based on autocorrelation and power spectra. The study of turbulence in flows is adapted to fractals. Turbulent methods based on chaos theory makes it possible to flows are chaotic and are very difficult to model correctly. A differentiate between two-kinds of tumors and these fractal representation of them helps engineers and physicists differences have biological justifications. to better understand complex flows. Fractal techniques are Another application of fractals is segmentation used in histopathology to interpret histological images to where the image is divided into a number of blocks and the make a diagnosis and selection of treatment. (Gabriel, fractal dimension of each block is computed. A histogram is 1996). Fractal models have been found to be useful in plotted for the fractal dimension values. The valley at which describing and predicting the location and timing of the histogram is broken is chosen as the threshold and the earthquakes (Hastings and Sugihara, 1993). Astronomy, in image is segmented. Besides these image processing particular cosmology, was one of the fields where fractals applications, fractals are used for classification of textures, were found and applied to study various phenomena. The determination of shape from texture , estimation of 3D largest subsystem studied by means of fractals is the roughness from image data, image compression etc. distribution of galaxies. The map of the distribution of the In this thesis, discussion is focused on fractal galaxies obeys the postulation of a power-law decreasing techniques in compression, analysis and classification behavior for the density in concentric spheres as a function methods. 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 1-D fractal analysis to 2-D patterns. It is possible to transform a 2-D pattern in such a way to obtain a 1-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 of the long axis, the fraction of 'white' pixels in the column orthogonal to the long axis, denoted as t [1,N] is calculated by X 1 (t) = M 1 (t) / M [0,1] ……….. (1.3) where M 1 (t) denotes the number of white pixels in the t-th column. The resulting series of N rational numbers X 1 (t) serves as input for the subsequent 'signal analysis' 1.6.1 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). 1.6.2 Fractal techniques in II) 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 of a function to the behaviour of its coefficients. The fractal analysis concerns itself with the measurement of local smoothness of the signals. The significant information in a signal does not result in its amplitude, but in the local variations of its irregularities. An important application of ID 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 112
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Fractal Geometry in Image Processing

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