Fractal Geometry in Image Processing

IRACST- International Journal of Research in Management & Technology (IJRMT), ISSN: 2249-9563 

Vol. 2, No. 1, 2012 

Fractal Geometry in Image Processing 

Prof. A. Annadhason, 

Head & Asst. Professor of Computer Science Department, St. Jude’s College, India. 

Abstract: The important characteristics of fractal geometry 

namely, fractal dimension is described. Resolution, being a 

primary cause of error in any image processing task, an 

analysis of the effect of resolution on fractal dimension has 

been made. For varying levels of brightness and contrast, 

the fractal dimension is evaluated. Also, the variation of 

fractal dimension on modified images such as high and low 

grey valued images, edge detected images and filtered 

images has been studied and results are presented. 

1.1 Introduction 

Fractal geometry is a new language used to 

describe, model and analyze complex forms found in nature. 

The term fractal is commonly used to describe the family of 

non-differentiable 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 irregular 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 irregular structures and the other is to 

generate data by recursively duplicating certain patterns. In 

image processing and analysis, fractal techniques are 

applied to image compression, image coding, modeling of 

objects, representation and classification of images ( 

Falconer, 1992 ; Sato et al., 1996 ; Li et al., 1997 ; Cochran 

et al., 1996 ; Lee and Lee, 1998 ; Luo, 1998). 

object made of parts which are similar to the whole. The 

notion of self similarity is the basic property of fractal 

objects. Self-similarity 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). 

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 fractals, topological 

dimension cannot be used; instead Hausdorff-Besikovitch 

dimension is used. The formal 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 

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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 

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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 

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characteristic feature of nonlinear (as opposed to linear) 

process is the interaction (coupling) of different modes, 

which may lead to non-random 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 of 

electrophysiological time series (Klonowski, 2000). 

b. 2-D fractals and image analysis 

A digitized image is a pattern 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 described as a 

set of techniques required to extract symbolic information 

from the image data. Different techniques are available for 

performing image analysis, of which fractal methods are 

promising in certain applications. The images can be studied 

by comparing the fractal dimensions of the original and 

transformed images. Another 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 al., (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. 

1.6.3. Fractal techniques in classification 


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the recognition of skin samples to determine the age of a 

person. They are also used in fingerprint recognition and 

fabrics processing. 

1.7 Conclusion 

The fractal theory is still descriptive rather than inferential. 

There are a lot of application for fractals in different fields. 

The basic concepts and applications of fractals are discussed 

in this chapter. 

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