Fractal Signals Characterization Using Fractal Dimension Approach
B. Vojnović, A. Maksimović
Laboratory for Stochastic Signals and Processes Research, Electronic Division
Ruđer Bošković Institute
Bijenička 54, Zagreb, Croatia
Phone/Fax: +385 1 4680 090, E-mail: vojnovic@irb.hr; maks@faust.irb
Abstract – In study of many natural phenomena, as well as
dynamic systems and processes, we have to analyze and
process obtained experimental informations (data). It appears
often, that these data could be beter approximated by fractal
structures (functions) rather by ordinary diferentiable
functions. By means of Mathematica program we have
extracted fractal sets from choosen complex picture formats.
The fractal dimension of extracted sets of points, calculated by
box-counting (BC) method, shows good agreement with values
calculated directly from known dynamics.
I. INTRODUCTION
In recent years, many natural phenomena, as well as
complex systems and processes, of interest to science and
technology, have been quantitatively analyzed and
characterized using idea of fractal [1, 2, 3]. It was shown
that these phenomena display fractal features, when plotted
as a function of time or expressed as two or threedimensional structures (pictures).
Mandelbrot, often considered as the father of fractal
geometry, derived word "fractal" from Latin "fractus",
which means "fragmented" or "irregular". Latin verb
"frangere" also means "to break" or "to create irregular
fragments". In his original essay he define a fractal to be a
set with fractal (Hausdorf) dimension, strictly greater than
its topological dimension.
The set F which is considered as a fractal has the
following features:
It has fine structure, which assures to see details on
arbitrary small scales.
F could be not described in classical geometry way,
due to high irregularity, both globaly and localy.
In most practical cases F is defined in a very simple
mathematical way.
Often it is characterized by self-similarity.
Ussualy, the fractal dimension of F is greather than
its topological dimension.
Most popularly known fractals are two-dimensional
«beautiful» pictures (Mandelbrot, Julia sets, Koch
snowflake, Serpinski triangle, «ferns» etc.). Zooming any
part of them, with arbitrary scale, we get a picture which
resembles the whole set.
Fractal functions (signals) have some common
characteristics with elementary "Euclidean" functions, they
have geometrical character, they can be represented by
formulas, and they can be computed. However they are
non-smooth, they have no derivative at any point and they
have noninteger fractal dimension.
Strictly speaking, there are no true fractals in nature (in
the mathematical sense), but it is the case with functions in
clasical geometry too.
Ther are two particular fractal structures of our interest:
1. Structures that could be represented and described by
"one dimensional" fractal functions in time (signals and
time-series). These functions are mostly derived from
experimental data, and are important in analysis of
complex processes and systems in electronics and
information techologies, economy and finances,
bioinformatics, physics and chemistry, etc.
The examples of processes and systems, described by
such functions include, among many others:
- noise in electronic devices and systems,
- "one-over-f noise" in many natural as well astechnical
systems and processes such as:
DNA sequences, heart-beat phenomena, music and
speech, traffic flow, financial data, neuro systems,
geophysical records, radioactive decay, written
language etc.
2. Two-dimensional fractal structures (images) that appear
in picture simulation and computer graphics, in study of
ecological systems, in fluidic dynamics ,in written language
analysis, etc.
The analysis of these structures includes feature
extractions, characterization and compression and pattern
recognition.
It could be emphasized, that the theory and practice of
fractals is complementary connected to chaos theory and
application, because many representations of chaotic
phenomena have a fractal structure. These totaly irregular
structures are often generated by nonlinear dynamic
processes and systems, characterized as deterministic ones,
whose anlytical expressions are known or not.
From the technical point of view, it is sufficient to know
some parameters of the processes and systems. One of the
most important parameter is the fractal dimension.
II. THE CALCULATION OF FRACTAL
DIMENSION
Fractal dimension, as the measure of the fractal structure
complexity, is an objective means to characterize and
compare fractals. It can be defined often in connection with
real-world-data and could be measured approximately
using the data from experiments.
For example, it is reported in heart-rate-variability
signal analysis that the fractal dimension reduction is
indication of loss of the heart-system complexity,
associated with orthostatic stress.
In the theory and practice of fractal structures analysis,
several definitions (approaches) to fractal dimension
calculation are in use.
Hausdorff dimension was defined by the minimum
number of balls N(ε) of radius ε, required to cover bounded
fractal set Θ.
Then the Hausdorf dimension is defined as
ln N  
 0
ln 
d     lim
(1)
Information dimension is expressed by
N
  lim
 P    ln P  
i
1
i
(2)
ln 
 0
where Pi is the probability of finding a point of fractal set
in the i-th cube of size ε.
Correlation dimension is defined through correlation
integral using the similar relation as in (1)
  lim
 0
ln C  
ln 
(3)
where the correlation integral is defined as
N 
 H   x
N
C    lim 1 N 2
i , j 1,i  j
i
 xj

(4)
where H denotes Heaviside step function.
Box-Counting dimension also called Capacity
dimension is of our dominant interest, because its
suitability for computing. We compute the box-counting
dimension from a grid of equally sized elements (boxes)
with edge size ε that is superimposed (covers) on a fractal
set (image). We than count how many boxes in the grid
contain part of the fractal. During the process of counting
the boxes gets smaller, but their number increases, covering
the same area of fractal image. The box-counting
dimension is than
ln N  
  0 ln 1 
D  lim
(5)
It should be noted , finally, that the following relation is
valid
   D
(6)
III. EXTRACTING FRACTAL SET FROM PICTURE
In practice, one often has a picture of complex structure in
some of the available electronic (digital) formats, such as
portable bitmap (pbm), CompuServe graphics interchange
format (gif), Microsoft windows bitmap (bmp) , or some
other well known format.
Pictures in one of these formats can be obtained by
various means, for example one can obtain the picture with
scanner or digital camera.
If the object in the picture is too complex, or if we need
only a segment of the picture, we can change or extract part
in some of the popular software for image analysis. For
example, we can convert a picture to a gray scale, or
change the format.
Mathematica program recognize wide range of a picture
formats, and one can import the picture with command
Import.
We wrote functions, which extract points from imported
picture with chosen color and save result in a list which
represents the coordinates of extracted points. The object
can be displayed with command ListPlot. We applied the
functions on the pictures of the few famous fractals in order
to obtain a corresponding data series. By using the BC (box
counting) software [4, 5], the fractal dimension was
calculated from the image of fractal object and compared
with values of the fractal dimension, obtained from the
corresponding dynamical systems. The fractal dimension D
of a set F can be obtained from the relation (4).
The format of the imported graphics depends on the
format of the original picture. The black and white portable
bitmap picture is in the form of a matrix which contain only
0 and 1 as the matrix elements. Position of a some point in
the matrix (row, column) determines relative coordinate of
that point. We take all relative positions with chosen value
in the matrix (0 or 1) and save result in the list as a set of
pairs of the (x, y) coordinates. The calculated list is
suitable for displaying with ListPlot command. Color
pictures instead of one value in the matrix have three
values, which represent color in the form {red, green,
blue}.
We use the function GetPictureCoordinates,which
returns a list of coordinates from the imported picture in
Mathematica, together with the Box-Counting package
(BC) to estimate fractal dimension of an object represented
as a black and white images or color image.
This approach allows implementation for wide range of
picture formats. Another benefit is that the data sets
obtained from images, can be manipulated as a List objects
and compared to the values calculated from corresponding
dynamical systems or maps.
The application of the function on the image in bmp
format gives us the data series of the fractal Sierpinski
triangle as the first example.
We calculated the capacity dimension using the same
parameters and procedure as before. The calculated
capacity dimension for IFS is D=1.5860.002. These
results are in good agreement with analytical calculation
for fractal dimension of the Sierpinski triangle which is
log(3)/log(2) = 1.58496.
The second example is Henon map,
xi 1  1  y i  ax 2 , y i 1  bx
Fig. 1. The Sierpinski triangle generated with ListPlot command
for the data series, obtained from picture of the attractor.
The data series are displayed in Fig. 1. with ListPlot
command. It is evident from Fig. 1. that there is no visible
difference with the original image of the Sierpinski triangle
fractal.
Using the function CountBox we obtain a list of
occupied boxes nb and coresponding probabilities for
length scales 1/ε, ε=2k, k=1, 2,......,12. The capacity
dimension and the figure of the data fit is obtained with the
command CapacityDim[nb,2,7], where nb is a listof
occupied boxes, while second and third argument
determine range of the length scales ε=2k, k=2,....,7 used in
the fit. Calculated capacity dimension is D= 1.610.02.
(see Fig.2)
lnN()
(8)
where a = -1.4 and b = 0.3. We iterate Henon map 10000
times from the initial point (0,0) and exclude first 500
transient points with the command iterate from the BC
package. The number of occupied boxes and probabilities
is obtained with the function CountBox for length scales
1/ε, ε = 2k , k = 1,....10.
The fit of the occupied boxes vresus ln(1/ε) for the range
of the length scales ε = 2k , k = 2,...9 determines the value
of the capacity dimension D = 1.210.01. The black and
white picture in the pbm format is generated with the
gnuplot program from the dataset obtained with function
iterate.
The data set isextracted with the function
GetPictureCoordinate from the image as in the previous
example (see Fig. 3.).
In Fig. 4. it was shown the least square fit of the number
of occupied boxes versus log( 1/ε ) for the coordinates
obtained from the picture. The value of the slope
determines fractal dimension D=1.210.02. The calculated
capacity dimension in this example is the same for the
dynamical system and the data series obtained from the
image.
ln(1/
)
Fig. 2. Plot of lnN(ε) versus ln(1/ε) for Sierpinski triangle. Full
line shows least square fit.
To compare estimated value of the fractal dimension
with the value calculated from the dynamical system, we
construct the fractal Sierpinski triangle by means of an
Iterated Function System (IFS). It is obtained by using the
following set of a three affine transformations:
xi 1  xi 2  ak , yi 1  yi 2  bk
(7)
with k = 1,2,3, where a1=a2=b1, a3=b2=50, and all
transformations were applied with equal probability.
One way to generate the Sierpinski triangle with IFS is
to use random iteration algorithm [4]. One of the mappings
is chosen at random and applied at point (0,0) producing a
new point. This random procedure is reapplied to the new
point and so on. Function ifs from the BC package [5]
implements this algorithm, which is also called chaos
game. Excluding first 1000 transient points with command
Drop, the data series are obtained with following command
Drop{ifs[mmap, 100000], 1000}, where mmap is affine
transformatiom [6].
Fig. 3. The Henon attractor, generated with ListPlot command for
the data series, obtained from the image of the attractor.
lnN()
ln(1/)
Fig. 4. Plot of lnN(ε) versus ln(1/ε) for Henon attractor
Our final example is Fern Attractor. The data series are
obtained from the picture in the jpg format. Fig. 5. shows
the Fern Attractor displayed with ListPlot command.
We applied the functions to the three well known
attractors, and by using be package the fractal dimension
was calculated. Although the precision depends on the
resolution and format of the picture, the values for fractal
dimension are in good agreement with values calculated
directly from dynamical system.
IV. CONCLUSION
Fig. 5. Fern attractor, generated for data series, obtained from jpg
picture.
Associated fractal dimension was calculated as in the
two previous examples and we got D = 1.768+0.02. (see
Fig. 6.)
lnN()
ln(1/)
Fig.6- Plot of lnN(ε) versus ln(1/ε) for Fern attractor
We
have
implemented
the
function
GetPictureCoordinates which extract the relative
coordinates for a points with desired color and save the
result in a list.
By means of Mathematica programming we
implemented a method for extracting a fractal set of points
from picture.
The method was applied to three well known attractors,
and by using BC package, the fractal dimension was
calculated.
Although the precision depends on the resolution and
format of the picture, the values for the fractal dimension
are in good agreement with values calculated directly from
associated dynamical systems.
LITERATURE
[1] K. Falconer: FRACTAL GEOMETRY-Mathematical
Foundations and Applications, John Willey and Sons Ltd.,1995.
[2] M. Barnsley: Fractals Everywhere, Academic Press, New
York, 1988.
[3] B. B. Mandelbrot: The Fractal Geometry of Nature, Freeman,
San Francisco, 1982.
[4] A. Maksimović, S. Lugomer, B Vojnović: Fast procedure for
estimatin capacity dimension of the fractal object by the box
counting, Fizika, B 4 (1995); 29.
[5] A. Maksimović, B Vojnović: Box Counting in Mathematica,
Proceedings PrimMath2001, 1st Meeting, Mathematica in
Science, Technology and Education, Zagreb, Croatia ,September
2001, pp.207-214.
[6] P. Grassberger: Generalized dimension of strange attractors,
Phys. Let. A 97 (1983), 6, 227.