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International Journal of Science and Technology Volume 1 No. 5, May, 2012

IJST © 2012 – IJST Publications UK. All rights reserved. 234

Disk and Box Dimensions: Selected Case Studies of Fractals

with Ifs Codes

Salau, T. A.O.1, Ajide, O.O.

2

1, 2 Department of Mechanical Engineering,

University of Ibadan, Nigeria

ABSTRACT

Disk and box count methods enjoyed higher popularity among known methods for estimating the fractal dimension of

computable fractals in dimensional Euclidean space for their relative ease of implementation among others. The present study

investigated the suitability of disk and box count methods using Monte Carlo approach for the estimation of fractal dimension

of some selected fractals with IFS.

Six (6) fractals with IFS were identified from literature. Algorithms (coded in FORTRAN) based on Monte Carlo approach

was developed for disk and box count methods. Common factors to all studied fractals are seed value for random number

generation (9876), start coordinates (1,0.5), transient solutions (1000), steady solutions (5000), total number of corresponding

scale of examinations (20) and ten (10) trial times. The FORTRAN programme computes both transient and steady solutions

of fractals with IFS, Estimated dimension and other relevant quantities of this study while graphs were plotted using

Microsoft office Excel 2003.

Programming for the disk overlay is less skill demanding than box overlay as experienced from this study. Estimated

dimensions vary from transient to steady for cases. Dimension variation transient from lower dimension value to higher

steady dimension value except for some cases investigated with box method. Estimated disk dimension was consistently on

the lower side of actual dimension with absolute relative error (%) range of 0.5 to 19.6 for cases. Similarly 66.7 percent of

estimated box dimensions were on the lower side of actual dimension with absolute relative error (%) range of 0.9 to 17.2 for

cases. The average absolute error (%) for disk and box methods was 6.7 and 6.8 respectively. Actual dimension was

sandwiched between estimated disk dimensions and box dimensions in 33.3% for cases.

Preference can be given to use of disk count method for solving fractal dimension problems for its capability to estimate

fractal dimension consistently and the fact that the method is averagely less error prone compared with box method.

Keywords: Fractal, Fractal Dimension, IFS Codes, Monte Carlo

1. INTRODUCTION Fractals are tenuous spatial objects whose geometric

characterization includes irregularity, scale-independence,

and self-similarity (Guoqiang, 2001). Fractal dimension

can be termed as the basic notion for describing structures

that have scaling symmetry. The application of fractals for

dynamics characterization is presently attracting more

researchers interest in almost all disciplines (engineering,

medicine, agriculture, languages e.t.c ).A critical study of

a book by Edward(1996) on invitation to dynamical

systems reinforces the interest in fractal characterisation.

Dimensionality curse and dimensionality reduction are

two key issues that have retained high interest for data

mining, machine learning, multimedia indexing and

clustering (Caetano et al, 2010). The authors presented

fast, scalable algorithms that quickly select the most

important dimensions for a given set of n-dimensional

vectors. The major idea of this research paper is to use the

fractal dimension of a dataset as a good approximation of

its intrinsic dimension and to drop attributes that do not

affect it. The authors applied this method on real and

synthetic datasets where it gave fast and correct results.

Speech signals can be described as being generated by

mechanical system with inherently nonlinear dynamics

(Martinez et al, 2003). The hallmark of Martinez et al

paper was to describe the complexity of speech signal

using the fractal dimension of variety of Spanish voiced

sounds (vowels, nasals) and unvoiced sounds (fricatives).

The authors concluded that the fractals measure expand

International Journal of Science and Technology (IJST) – Volume 1 No. 5, May, 2012


the distinguishing features in characterizing voiced and

unvoiced sounds that lead to better speech recognition

performances. Jorge (2010) investigated the fractal

dimension of the leaf vascular system of three Relbunium

species. Tree ramification, root and venation systems are

some examples of patterns studied using this geometry

and fractal dimension has quantified and graded the

complexity of these structures. The paper focus on

determination of fractal dimension of the leaf vascular

system in three species of Relbunium. The results of the

study showed that significant differences in the fractal

dimension of three species (1.387 in R.megapotamicium,

1.561 in R.hirtum and 1.763 in R.hypocarpium). This

implies that this type of measurement can be used as a

taxonomic character to differentiate species and to

quantify and grade venation of leaves. According Aura

and Felipe (2005), fractal codification of images is based

on self-similarity and self affine sets. The codification

process consists of construction of an operator which will

represent the image to be encoded. The authors observed

that the major disadvantage of the automatic form of

fractal compression is its encoding time. Most of the time

spent in construction of such operator is on finding the

most suitable parts of the image to be encoded. The paper

showed how a new idea for decreasing the encoding time

can be implemented. A vivid description and quantitative

measurement of irregular fragments or complex shapes of

materials can be done using fractal. Alabi et al (2007) did

a fractal analysis in order to characterize the surface finish

quality of machined work pieces. The results of the study

showed an improvement in the characterization of

machine surfaces using fractal. Bozica has previously

used fractal analysis in the characterization of grinded

ceramics surface textures by surface profile fractal

dimension .In the recent study (Bozica et al, 2009),the

authors did a fractal analysis for biosurface comparison

and behaviour prediction. Skyscrappers method was

employed for calculating fractal dimension of surface

using the image processing toolbox as well as a

customized algorithm of matlab environment. Scanning

probe microscope was used in recording the surface as an

image. The results of the study showed that fractal

dimension values confirm changes of the surface

roughness during cleaning and wearing process. The

authors concluded that examination of real surface

roughness could provide comparison and functional

behaviour prediction. David (1999) employed wavelet

packet transforms to develop an engineering model for

multi-fractal characterization of mutation dynamics in

pathological and non-pathological gene sequences. The

work examines the model’s behaviour in both

pathological (mutations) and non-pathological (healthy)

base pair sequences of the cystic fibrosis gene. The results

of the study suggest that there is scope for more multi-

fractal models to be developed. Hagiwara et al (1999)

research paper is based on the fractal application of the

fractal dimension in the evaluation of cutting ability of

grain edge. The authors observed earlier that grain edge

shape plays an important role in cutting ability and

microscopic evaluation of grain edge shape which must

be taken into consideration for better abrasive machining.

In this study, the role of fractal geometry of abrasive grain

in lapping is evaluated. The results of the study showed

that when grain surface consists of micro edges and has a

large fractal dimension, the surface finish becomes better.

The paper concluded that fractal dimension that

characterizes the grain edge geometry can be a better

index of cutting ability. Alabi et al in 2008 did a study on

fractal dynamics of a bouncing ball on accelerating lift

tabletop with both constrained to vertical motion. The

results of this study further show case how fractal can be

used in characterization of engineering dynamics. It is

generally accepted that DC positive corona discharge is a

complex phenomenon and its discharge figures undergoes

a great change when compared to negative corona

discharge (Sato et al,1999).The authors research paper

describes the experimental results and analysis of the

fractal dimension of DC positive corona discharge figure

using the fractal theory. It is evident from the study that

the fractal dimensions of discharge figures is estimated

from about 1.1 to 1.6.The fractal behaviour of the

tungsten phase boundary of WNiFe presintered materials

and powder allgomerates can be characterized by

developing a box dimension method (Marchionni and

Chaix, 2003).Zhao et al ( 2011) explained that the service

performance of precision parts has close relationship with

the surface micro-topography. In this recent paper of the

authors, fractal theory is applied to describing the ground

surface quality in ultrasonic vibration grinding and the

influence of different grinding parameters on the fractal

dimension. Equally, the relationship between the fractal

dimension and surface roughness is analyzed. The

experimental results revealed that there is a close

relationship between the fractal dimension and surface

roughness. The author concluded that the reliability of

surface quality described by the fractal dimension is

verified through measurement of the bearing ratio of the

surface profile. Mazza et al (2011) presented a method for

the quantitative determination of a morphology descriptor

of free clusters with complex nanostructure. The finding

of the authors showed that the clusters have an open

fractal-like structure with fractal dimension depending on

their thermal history during growth and evolving towards

softer aggregates for longer residence times where lower

temperature conditions characterize the growth

environment.

Existing literature survey shows that disk and box

methods are among the most popular techniques for



estimating the fractal dimension of computable fractals in

dimensional Euclidean space. The major reason attributed

to this popularity is relative ease of implementation.

Despite this popularity, there is still a research gap. The

pertinent question on the suitability of box and disc

techniques for characterizing fractals with IFS codes has

not been addressed.

The objective of this paper is to demonstrate the

suitability of disk and box dimension methods for

estimating the dimension of some selected fractals with

IFS codes. The implementation of disk and box counts

methods are achieved using Monte Carlo approach.

II. MATERIALS AND METHOD

Six (6) fractals with known IFS functions were identified

from literature (see table 1 for their details). The Actual

dimension of these fractals indicated in table 1 was picked

from literature. In view of the fact that fractal properties

can be obtained from power related function informed

equation (2) given below:

XD

Y

(1)

Proportional related equation (1) can be re-written as in

equation (2) below.

XD

KY (2)

Where for the present study we let:

X = Number of Disks or Boxes (same size) used to

overlay the characteristic length (AB) of a fractal image in

2-dimensional Euclidean space.

Y = Number of either Disk or Box required to overlay

fractal image with corresponding characteristic length

(AB) in 2-dimension Euclidean space.

D = Fractal dimension. This will be referred in this study

as Estimated Disk dimension for Disk count method and

Estimated Box dimension for Box count method.

K = Constant of proportionality

Take logarithm (any base) of both sides of equation (2) to

make it a linear function, this yields equation (3).

CDxy (3)

Where y, x, and C are logarithms of Y, X, and K

respectively.

Two separate algorithms based on disk count and box

count method and which incorporated equation (3) was

developed to enable estimation of dimension for selected

fractals of this study. The algorithms was coded in

FORTRAN language and can compute transient and

steady solutions of fractals with IFS, implement disk and

box count methods using Monte Carlo approach,

prescribe fractal dimension based on equation (3) and

other relevant quantities required for results reported in

this study. Microsoft office Excel 2003 was used for the

plot of all graphs used in this report.

III. RESULTS AND DISCUSSION

Figures and tables were used to present the results of this

study.

Common Parameters: Iseed =9876=seed value for

generating random number

Coordinate of start point = (1, 0.5)

Transient solution=first 1000 solution points

Steady solution=First 5000 solution points after transient

solution.

Table 1: Fractals and Associated IFS Codes

Function Koch

1 0.3333 0.0000 0.0000 0.3333 0.0000 0.0000

2 0.1667 -0.2887 0.2887 0.1667 0.3333 0.0000

3 0.1667 0.2887 -0.2887 0.1667 0.5000 0.2887

4 0.3333 0.0000 0.0000 0.3333 0.6667 0.0000

Function Sierpinski Triangle

1 0.5000 0.0000 0.0000 0.5000 0.0000 0.0000

2 0.5000 0.0000 0.0000 0.5000 0.5000 0.0000

3 0.5000 0.0000 0.0000 0.5000 0.2500 0.5000

Function Gasket



1 0.5000 0.0000 0.0000 0.5000 0.0000 0.0000

2 0.5000 0.0000 0.0000 0.5000 0.5000 0.0000

3 0.5000 0.0000 0.0000 0.5000 0.2500 0.5000

Function Xmas Tree

1 0.3333 0.0000 0.0000 0.3333 0.0000 0.0000

2 0.3333 0.0000 0.0000 0.3333 0.6667 0.0000

3 0.3333 0.0000 0.0000 0.3333 0.0000 0.6667

4 0.3333 0.0000 0.0000 0.3333 0.6667 0.6667

5 0.3333 0.0000 0.0000 0.3333 0.3333 0.3333

Function Fractal-T

1 0.5000 0.0000 0.0000 0.5000 0.0000 0.0000

2 0.5000 0.0000 0.0000 0.5000 0.5000 0.5000

3 0.2500 0.0000 0.0000 0.2500 0.0000 0.7500

4 0.2500 0.0000 0.0000 0.2500 0.2500 0.5000

5 0.2500 0.0000 0.0000 0.2500 0.5000 0.2500

Function Fractal-L

1 0.5000 0.0000 0.0000 -0.5000 0.0000 0.5000

2 -0.5000 0.0000 0.0000 -0.5000 1.0000 1.0000

3 0.3333 0.0000 0.0000 0.3333 0.6667 0.0000

4 0.2500 0.0000 0.0000 0.2500 0.2500 0.5000

Length (AB) represents the longest distance between

coordinate pair of attractors. It characterises attractor and

thus vary from one attractor to the other. Referring to

Xmas tree (one of the fractal images studied) length (AB)

was found to be 1.4079unit. Thus a size 4 disk and a size

4 box for fractal image (Xmas Tree) has radius 0.1760unit

and a square length 0.3520unit respectively as can be seen

in figure 1.

Figure 1: Scattered Representation of a Disk and a Box with Common Centre (Xmas Tree).

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

-0.20 -0.10 0.00 0.10 0.20

A Disk and a Box with a Common Centre



Figure 2: Scattered Characteristic Length (AB) of Xmas Overlaid with Disks.

Figure 2 represents the characteristic length (AB

=1.4079unit) of fractal image (Xmas Tree) overlaid with

four (4) disks of radius 0.1769unit each. This disk was

referred in this study as size 4 for corresponding fractal

images studied.

Figure 3: Scattered Characteristic Length (AB) of Xmas Overlaid with Boxes.

Figure 3 represents the characteristic length (AB

=1.4079unit) of fractal image (Xmas Tree) overlaid with

four (4) square boxes of length 0.3520unit each. This box

was referred in this study as size 4 for corresponding

fractal images studied.

-0.20

-0.10

0.00

0.10

0.20

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Characteristic Length (AB) of Xmas Tree Overlaid with Disks

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Characteristic Length (AB) of Xmas Tree Overlaid with Square Boxes



Figure 4: Fractal Image (Xmas Tree) overlaid optimally with size 4 Disks

Figure 5: Fractal Image (Xmas Tree) overlaid optimally with size 4 Boxes

Xmas Tree Overlaid with 18-Disks of Size 4

Xmas Tree Overlaid with 14-Boxes of Size 4



Shown above in figures 1 to 5 is one sample each for Disk and Box methods that leads to generation of results of this study.

Table 2: Disk Count and Logarithm of Disk Count for Xmas Tree

Count of Required Disk for Xmas

Tree (AB) =X

Count of Required Disk for

Xmas Tree =Y

Logarithm of

X

Logarithm

of Y

1 3 0.0000 1.0986

2 6 0.6932 1.7918

3 12 1.0986 2.4849

4 18 1.3863 2.8904

5 24 1.6094 3.1781

6 30 1.7918 3.4012

7 39 1.9459 3.6636

8 50 2.0794 3.9120

9 67 2.1972 4.2047

10 79 2.3026 4.3695

11 83 2.3979 4.4188

12 88 2.4849 4.4773

13 102 2.5650 4.6250

14 108 2.6391 4.6821

15 113 2.7081 4.7274

16 119 2.7726 4.7791

17 134 2.8332 4.8978

18 148 2.8904 4.9972

19 169 2.9444 5.1299

20 183 2.9957 5.2095

The log-log plot and equation of line of best fit for

variables (X and Y) in table 2 is shown in figure 6.

Referring to figure 6 the disk dimension for Xmas Tree

estimated over twenty (20) scales of observation was

1.4088 with coefficient of fitness (R2=0.995).



Figure 6: Log=Log Plot of Disks Required for overlaying of Xmas Tree

Table 3: Box Count and Logarithm of Box Count for Xmas Tree

Count of Required Box for Xmas

Tree (AB) =X

Count of Required Box for

Xmas Tree =Y

Logarithm of

X

Logarithm

of Y

1 1 0.0000 0.0000

2 4 0.6932 1.3863

3 8 1.0986 2.0794

4 14 1.3863 2.6391

5 20 1.6094 2.9957

6 23 1.7918 3.1355

7 34 1.9459 3.5264

8 40 2.0794 3.6889

9 51 2.1972 3.9318

10 64 2.3026 4.1589

11 76 2.3979 4.3307

y = 1.4088x + 0.9648 R² = 0.995

0.0000

1.0000

2.0000

3.0000

4.0000

5.0000

6.0000

0.0000 0.5000 1.0000 1.5000 2.0000 2.5000 3.0000 3.5000

Logarith

m o

f Y

Logarithm of X

Xmas Tree Disk



12 81 2.4849 4.3945

13 88 2.5650 4.4773

14 93 2.6391 4.5326

15 99 2.7081 4.5951

16 108 2.7726 4.6821

17 111 2.8332 4.7095

18 117 2.8904 4.7622

19 120 2.9444 4.7875

20 136 2.9957 4.9127

The log-log plot and equation of line of best fit for

variables (X and Y) in table 3 is shown in figure 7.

Referring to figure 6 the box dimension for Xmas Tree

estimated over twenty (20) scales of observation was

1.6037 with coefficient of fitness (R2=0.9887).

Figure 7: Log-Log Plot of Boxes required for overlaying of Xmas Tree

y = 1.6037x + 0.2915 R² = 0.9887

0.0000

1.0000

2.0000

3.0000

4.0000

5.0000

6.0000

0.0000 0.5000 1.0000 1.5000 2.0000 2.5000 3.0000 3.5000

Lo

gari

thm

of

Y

Logarithm of X

Xmas Tree (Box)



Table 4: Variation of Estimated Disk Dimension with Increasing Number of Scale

Number of

Scales

Attractors

1 2 3 4 5 6

2.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.2224

3.0000 1.0000 1.2337 1.2337 1.2337 1.3293 1.3226

4.0000 1.0000 1.3020 1.3020 1.3020 1.4921 1.3675

5.0000 1.0000 1.3846 1.3846 1.3224 1.4991 1.3944

6.0000 1.0000 1.4362 1.4362 1.3253 1.5332 1.4047

7.0000 1.0179 1.4598 1.4598 1.3418 1.5378 1.4173

8.0000 1.0498 1.4726 1.4726 1.3662 1.5552 1.4260

9.0000 1.0557 1.4837 1.4837 1.4057 1.5683 1.4335

10.0000 1.0713 1.4916 1.4916 1.4342 1.5717 1.4364

11.0000 1.0894 1.5010 1.5010 1.4410 1.5713 1.4362

12.0000 1.1033 1.5061 1.5061 1.4371 1.5731 1.4406

13.0000 1.1139 1.5118 1.5118 1.4380 1.5714 1.4463

14.0000 1.1250 1.5148 1.5148 1.4335 1.5716 1.4496

15.0000 1.1357 1.5165 1.5165 1.4246 1.5723 1.4525

16.0000 1.1417 1.5191 1.5191 1.4140 1.5737 1.4558

17.0000 1.1462 1.5187 1.5187 1.4083 1.5751 1.4587

18.0000 1.1527 1.5194 1.5194 1.4052 1.5768 1.4618

19.0000 1.1562 1.5207 1.5207 1.4069 1.5763 1.4615

20.0000 1.1612 1.5216 1.5216 1.4088 1.5764 1.4624



Figure 8: Variation of Estimated Disk Dimension for Attractors.

Referring to figure 8 the estimated disk dimension varies

from transient to steady for all the attractors studied. All

cases transient from lower disk dimension to higher

steady disk dimension. This graph shows that Xmas tree

has the longest transient period.

Table 5: Variation of Estimated Box Dimension with Increasing Number of Scale

Number of

Scales

Attractors

1 2 3 4 5 6

2.0000 0.5850 1.3219 1.3219 2.0000 2.0000 1.5850

3.0000 0.8072 1.4496 1.4496 1.9043 2.0000 1.4779

4.0000 0.9758 1.5007 1.5007 1.8941 1.8430 1.5168

5.0000 0.9886 1.5454 1.5454 1.8642 1.7911 1.5316

6.0000 1.0005 1.5661 1.5661 1.7883 1.7698 1.5094

7.0000 1.0098 1.5652 1.5652 1.7800 1.7442 1.5213

8.0000 1.0309 1.5643 1.5643 1.7590 1.7117 1.5279

9.0000 1.0532 1.5680 1.5680 1.7529 1.7000 1.5199

10.0000 1.0808 1.5665 1.5665 1.7558 1.6922 1.5256

11.0000 1.0969 1.5711 1.5711 1.7585 1.6833 1.5243

12.0000 1.1059 1.5700 1.5700 1.7487 1.6752 1.5235

13.0000 1.1063 1.5725 1.5725 1.7348 1.6659 1.5212

14.0000 1.1155 1.5677 1.5677 1.7165 1.6583 1.5168

15.0000 1.1208 1.5680 1.5680 1.6968 1.6554 1.5119

16.0000 1.1282 1.5678 1.5678 1.6793 1.6485 1.5109

17.0000 1.1326 1.5690 1.5690 1.6590 1.6404 1.5113

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

1.80

0 5 10 15 20 25

Dim

en

sio

n

Number of Scale

Variation of Estimated Disk Dimension

Koch

S.Triangle

Gasket

Xmas Tree

Fractal-T

Fractal-L



18.0000 1.1366 1.5696 1.5696 1.6392 1.6360 1.5110

19.0000 1.1448 1.5683 1.5683 1.6182 1.6342 1.5131

20.0000 1.1461 1.5681 1.5681 1.6037 1.6314 1.5139

Figure 9: Variation of Estimated Box Dimension for Attractors

Referring to figure 8 the box estimated dimension varies

from transient to steady for all the attractors studied.

Some cases transient from lower disk dimension to higher

steady disk dimension and vice versa. This graph shows

that Xmas tree has the longest transient period.

Table 6: Actual and Estimated Dimensions

Attra

ctor

Estima

ted

Disk

Dimen

sion

Actual

Dimen

sion

Estima

ted

Box

Dimen

sion

Absolute

Relative Error

(%)

Is

Dis

k

Err

or

low

er

tha

n

Box

Err

or?

Estima

ted

Disk

Dimen

sion

Estima

ted

Box

Dimen

sion

Koch 1.1612 1.2619 1.1461 8.0 9.2 Yes

Trian

gle 1.5216 1.5850 1.5681 4.0 1.1 No

Gask

et 1.5216 1.8928 1.5681 19.6 17.2 No

Xmas

Tree 1.4088 1.4650 1.6037 3.8 9.5 Yes

0.00

0.50

1.00

1.50

2.00

2.50

0 5 10 15 20 25

Dim

en

sio

n

Number of Scale

Variation of Estimated Box Dimension

Koch

S.Triangle

Gasket

Xmas Tree

Fractal-T

Fractal-L



Fract

al-T 1.5764 1.5850 1.6314 0.5 2.9 Yes

Fract

al-L 1.4624 1.5284 1.5139 4.3 0.9 No

Average absolute error (%) 6.7 6.8

Table 6 refer the estimated disk dimension was

consistently on the lower side of the actual dimension and

the absolute relative percentage error was between 0.5 and

19.6. The average absolute error (%) for the disk method

was 6.7 while it was 6.8 for the box method.

Estimated box dimension was four (4) times out of six (6)

on the lower side of actual dimension and two (2) times

out of six (6) on the high side. Actual dimension was two

(2) times sandwiched between estimated disk and box

dimensions which amount to 33.3% chance. In term of

relative absolute percentage error disk and box methods

are equally good for use to estimate dimension of the

selected attractors.

CONCLUSIONS

This study has established that disk method is highly

consistent and averagely less error prone compared with

its counterpart box method. Estimated dimension using

box method is consistently on the lower side of actual

dimension with absolute relative error range of 0.5% to

19.6%. The worst error being 19.6% and it’s occurred

only once in six cases.

REFERENCES

[1] Alabi, B.; Salau, T.A.O. and Oke,

S.A.2007.Surface Finish Quality

Characterization of Machined Work Pieces

Using Fractal Analysis. MECHANIKA, Nr.2

(64), pp. 65-71, ISSN 1392-1207.

[2] Alabi, B.; Salau, T.A.O. and Oke, S.A.2008.

Fractal Dynamics of a Bouncing Ball on

Accelerating Lift Tabletop with both Constrained

to Vertical Motion .ISSN 1392-

1207.MECHANICKA.Nr.1 (69).

[3] Aura, C. And Felipe, R.A.2005.Fractal Coding

Base on Image Local Fractal Dimension.

Computational & Applied Mathematics.ISSN

1807-0302, Vol.24, No.1.

[4] Bosica, A.B.; Zoran,D.M.;Bojan,R.B. and

Duro,L.J.K.2009.Fractal Analysis for Biosurface

Comparison and Behaviour

Prediction.University of Belgrade, Faculty of

Mechanical Engineering, Belgrade, Republic of

Serbia.

[5] T.; Agma, T.; Leejay W. and Christos

F.2010.Fast Feature Selection Using Fractal

Dimension. Journal of Information and Data

Management.Vol.1, No.1.pg 3-16.

[6] David, L.W.1999.Use of Wavelet Packet

Transitions to Develop an Engineering Model for

Multi-fractal Characterization of Mutation

Dynamics in Pathological and Non-pathological

Gene Sequences. Doctor of Philosophy

Dissertation Submitted to the College of

Engineering and Mineral Resources at West

Virginia University.

[7] Edward R. S. (1996): Invitation to Dynamical

Systems, Prentice Hall, Inc. A Simon & Schuster

Company, Upper Saddle River, New Jersey

07458, pp.231-316.

[8] Guoquian, S.2001. Fractal Dimension and

Fractal Growth of Urbanized Areas. International

Journal of Geographical Information

Science.ISSN-1365-8815.Print ISSN 1362-

3087.http://www.tandf.co.uk/journals.

[9] Haggiwara, S.; Obikawa, T. And Tanase,

R.1999.Evaluation of Cutting Ability of Grain

Edge Based on Fractal Dimension. Journal of

Japan Society for Precision

Engineering.F0268A, ISSN-0912-0289, Vol.65,

No.9.pg.1350-1354.

[10] Jorge, E.A.M.; Rinaldo, P. and Ricardo,

P.2010.Fractal Dimension of the Leaf Vascular

System of three Relbunium Species

(Rubiacease).

[11] Marchionni, C. And Chaix, J.M.2003. A Fractal

Dimension Method for Characterizing WNiFe

Presintered Materials.

[12] Martinez, F.; Guillamon, A. and Martinez, J.J.

2003.Vowel and Consonant Characterization

Using Fractal Dimension in Natural Speech.

Departmento de Matematica Applicada y

Estadistica, Universidad Politenica de

Cartgena,Spain. [email protected].

[13] Mazza, T.; Devetta, P.M.; Bongioruo, G. and

Coreno, M.2011. Accessing the Fractal

mailto:[email protected]



Dimension of Free Clusters in Supersonic

Beams. New Journal of physics.

http://www.njp.org/

[14] Sato, Y.; Okumura, K. and Kawamura, T.

1999.Prediction of Fractal Dimensions of DC

Positive Corona Discharge by Fractal Theory.

Paper of Technical Meeting on Electrical

Discharge, IEE, Japan.Vol.ED-99, No.60-68, pg

13-18.

[15] Zhao, B.; Zhao,M. and Wang,Y.2011.Research

on the Surface Quality in Two Dimensional

Ultrasonic Vibration Grinding Based on Fractal

Theory. Advanced Science Letters, Vol. 4,

Number 3 pg702-707(6).American Scientific

Publishers.

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