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FRACTAL ANALYSIS OF THE IMAGES USING WAVELET
TRANSFORMATION
PETRA JERABKOVA, OLDRICH ZMESKAL, JAN HADERKA
Institute of Physical and Applied Chemistry, Faculty of Chemistry, Brno University of Technology,
Purkynova 118, 612 00 Brno, Czech Republic
This article describes a new method of determining fractal dimension and fractal measure using
integral (e.g. wavelet – Haar) transformations. The main advantage of the new method is the ability
to analyse both, black & white and grey scale (or colour) images. The fractal dimension (box
counting fractal dimension) evaluated using the Haar transformation offers in difference to the
classic box counting method much wider range of usability. It can be also used to determine the
fractal parameters of surfaces or volume of three-dimensional structures e.g. distribution of the mass
or electrical charge in the surface or space. The theory of the fractal structures in E-dimensional
Euclidean space is described more closely in our previous work.
1 Introduction
Fractal analysis has become one of the fundamental methods of the image analysis.
Fractal dimension D and fractal measure K [1] are the two characteristic values, which
can be determined by fractal analysis. Fractal measure characterizes the properties of
surfaces and edges of image structure, whereas the fractal dimension shows the trends of
surfaces and edges changes in relation to the details of structures themselves.
Fractal dimension value lies in the range D ∈ 0,
E , where E stands for Euclidean
space dimension (E = 2). The limits of the range of the fractal dimension values can be
also described as:
– for D → 0 the change of the structure as a function of the size of its measure is
maximum (this is characteristic for image structures where analysed structure forms
just small part of the complete image, e.g. some colour has very minor representation
in the image or the length of the edge between different colours is very short),
– for D → 2 the change of the structure doesn’t depend on the change of measure (the
analysed structure fills image almost completely, e.g. one colour has majority in the
image or the edge fills whole image – Peano curve).
Fractal measure lies in the range K ∈ 0, K max , where Kmax is total count of pixels
within the image. Using the fractal measure we are able to determine how much of the
image is covered by the analysed structure. For example fractal measure of Peano curve
is equal to the total count of the pixels within the image (or we can say that the coverage
by the edge is 100%, K/Kmax = 1 respectively). The fractal dimension of the edge between
black and white within the image is then DBW = 2.
The method most often used to determine the fractal parameters of the image is
called box counting method. This method allows analysing of black & white pictures
created by changing of threshold value of colourful image structure by specified criteria
1

2
(e.g. on RGB, HSV, HSB space) – the thresholding. The results of fractal analysis (using
the box counting method) are box counting fractal dimension and box counting fractal
measure.
This article describes a new method of determining fractal dimension and fractal
measure using integral (wavelet – Haar) transformation. The main advantage of the new
method is the ability to analyse grey scale (or colour) images without lengthy
thresholding and calculating the box counting fractal dimension D and fractal measure K
for each thresholded image in the set.
More detailed results of the analysis using this new method can be found on our web
pages http://www.fch.vutbr.cz/lectures/imagesci. The interpretation of the results
obtained from the black & white image analysis (using the new method) remains the
same as from the box counting method of black & white pictures. The main disadvantage
of the new method may be seen in smaller set of data used for regression analysis as in
box counting method. There can be obtained only 10 independent values per image with
1024 × 1024 pixels matrix that can be used to determine the fractal dimension D and
fractal measure K. Such disadvantage can be turned into an advantage as removing
redundant use of pixels minimizes the error introduced by reusing the values of wrongly
determined values of pixels.
2 Box counting method
Analysis of black & white images is usually performed by box counting method [2].
Figure 1 shows how is the method used. A net of different size (1 × 1, 2 × 2, 3 × 3, 4 × 4,
etc. pixels) is placed on the analysed image and the total count of black boxes NB, white
boxes NW and black&white NBW boxes is determined.
Figure 1. Determination of the fractal measure and fractal dimension using the box counting method.
The function of the count of pixels on the net size ε = 1/n, where ε is the size of
pixels and n is the scale, is used to determine three different types of fractal dimension:
black area, white area and their edge (DBBW, DWBW, DBW). To calculate the dimension
DBBW and DWBW, the count of black and black & white (BW) N BBW = NB
+ NBW
or
white and black & white (BW) N WBW = NW
+ NBW
boxes is utilised. The variation of
fractal dimension with the count of boxes can be written

N
−D
D
( ε ) = K ⋅ε
N( n)
= K ⋅n
,
3
, (1)
respectively, where K is the fractal measure and D is the fractal dimension (without their
indexes). From these terms the fractal dimension is given by the slope of function (1)
d ln N
D = −
d lnε
( ε ) d ln N(
n)
=
d ln n
The area filled by the examined set (points) can be expressed as
S
( n)
and the length of lines in figure as
L
( n)
N
=
n
( n)
D−2
2
N
=
n
= K ⋅ n
( n)
D−1
= K ⋅ n
. (2)
respectively.
The Figure 2 shows the dependency obtained from the model image of a tree for
black & white squares. The image shows that the function is almost linear (structure of
the image is fractal). There is clearly a visible error for small sizes of the boxes
(introduced by the interpolation of values) and also error for the very big sizes of the
boxes (introduced by saturation of the number of boxes).
log(N BW)
8
7
6
5
4
3
2
1
0
Square numbers
WM
BCM
-6 -5 -4 -3 -2 -1 0
log(n )
Figure 2. Determination of the fractal measure and fractal dimension using the box counting (see Figure 1) and
for wavelet method (see Figure 3 - Figure 5).
Acquired fractal measures KBBW, KWBW, KBW can be used for determination of the
black – SB (white – SW respectively) areas of threshold figure
S
B
=
K
BBW
KBBW
− KBW
+ K − K
WBW
BW
,
W
=
K
and the length of divided lines L of threshold figure
S
K WBW − KBW
+ K − K
BBW
WBW
BW
(3)
(4)
(5)

4
L =
K
BBW
K
+ K
BW
WBW
− K
BW
. (6)
The results are taken in relative units (multiplied by 100 in percent). Results in
pixels can be taken after multiplying this number by area size of the figure (Kmax).
3 Wavelet method
The description of the box counting method shows that it is variation of integral methods.
Using that knowledge the authors tried to find the best integral transformation that can be
used to evaluate the fractal parameters of image structures. The driving force of our
effort was to simplify the algorithm for determination of the fractal dimension and to
extend the range of possible interpretation of the results. It was found that already very
simple wavelet transformation – Haar transformation [3] gives us the identical results of
values of fractal parameters (for threshold figure) as the box counting method.
Haar transformation (HT) is linear orthogonal transformation with signum function
(rectangle function) base. The HT transforms real image f(m, n) into discreet spectrum
represented by real function F(k, l)
( , ) = l k F
− N 1 N −1
∑∑
m=
0 n=
0
( , ) h n m f
m, k n,
l h
where hn,k and hm,l are the coefficients of so called Haar matrix HN
H
0
⎡1
1⎤
= , H1
= ⎢ ,
1 1
⎥ H
⎣ − ⎦
1 2
⎡1
⎢
1
= ⎢
⎢1
⎢
⎣0
1
1
−1
Haar matrix Hn is the order of 2 n (2 n dimensional matrix).
0
, (7)
1
−1
0
1
1⎤
−1
⎥
⎥
0⎥
⎥
−1⎦
2 × 2 4 × 4 8 × 8
, ...
. (8)
Figure 3. Spectra of image structure determined by Haar transformation of image from Figure 1 (wavelet
method)
The method requires (like the Fast Fourier Transformation) a square matrix of data
given by square of two. Figure 3 show the resulting spectra and images of the tree
(Figure 1) of size 1024 × 1024 pixels. The spectra are calculated using the matrix of
Haar transformation (transformation base) of size (matrix and image) 1024 × 1024

pixels. Figure 3 shows top left corner cut from the complete matrix (2 × 2, 4 × 4, 8 × 8
pixels). The top left item of the matrix stores information about amount of black pixels
within the image. The information about count of black pixels in each quarter of the
image can be then determined from the matrix obtained by the reverse Haar
transformation (such matrix is then half of the size of the original matrix), see Figure 4.
The Figure 5 shows information about count of black pixels in the quarter of the
matrix. Shades of grey on Figure 4 are equivalent to the count of black & white (BW)
pixels in the quarter. Fractal dimensions of black, white and black & white can be
determined as it was possible when using box counting method (DBBW, DWBW, DBW). The
parameters obtained by our method are identical to parameters determined by the box
counting method for matrixes of 2 n × 2 n (1 × 1, 2 × 2, 4 × 4, 8 × 8, etc.) pixels.
64 × 64 32 × 32 16 × 16
Figure 4. Determination of the fractal measure and fractal dimension using wavelet method (grey scale squares)
64 × 64 32 × 32 16 × 16
Figure 5. Determination of the fractal measure and fractal dimension using wavelet method (tresholded squares)
The function of count of black & white squares is displayed in Figure 2 (these
determine the characteristic of the interface between black and white areas in the image).
The fractal dimension determined by the box counting method was DBW = 1.1053. The
graph displayed shows that the function is not linear. The exact derivative of the function
is shown in Figure 2 right. Error in the big squares area (on the right side of the graph) is
caused by error of applying grid on the image (error is approximately same for both
methods). Error in the small squares area (left side of the graph) is caused by wrong
interpolation of the data. Inflection point of the graph curve is exactly at the point equal
to the value of the fractal dimension of the structure.
The acquired fractal measures KBBW, KWBW, KBW can be used for determination of
black and white areas of figure and the dividing lines lengths using the same equations as
for box counting method, see Eqs. (5) and (6).
5

6
4 Algorithm of fractal parameters calculation using the Haar transformation
The method of fractal parameters calculation described in the previous chapter is
relatively time-consuming. It requires performing the direct Haar transformation of
analysed picture (with size 2 n × 2 n pixels) and (n−1) reverse Haar transformations of
spectrums, which were previously filtered by low-pass filtering (n−1) × (n−1), (n−2) ×
(n−2), ... pixels. The fractal parameters are then calculated using standard technique (like
box-counting method) from the sequence of the obtained pictures with flattened details.
It was found by detail problem analysis that the algorithm above can be simplified
by simple adding of four adjacent pixels (which have values 0 and 1 in the original
thresholded picture). After the first step we obtain picture of size 2 n−1 × 2 n−1 pixels with
values from 0 to 4, after the second step the picture will have its size 2 n−2 × 2 n−2 pixels
with values from 0 to 16, etc. The NB of the picture is then the number of pixels with
value 0, the NW is number of pixels with maximal value (i.e. 1, 4, 16, ... , 2 2n ) and the
NBW is the number of the rest of pixels. The log N = f(log n) dependency allows to
determine the fractal dimension and fractal measure of the tresholded picture using the
standard techniques. The analysis of a black box gives fractal dimension DW = 0 (number
of white pixels is always zero) whereas the analysis of a box containing only white pixels
results in DW = 2.
5 Analysis of three - dimensional structures
In the case of analysing a grey scale picture, it is supposable, that the shade represents a
profile of a three-dimensional structure (see e.g. Figure 6). In this case the particular
pixels have values from 0 (black) to 255 (white). Adding of values of foursome pixels
results in picture of 2 n−1 × 2 n−1 pixels size with values from 0 to 1020. After the second
iteration a 2 n−2 × 2 n−2 pixels sized picture with values from 0 to 2040 is obtained, etc.
2D projection 3D projection
Figure 6. Determination of the fractal measure and fractal dimension of 3D structure (fractal dimension of
surface) using wavelet method
The NB of each individual picture is then the number of pixels with value 0, the NW
is the number of pixels with maximal value (i.e. 1020, 4080, ... , 255 · 2 2n ) and the NBW is

the number of the remaining of pixels. The log N = f(log n) dependency allows to
determine the fractal dimension and fractal measure of the tresholded picture using the
standard techniques. The analysis of a black box gives fractal dimension DW = 0 (number
of white pixels is always zero) whereas the analysis of a box containing only white pixels
results in DW = 3.
The volume filled by the examined set (points) can be expressed as
V
( n)
N
=
n
( n)
D−3
3
= K ⋅ n
. (9)
The following equations determines the areas under (VB) and above (VW) of the 3D
object surface
V
and area size S
B
=
K
BBW
K BBW − K BW
+ K − K
WBW
S =
K
BW
BBW
V
,
K
+ K
W
BW
WBW
=
K
− K
K WBW − K BW
+ K − K
in relative units (in percent respectively). Volumes and areas in volume pixels (voxels)
can be obtained by multiplying the results by the maximal volume. It depends on the area
size of figure and its colour depth (e.g. for eight bit depth is given by multiplying by the
number 2 1 255 ). Coefficient which determines how many times is the fractal
surface area higher then the horizontal surface area (size of figure) can be also calculated.
8
− =
log(N GRAY)
18
16
14
12
10
8
6
4
2
0
-2
BBW
BW
Square numbers
WBW
-6 -5 -4 -3 -2 -1 0
log(n )
Figure 7. Determination of the fractal measure and fractal dimension of 3D structure (fractal dimension of
surface) using wavelet method
Fractal dimension of surface of a fractal structure in Figure 6 is DBW = 2.263. Fractal
dimension of the structure volume was found to be DBBW = 2.877 and fractal dimension
of the surrounding volume is DWBW = 2.900. Volumes of the object and surroundings
have the fractal dimension near the value of the Euclidean dimension of volume E = 3.
BW
WM
7
(10)
(11)

8
This implies that the object has nearly the Euclidean shape. Fractal dimension of the
surface is significantly bigger than the Euclidean dimension of plane E = 2 and has
therefore evidently fractal nature. The fractal measure of the interface determines the size
of the fractal structure surface (in pixels) and is KBW/Kmax = 362.79 % of the image’s
planar surface, where Kmax = (512 × 512) pixels. Fractal measure of the structure volume
is KBBW/Kmax = 40.97 % and its surroundings KWBW/Kmax = 59.03 %, where
Kmax = (512 × 512 × 255) voxels (volume pixels). The fractal measure of the interface
tells us therefore that the surface of the fractal structure is more than three times larger
than the surface of plane.
6 Conclusion
It is worth mentioning that the fractal dimension of the deterministic fractal structures
(e.g. Sierpinsky carpet) is immutable in the whole range of sizes of the net applied on the
image using box counting method (or level of filter using wavelet – Haar –
transformation) and is the same as the theoretical fractal dimension value. Similar fractal
dimension values were also obtained using other types of integral, e.g. the Fourier
transformations. In this case the values of fractal parameters are influenced by broader
vicinity (step function is replaced by harmonic function).
The fractal dimension evaluated using the Haar transformation offers in contrast to
the classic box counting method a much wider range of usability. The method can also be
used to determine the fractal dimension of colour structures (e.g. greyscale or full colour
images). This also allows determining the fractal dimensions of surfaces specified by
shades of grey or colour components. This can be also used to determine the fractal
parameters of surfaces or three-dimensional structures like distribution of the mass in the
space or electrical potential in space. The theory of the fractal structures in Edimensional
Euclidean space is described more closely e.g. in [4, 5].
Software for fractal analysis of the images is provided free on the Internet
http://www.fch.vutbr.cz/lectures/imagesci.
Literature
1 . B. B. Mandelbrot, Fractal geometry of nature. New York: W.H. Freeman and Co.
(1983)
2 . M. J. Barnsley, Fractals everywhere, New York:Academic Press Inc. (1993)
3 . G. Strang, Wavelets Transforms versus Fourier Transforms, Bulletin of the
American Mathematical Society, 28, 288 (1993)
4 . O. Zmeskal, M. Nezadal, M. Buchnicek, Chaos, Solitons & Fractals, 17, 113 (2003)
5 . O. Zmeskal, M. Nezadal, M. Buchnicek, Chaos, Solitons & Fractals, 19, 1013 (2004)