Fractals a Useful Beauty in Computation

8/10/2019 Fractals a Useful Beauty in Computation

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CVPR 2013 Submission #236. CONFIDENTIAL REVIEW COPY. DO NOT DISTRIBUTE.

National Conference on Technological Trends (NCTT)-2014

College of Engineering Trivandrum

August 2014

Fractals: Useful beauty in Computation

Paper ID 236

Abstract

Fractals are the extremely complex mathematical struc-

tures found in nature. These are beautiful, infinitely detailed

self-similar mathematical constructs. Complexity hidden in

this beauty are useful in many practical areas of compu-

tation including Communication, Compression technology,

Information security, Artistic graphics ,Health care etc.

This paper is a brief study of Applications of Fractals in

some of the areas in computation.

1. Introduction

From ancient days on wards we used mathematics to cre-

ate geometrical shapes, buildings, and study regular motions

of planets. This geometry was recognized as Euclidean ge-

ometry. Using this geometry we created circles,rectangles,

pyramids,smooth surfaces,etc. The basic assumptions that

underlies classical mathematics is that everything is ex-

tremely regular that You can reduce everything to straight

lines,circles ,triangles, pyramids smooth surfaces etc. But

Classical mathematics is only suited to study the world we

have created.The things we built using classical mathemat-

ics.The patterns in nature, the things that already were there

before we came on to the planet the trees, the plants, the

clouds the weather system those were outside of mathemat-

ics. They were different structures that classical mathemat-

ics cant explain. At this point nature and text books dif-

fer.[1]

This was the scenario until the 1970s when Benoit man-

delbrota German Mathematician introduced his new geom-

etry in his book Fractal Geometry of Nature where he

introduced a new kind of geometry, the Fractal Geometry.

He opened roughness for investigation, found order in dis-

order.According to him all we need to do is look at these

patterns of nature in the right way and we can apply math-

ematics.You can write down formulas that can describe the

clouds the flowers and plants its just they are different kinds

of formulas and they give you a different kinds of geometry.

Being one of natures biggest design secrets, fractals

found many more applications in our day today life. You

can find it in the rain forest, in medical research, artistic

graphics textile designs, in the movies and it is all over the

world of wireless communications.Fractals are in our lungs,

kidneys and blood vessels,flowers, plants weather systems

in the rhythms of the heart, in the very essences of life.[2][3]

This paper is a brief study of Fractals and its application

in the field of computation. Rest of the sections are orga-

nized as follows: Section-2 introduces basic ideas around

fractal theory. Section-3 is the study of Famous fractal

structures Julia and Mandelbrot Fractals. Section -4 is the

Applications of Fractals in different contexts and Section -5

concludes the paper.

2. Fractals

Mandelbrot coined the name Fractals to denote irregu-

lar jagged shapes found in nature. Fractals, at its origin, is

based on pure mathematics. A fractal is an object or quan-

tity that displays self-similarity, on all scales. The object

need not exhibit exactly the same structure at all scales, but

the same type of structures must appear on all scales.

2.1. Fractals Brief History

In 1958 IBM was looking for solution of a problem of

great concerns to the company.IBM engineers were trans-mitting computer data over phone lines. But sometimes the

information was not getting through.sometimes The lines

become extremely noisy.Mandelbrot being one mathemati-

cian assigned to study the cause, graphed the noise data

and what he saw surprised him. Regardless of time scale

the graph looked similar.One day, one hour, one second it

didnt matter it looked about the same. The strange pat-

tern reminded him of something he seemed when he was

a young man.Mathematical mystery that dated back nearly

a 100 years. The German mathematician Georg cantor

created the first of them in 1883. He just took a straight

line, broke that line into 3rd and erased the middle third. So

we are left with two line at each end.Then take those two

lines take out the middle third.And we will do it again and

again. Our common intuition is that if we throw everything

away there will be nothing left.its not the case with Cantor

Set(given in Fig.1).There are infinitely many points left.As

you zoom in onto the cantor set the pattern remains the same

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much like the noise pattern Mandelbrot seen in IBM.

Figure 1. Cantor Set

Another strange shape was found put forward by the

Swedish Mathematician Helge von Koch in a 1904 pa-

per titled On a continuous curve without tangents, con-

structable from elementary geometry[4].Koch said that

when you start with an equilateral triangle, one of the classi-

cal Euclidean geometrical figures, by taking each side sub-

stitute two pieces.They are now longer than the original

piece.For those pieces substitute two pieces each figure will

be longer than the original.If this process is repeated over

and over again,you get the strange shape. But now each linehas a triangular bump on it.Each time we break it the line

gets longer.If we repeat it infinitely many times what you

end up with is something that is infinitely long.This struc-

ture was named Koch Snowflake(given in 2)It was a paradox

to traditional mathematics.To the eye the curve appears to

be perfectly finite.But mathematically it is infinite. Which

means it cant be measured.Because it made no sense people

were thinking measurements in terms Euclidean geometry.

But the Koch curve turns out to be crucial to a measurement

problem, the length of the coast line.

Figure 2. Koch Snowflake

In the 1940s British scientist Luis Richardson had ob-

served that there can be great variations between measure-

ments of a coast line. That depends on how long your yard

stick is.If you measure coast line with 1 mile yard stick,

you get so many yard sticks, gives you so many miles.If

you measure with one foot yard stick,it turns out that they

are longer. Every time you use a shorter yard stick you get

a longer number, because you could always find finer in-

dentations.Mandelbrot saw that the finer indentations in the

Koch curve were precisely what was needed to model coast

lines. He wrote a very famous article in science magazine

called how long is the coast line of Britain?[5]. Man-

delbrot said in the paper a coast line in geometric terms

is a fractal.And though he know he couldnt measure the

length, he suspected he could measure something else, its

roughness. To do that require rethinking one of the basic

concepts in mathematics- dimension. What we would thinkof dimension as in normal geometry, one dimension is the

straight line, two dimension is the box and three dimension

is a cube.But there are other structures that have dimension

in between two and three, the fractional value for dimen-

sion. The rougher the objects are the higher there fractal

dimension.

Mandelbrot extended his study to another set of mathe-

matical monsters, a problem introduced during world war-

I by a French mathematician Gaston Maurice Julia. This

study created a new era in Fractal Geometry.

3. Julia sets and Mandelbrot Set

3.1. Julia Sets

The Julia set is named after the French mathematician

Gaston Julia. Julias concept was to observed the behavior

of the orbit of a complex number under iteration of a special

function in Complex plane. That is, start with a complex

number z0 , visualized as a point in the plane, and apply f

to z0 . The resulting value is fed back into the same func-

tion f to obtain a new complex number z1. This again is fed

back into f to obtain z2, and so on. The resulting sequence

of complex numbers z0 , z1 , z2 , ... is called the orbit of

z0 under f. This study was not recognized until 1980 when

Mandelbrot explored Julia set with the help fast computa-

tional machines to compute Julia sets.

3.2. Mandelbrot Set

Mandelbrot observed the question what happens when

you perform infinitely many iterations in Julia Set.He en-

ters the numbers from the julia set to into the points on a

graph, got 100s of pictures.Those images brought Mandel-

brot to a breakthrough, in 1980 he created an equation on

his own.One that combined all of the Julia sets into a single

image.When Mandelbrot iterated his equations he got his

own set of numbers graphed on a computer it was a kind of

roadmap of all the Julia sets and quickly become famous as

the emblem of Fractal geometry- the Mandelbrot Set.

Mandelbrot set is a mathematical set of points whose

boundary is a distinctive and easily recognizable two-

dimensional fractal shape. Images of the Mandelbrot set

are made by sampling complex numbers and determining

for each point in complex plane whether the result tends to-

wards infinity when a particular mathematical operation is

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iterated on it. Treating the real and imaginary parts of each

number as image coordinates, pixels are colored according

to how rapidly the sequence diverges, if at all.

More precisely, the Mandelbrot set is the set of values of

C in the complex plane for which the orbit under iteration

of the complex quadratic polynomial,

Zn+1 = Z2n + C (1)

remains bounded. That is, a complex number C is part of

the Mandelbrot set if, when starting with

Z0 = 0 (2)

and applying the iteration repeatedly, the absolute value of

Znremains bounded however large n gets.

Figure 3. Mandelbrot Set

For example, letting C= 1 gives the sequence 0, 1, 2,

5, 26, . . . which tends to infinity. As this sequence is un-

bounded, 1 is not an element of the Mandelbrot set. On the

other hand , c = i (where i is defined as i 2 = -1) gives the

sequence 0, i, (-1 + i), -i, (-1 + i), -i, . . .which is bounded,

and so i belongs to the Mandelbrot set.Images of Mandelbrot set shows finer details as we zoom

in. The structures obtained after zooming into particular

portions of Mandelbrot set is similar to the structure of the

entire image. It shows self similarity of fractals(part is sim-

ilar to whole) in a very detailed way.Fig. 3 illustrates ba-

sic shape of Mandelbrot Set generated in the above proce-

dure. Images of the Mandelbrot set display an elaborate

boundary that reveals progressively ever-finer recursive de-

tail at increasing magnifications. The style of this repeat-

ing detail depends on the region of the set being examined.

The sets boundary also incorporates smaller versions of the

main shape, so the fractal property of self-similarity applies

to the entire set, and not just to its parts.

4. Applications of Fractals in Computation

Fractals have vast amount of application in our day to day

life.Here some of the main areas of Applications of Fractals

are introduced. These are:

4.1. Image Compression

One of the area which Fractals theory revolutionized is

Image Compression. This area is based on Iterative Func-

tion System, which is a method of generation of Fractals.

Using IFS a lossy image compression scheme is developed.

4.1.1 Fractal Image Compression Principle(Iterated

Function System)

Its basic principle is to try and consider a given image I as

the fixed point (or attractor) of a geometrical transform T

defined on the set of all images with same size. It is de-

duced from one of the usual ways some synthetic images

are generated, fractal images. In this case, the image is de-

fined as the attractor of a given system of iterated functions

(IFS). The fixe point is obtained from any initial image, as

the limit of an image sequence that is iteratively defined.

Here we face the inverse problem, the goal is to recover

from a known image, one system that would precisely lead

to this image as the fixe point. Before we present the com-

pression method, the definition of an IFS has to be given in

a more precise way.

An IFS is a system of transforms that allows to gener-

ate a single fractal image. It is built as a set of contractive

transforms. These transforms make possible the definition

of a function T defined on the set of images with the same

size. The result obtained by applying the T transform on

an image I is computed.The image thatis obtained has some

specific properties, in particular it is autosimilar. But this

property does not always hold in any image. In order to

adapt to this default present in normal images, in the appli-

cations, the transforms Ti are limited to applications high-

lighting similarities between parts of the image. Then, in

this case, the model that is chosen is a system of partitioned

iterative functions (PIFS).

4.1.2 Compression step

From a practical point of view, the aim of the compression

phase is to determine the transforms that are part of the PIFS

having the initial image as unique fixed point( in fig4). To

achieve this the image is partitioned in sub-images R that

are called Ranges. These ranges have to be interpreted as

the result of the transform through a geometrical affine and

contractive transform T of domains D that will be called the

Domains. These Domains have to be themselves sub im-

ages of the intitial image. In a usual way, the Domains D

are twice the size of the corresponding Ranges. We have:

R = T(D). The coefficients a, b, c and d determine the geo-

metrical transforms, e and f determine the gray levels; o the

contrast and s the luminosity.Starting from an initial image,

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fractal compression process replaces the image by a system

of transformations that are each defined by 8 parameters.

Figure 4. Fractal Image Compression

4.1.3 Decompression step

In the decompression step the set of transforms that have

been previously defined is iteratively applied to any image

till the fixed point is obtained. It is assumed to be obtained

when the difference between two successive images of the

sequence is small enough.

4.1.4 Advantages and Applications of Fractal Com-

pression

Fractal image compression does not suffer from edge depen-

dency like that in JPEG compression. Furthermore, the frac-

tal method has the benefit of faster decompression speed,

having done most of the computation during the compres-

sion step, while giving equal or better compression ratio.

These advantages mean that fractal image compression is

well suited for applications requiring fast access to high-

quality images. The most notable of applications using frac-

tal compression is Microsofts best-selling Encarta multi-

media encyclopedia.[5] Encyclopedia Encarta, includes on

one CD-ROM seven thousand color photographs which may

be viewed interactively on a computer screen. The images

are diverse; they are of buildings, musical instruments, peo-

ples faces, baseball bats, ferns, etc. What most users do not

know is that all of these photographs are based on fractals

and that they represent a (seemingly magical) practical suc-

cess of mathematics.[6]

However, fractal image compression is not without its

limitations. For example, its lengthy compression step will

preclude it from being used in applications where it is es-

sential to be able to send out the compressed images with

minimal delay, such as live broadcast of video over a net-work, tele-conferencing, and videophone.[5]

4.1.5 Other Implementations

Using this Fractal image Compression scheme a paper is

proposed in 2002 as Writers Authentication and Fractal

Compression. In this paper unique identity of Writers are

recognized using IFS.[7] In A Secure Fragile Watermarking

Algorithm Based on Fractal Compression and Differentials

Record Theory a new watermarking scheme was introduced

based on this compression theory.[7]

4.2. Cryptography

Complexity within simplicity is the key feature of Man-

delbrot Set. This feature can be exploited for the purpose

of creation of Cryptographic algorithms. In cryptography a

new algorithm for public key encryption is proposed based

on Complexity of Mandelbrot Fractal Set named Public

Key Cryptographic system using Mandelbrot Sets.[8] In

this scheme relation between Mandelbrot set and Julia set

is utilized for Public key cryptosystem design. One major

advantage of Mandelbrot set based cryptosystem is that it

is infeasible to mount attack, since there is no way to relate

output of encryption to the key.

In 2009 an image encryption algorithm is introduced

based on the principle of complexity inherent in the pro-duction of Mandelbrot Set, by Rozouan Valerji in Modulo

Image Encryption Algorithm Utilizing Mandelbrot Set[9].

He proposed a scheme of new image encryption technique

which utilized Mandelbrot set parameters as the key to en-

crypt the image. This image encryption technique may find

several applications in Image encryption in smart phones

etc.

In E-Banking Security: Mitigating Online Threats Us-

ing Message Authentication Image (MAI) Algorithm[10] a

new digital signature scheme named Message Authentica-

tion Image(MAI) is introduced for providing effective secu-

rity to the e-banking environment.

4.3. Communication

In communication area a paper is presented in 2009 la-

beled On the Fractal Characteristics of Internet Network

Traffic and its Utilization in Covert Communications and

On Fractal Quantizers for a Class of Chaotic Signals Used

in Digital Communication.

5. Conclusion

Fractals being one of natures biggest design secrets, it

inherits simplicity and complexity at the same time. It has

wast amount of applications in many of the areas in our

day to day life, especially in the field of computation. In

Image compression and Cryptographic perspective fractals

have large set of applications.It found application in many

more fields including health care,artistic graphics,films and

entertainment,etc. Its relation with Cellular automata like

models of computation gives clues of its relation to natures

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models of creation. It is the area which needed to be further

explored.

References

[1] Benoit.B. Mandelbrot, The Fractal Geometry of Na-

ture, W.H. Freeman,New York, 1983.

[2] Benoit.B. Mandelbrot, Fractals: Form,Chance and

Dimension, W.H. Freeman,New York, Sept. 1977.

[3] Michael Fielding Bransley, Fractals everywhere,

Academic Press Professional,New York,2000

[4] Helge von Koch, On a continuous curve without tan-

gents, constructible from elementary geometry, 1904

[5] Benoit.B. Mandelbrot, How long is the coast of

Britain?Statistical self-similarity and fractional dimen-

sion, Science, 1967

[6] A. SEROPIAN,Pr. N. VINCENT,Writers Authentica-tion and Fractal Compression, Proceedings of the

Eighth International Workshop on Frontiers in Hand-

writing Recognition (IWFHR02)0-7695-1692-0/02

[7] Whitfield Diffie and Martin E.Hellman, New Direc-

tions in Cryptography,

[8] Antonio San Martino, Xavier Perramon , Defending

E-Banking Services: Antiphishing Approach, The

Second International Conference on Emerging Security

Information, Systems and Technologies.

[9] C.Ronchi, A.Khodjanov, M.Mahkamov, S.Zakhidov,

Security, Privacy and Efficiency of Internet Banking

Transactions,

[10] Bernard Menezes,Network Security and Cryptogra-

phy, Cengage learning, 154-161 .

[11] William Stallings,Cryptography and Network Secu-

rity: Principles and Practice, Prentice Hall- 5th

Edition, 15-19 .

[12] Charlie Kaufman,Radia Perlman, Mike Speciner,

Network Security: Private Communication in a Public

world, Prentice Hall, 235-238 .

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Fractals a Useful Beauty in Computation

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